석사 학위논문

Master’s Thesis

고자장뇌기능자기공명영상압축센싱

Compressed Sensing for fMRI at High Field

Han, Paul Kyu

바이오및뇌공학과

Department of Bio and Brain Engineering

KAIST

2013

고자장뇌기능자기공명영상압축센싱

Compressed Sensing for fMRI at High Field

Compressed Sensing for fMRI at High Field

Major Advisor : Professor Jong Chul Ye

Co-Advisor : Professor Sung-Hong Park

by

Han, Paul Kyu

Department of Bio and Brain Engineering

KAIST

A thesis submitted to the faculty of KAIST in partial fulfillment of

the requirements for the degree of Master of Science in Engineering in the

Department of Bio and Brain Engineering . The study was conducted in

accordance with Code of Research Ethics1.

2012. 12. 05.

Approved by

Professor Sung-Hong Park

[Co-Advisor]

1Declaration of Ethical Conduct in Research: I, as a graduate student of KAIST, hereby declare that

I have not committed any acts that may damage the credibility of my research. These include, but are

not limited to: falsification, thesis written by someone else, distortion of research findings or plagiarism.

I affirm that my thesis contains honest conclusions based on my own careful research under the guidance

of my thesis advisor.

고자장뇌기능자기공명영상압축센싱

Han, Paul Kyu

위 논문은 한국과학기술원 석사학위논문으로

학위논문심사위원회에서 심사 통과하였음.

2012년 12월 05일

심사위원장 예 종 철 (인)

심사위원 박 성 홍 (인)

심사위원 박 현 욱 (인)

MBIS

20104496

Han, Paul Kyu. Compressed Sensing for fMRI at High Field. 고자장 뇌기능 자기공

명영상 압축 센싱. Department of Bio and Brain Engineering . 2013. 75p. Advisors

Prof. Jong Chul Ye, Prof. Sung-Hong Park. Text in English.

ABSTRACT

Conventional functional magnetic resonance imaging (fMRI) technique known as gradient recalled

echo (GRE) echo-planar imaging (EPI) is too sensitive to image distortion and degradation caused by

local magnetic field inhomogeneity at high magnetic fields. Pass-band balanced steady state free preces-

sion (bSSFP) has been proposed as an alternative high-resolution fMRI technique, however, the temporal

resolution of bSSFP fMRI is lower than the typically used GRE-EPI fMRI. One potential approach to

improve the temporal resolution of bSFFP fMRI is to use compressed sensing (CS). Recently, several

fMRI studies have applied CS to GRE-EPI and spiral scan, although it is known that GRE-EPI gener-

ally suffers from the contribution of magnetic field inhomogeneity which can degrade the performance

of CS algorithms. Although suffering from banding artifacts, bSSFP utilizes different radio frequency

(RF) excitations for each K-space lines, thus may work better with CS algorithms than GRE-EPI. In

this study, we tested the feasibility of a CS algorithm, called k-t FOCUSS, for both GRE and bSSFP

fMRI at 9.4T using the model of rat somatosensory stimulation. Experimental results show k-t FOCUSS

algorithm with sampling reduction by a factor of 4 works well for both GRE and bSSFP fMRI at high

field. The combination of CS algorithm with bSSFP may be a good solution for improving the temporal

resolution of fMRI at high field.

Keywords: High Magnetic Field; Pass-Band bSSFP; fMRI; Compressed Sensing

i

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Chapter 1. Introduction 1

Chapter 2. Literature Review 4

2.1 fMRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 bSSFP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Compressed Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 FOCUSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 k-t FOCUSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.1 k-t FOCUSS with Temporal FT . . . . . . . . . . . . . . 15

2.5.2 k-t FOCUSS with KLT . . . . . . . . . . . . . . . . . . . . 18

Chapter 3. Methodology 20

3.1 Data Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Sampling Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 k-t FOCUSS Parameters . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Region of Interest Selection . . . . . . . . . . . . . . . . . . . . . 21

3.5 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 4. Results and Discussion 26

4.1 k-t FOCUSS with Temporal FT . . . . . . . . . . . . . . . . . . . 26

4.2 k-t FOCUSS with KLT . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Practical Implications . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 5. Conclusion 72

References 73

Summary (in Korean) 76

ii

List of Tables

4.1 Comparison of T values, number of activation pixels, and percent signal changes of ROI

in original image data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Comparison of T values, number of activation pixels, and percent signal changes of ROI

in image data with only low frequency components. . . . . . . . . . . . . . . . . . . . . . 44

4.3 Comparison of T values, number of activation pixels, and percent signal changes of ROI

in image data reconstructed from Gaussian-weighted random down-sampling scheme with

full sampling of k-space center 8 lines using k-t FOCUSS with temporal FT. . . . . . . . 44

4.4 Comparison of T values, number of activation pixels, and percent signal changes of ROI

in image data reconstructed from random down-sampling scheme with full sampling of

k-space center 8 lines using k-t FOCUSS with temporal FT. . . . . . . . . . . . . . . . . 44

4.5 Comparison of T values, number of activation pixels, and percent signal changes of ROI

in image data reconstructed from Gaussian-weighted random down-sampling scheme with

full sampling of k-space center 1 line using k-t FOCUSS with temporal FT. . . . . . . . . 45

4.6 Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from Gaussian-weighted random down-sampling scheme using

k-t FOCUSS with temporal FT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.7 Comparison of T values, number of activation pixels, and percent signal changes of ROI

in image data reconstructed from random down-sampling scheme using k-t FOCUSS with

temporal FT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.8 Comparison of T values, number of activation pixels, and percent signal changes of ROI

in image data reconstructed from Gaussian-weighted random down-sampling scheme with

full sampling of k-space center 8 lines using k-t FOCUSS with KLT. . . . . . . . . . . . . 62

4.9 Comparison of T values, number of activation pixels, and percent signal changes of ROI

in image data reconstructed from random down-sampling scheme with full sampling of

k-space center 8 lines using k-t FOCUSS with KLT. . . . . . . . . . . . . . . . . . . . . . 62

4.10 Comparison of T values, number of activation pixels, and percent signal changes of ROI

in image data reconstructed from Gaussian-weighted random down-sampling scheme with

full sampling of k-space center 1 line using k-t FOCUSS with KLT. . . . . . . . . . . . . 62

4.11 Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from Gaussian-weighted random down-sampling scheme using

k-t FOCUSS with KLT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.12 Comparison of T values, number of activation pixels, and percent signal changes of ROI

in image data reconstructed from random down-sampling scheme using k-t FOCUSS with

KLT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

iii

List of Figures

2.1 Pulse sequence diagram of balanced steady-state free precession. The sum of all gradients

in each of the three directions (slice-selection, phase-encoding, and frequency-encoding) is

zero. The phase of the radio frequency pulse, denoted as the phase-cycling angle (θ), is

incremented after every repetition time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Magnetization spin precession of bSSFP sequence. Magnetization spin precession of

bSSFP as viewed from the laboratory reference frame (a) and during steady-state for

bSSFP with phase-cycling angle θ = 180◦ (b) is shown. The z axis denotes the B0-field

direction and the x and y axes denote the direction of a pair of orthogonal vectors in a

plane normal to the B0-field. M denotes the magnetization spin, Mz denotes the longi-

tudinal component of the magnetization spin M, Mxy denotes the transverse component

of the magnetization spin M, and α denotes the flip angle of RF pulse. Notice that the

magnitude of the transverse magnetization component is maintained once bSSFP signal

reaches steady-state with RF flip angles of alternating sign. . . . . . . . . . . . . . . . . . 8

2.3 The balanced steady-state signal profile as a function of off-resonance frequency in one

repetition time. The bSSFP magnetization signal magnitude and phase responses are

shown in response to off-resonance precession frequency for one repetition time. Balanced

steady-state signal profile for phase-cycling angle of 0◦ (a), 90◦ (b), 180◦ (c), and 270◦ (d)

are shown. Notice that signal magnitude profile is periodic. Also notice the shift of signal

magnitude profile as phase-cycling angle changes. . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Phase evolution of bSSFP transverse magnetization. The transverse magnetization off-

resonance phase shift due to magnetic field inhomogeneity (ψ) and phase-cycling angle (θ)

(a) and the effect of total phase shift (φ) in the acquisition of cartesian k-space lines (b)

are shown. Notice the increment of phase shift (φ) in the acquisition of each consecutive

k-space lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Masks of five different sampling patterns with down-sampling factor of 4. Patterns of

Gaussian-weighted random sampling with full sampling of center 8 lines (a), random

sampling with full sampling of center 8 lines (b), Gaussian-weighted random sampling with

full sampling of center 1 line (c), Gaussian-weighted random sampling (d), and random

sampling (e) are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Selection of ROI. Functionally active region is selected as the ROI from the t-statistics

map of original images acquired using GRE-EPI. Images of t-statistics map before ROI

selection (a), ROI mask (b), and selected ROI region (c) obtained for a representative

animal is shown. New ROIs were defined for each different rats. This ROI is used for

further quantitative analysis such as calculation of T values, calculation of percent signal

changes and roi time course plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

iv

4.1 Comparison of baseline images after down-sampling. Baseline images after down-sampling

with Gaussian-weighted random down-sampling pattern with full sampling of k-space cen-

ter 8 lines (a), random down-sampling pattern with full sampling of k-space center 8 lines

(b), Gaussian-weighted random down-sampling pattern with full sampling of k-space cen-

ter 1 line (c), Gaussian-weighted random down-sampling pattern (d), and random down-

sampling pattern (e) are shown. The 25th and 12th image slice of bSSFP and GRE is

shown, respectively. Down-sampling pattern and acquisition type or phase-cycling angle

are shown on the top and left-hand side of the images, respectively. . . . . . . . . . . . . 32

4.2 Comparison of reconstructed baseline images from sampling schemes with full acquisition

of k-space center 8 lines using k-t FOCUSS with temporal FT. Original baseline images

from fully-sampled k-space data (a), baseline images with only low frequency information

(b), reconstructed baseline images from Gaussian-weighted random down-sampling scheme

with full sampling of k-space center 8 lines (c), and reconstructed baseline images from

random down-sampling scheme with full sampling of k-space center 8 lines (d) are shown.

The 25th and 12th image slice of bSSFP and GRE is shown, respectively. Down-sampling

pattern and acquisition type or phase-cycling angle are shown on the top and left-hand

side of the images, respectively. Notice the disappearance of DC artifact (white arrow) in

all of the reconstructed phase-cycled bSSFP images. . . . . . . . . . . . . . . . . . . . . . 33

4.3 Comparison of reconstructed baseline images from Gaussian-weighted random sampling

scheme with full acquisition of k-space center 1 line and Gaussian-weighted random sam-

pling scheme using k-t FOCUSS with temporal FT. Original baseline images from fully-

sampled k-space data (a), baseline images with only low frequency information (b), recon-

structed baseline images from Gaussian-weighted random down-sampling scheme with full

sampling of k-space center 1 line (c), and reconstructed baseline images from Gaussian-

weighted random down-sampling scheme (d) are shown. The 25th and 12th image slice

of bSSFP and GRE is shown, respectively. Down-sampling pattern and acquisition type

or phase-cycling angle are shown on the top and left-hand side of the images, respec-

tively. Notice the disappearance of DC artifact (white arrow) in all of the reconstructed

phase-cycled bSSFP images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Comparison of reconstructed baseline images from random sampling scheme using k-t

FOCUSS with temporal FT. Original baseline images from fully-sampled k-space data (a),

baseline images with only low frequency information (b), reconstructed baseline images

from random down-sampling scheme (c) are shown. The 25th and 12th image slice of

bSSFP and GRE is shown, respectively. Down-sampling pattern and acquisition type or

phase-cycling angle are shown on the top and left-hand side of the images, respectively.

Notice the disappearance of DC artifact (white arrow) in all of the reconstructed phase-

cycled bSSFP images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5 Comparison of MSE plots of reconstructed image using k-t FOCUSS with temporal FT.

MSE plots of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative

rat are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

– v –

4.6 Comparison of reconstructed fMRI maps from sampling schemes with full acquisition of

k-space center 8 lines using k-t FOCUSS with temporal FT. Original fMRI maps from

fully-sampled k-space data (a), fMRI maps with only low frequency information (b), re-

constructed fMRI maps from Gaussian-weighted random down-sampling scheme with full

sampling of k-space center 8 lines (c), and reconstructed fMRI maps from random down-

sampling scheme with full sampling of k-space center 8 lines (d) are shown for significance

level of α = 0.05. Down-sampling pattern and acquisition type or phase-cycling angle are

shown on the top and left-hand side of the images, respectively. . . . . . . . . . . . . . . 37

4.7 Comparison of reconstructed fMRI maps from Gaussian-weighted random sampling scheme

with full acquisition of k-space center 1 line and Gaussian-weighted random sampling

scheme using k-t FOCUSS with temporal FT. Original fMRI maps from fully-sampled k-

space data (a), fMRI maps with only low frequency information (b), reconstructed fMRI

maps from Gaussian-weighted random down-sampling scheme with full sampling of k-

space center 1 line (c), and reconstructed fMRI maps from Gaussian-weighted random

down-sampling scheme (d) are shown for significance level of α = 0.05. Down-sampling

pattern and acquisition type or phase-cycling angle are shown on the top and left-hand

side of the images, respectively. Notice the reconstruction of activation foci shift (white

arrow) in the phase-cycled bSSFP data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.8 Comparison of reconstructed fMRI maps from random sampling scheme using k-t FOCUSS

with temporal FT. Original fMRI maps from fully-sampled k-space data (a), fMRI maps

with only low frequency information (b), reconstructed fMRI maps from random down-

sampling scheme (c) are shown for significance level of α = 0.05. Down-sampling pattern

and acquisition type or phase-cycling angle are shown on the top and left-hand side of the

images, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.9 Comparison of mean ROI time course plot of original image and reconstructed images using

k-t FOCUSS with temporal FT. Mean ROI time course plot of reconstructed images from

Gaussian-weighted random down-sampling pattern with full sampling of k-space center 8

lines (a), random down-sampling pattern with full sampling of k-space center 8 lines (b),

Gaussian-weighted random down-sampling pattern with full sampling of k-space center

1 line (c), Gaussian-weighted random down-sampling pattern (d), and random down-

sampling pattern (e) are shown. Time course plots of bSSFP were obtained with phase-

cycling angle of 180◦ of a representative rat. . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.10 Comparison of FDR of reconstructed fMRI map using k-t FOCUSS with temporal FT.

FDR of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat

are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.11 Comparison of Type II error of reconstructed fMRI map using k-t FOCUSS with temporal

FT. Type II error of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a

representative rat are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.12 Comparison of ROC curve of reconstructed fMRI map using k-t FOCUSS with temporal

FT. ROC curves of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a repre-

sentative rat are shown. The ROC curve with the highest area under curve (AUC) value

is indicated by the red-dashed box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

– vi –

4.13 Comparison of reconstructed baseline images from sampling schemes with full acquisition

of k-space center 8 lines using k-t FOCUSS with KLT. Original baseline images from fully-

sampled k-space data (a), baseline images with only low frequency information (b), re-

constructed baseline images from Gaussian-weighted random down-sampling scheme with

full sampling of k-space center 8 lines (c), and reconstructed baseline images from random

down-sampling scheme with full sampling of k-space center 8 lines (d) are shown. The

25th and 12th image slice of bSSFP and GRE is shown, respectively. Down-sampling

pattern and acquisition type or phase-cycling angle are shown on the top and left-hand

side of the images, respectively. Notice the disappearance of DC artifact (white arrow) in

all of the reconstructed phase-cycled bSSFP images. . . . . . . . . . . . . . . . . . . . . . 51

4.14 Comparison of reconstructed baseline images from Gaussian-weighted random sampling

scheme with full acquisition of k-space center 1 line and Gaussian-weighted random sam-

pling scheme using k-t FOCUSS with KLT. Original baseline images from fully-sampled

k-space data (a), baseline images with only low frequency information (b), reconstructed

baseline images from Gaussian-weighted random down-sampling scheme with full sampling

of k-space center 1 line (c), and reconstructed baseline images from Gaussian-weighted ran-

dom down-sampling scheme (d) are shown. The 25th and 12th image slice of bSSFP and

GRE is shown, respectively. Down-sampling pattern and acquisition type or phase-cycling

angle are shown on the top and left-hand side of the images, respectively. Notice the dis-

appearance of DC artifact (white arrow) in all of the reconstructed phase-cycled bSSFP

images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.15 Comparison of reconstructed baseline images from random sampling scheme using k-t FO-

CUSS with KLT. Original baseline images from fully-sampled k-space data (a), baseline

images with only low frequency information (b), reconstructed baseline images from ran-

dom down-sampling scheme (c) are shown. The 25th and 12th image slice of bSSFP and

GRE is shown, respectively. Down-sampling pattern and acquisition type or phase-cycling

angle are shown on the top and left-hand side of the images, respectively. Notice the dis-

appearance of DC artifact (white arrow) in all of the reconstructed phase-cycled bSSFP

images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.16 Comparison of MSE plots of reconstructed image using k-t FOCUSS with KLT. MSE plots

of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat are

shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.17 Comparison of reconstructed fMRI maps from sampling schemes with full acquisition of k-

space center 8 lines using k-t FOCUSS with KLT. Original fMRI maps from fully-sampled

k-space data (a), fMRI maps with only low frequency information (b), reconstructed

fMRI maps from Gaussian-weighted random down-sampling scheme with full sampling

of k-space center 8 lines (c), and reconstructed fMRI maps from random down-sampling

scheme with full sampling of k-space center 8 lines (d) are shown for significance level of

α = 0.05. Down-sampling pattern and acquisition type or phase-cycling angle are shown

on the top and left-hand side of the images, respectively. . . . . . . . . . . . . . . . . . . 55

– vii –

4.18 Comparison of reconstructed fMRI maps from Gaussian-weighted random sampling scheme

with full acquisition of k-space center 1 line and Gaussian-weighted random sampling

scheme using k-t FOCUSS with KLT. Original fMRI maps from fully-sampled k-space data

(a), fMRI maps with only low frequency information (b), reconstructed fMRI maps from

Gaussian-weighted random down-sampling scheme with full sampling of k-space center 1

line (c), and reconstructed fMRI maps from Gaussian-weighted random down-sampling

scheme (d) are shown for significance level of α = 0.05. Down-sampling pattern and

acquisition type or phase-cycling angle are shown on the top and left-hand side of the

images, respectively. Notice the reconstruction of activation foci shift (white arrow) in the

phase-cycled bSSFP data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.19 Comparison of reconstructed fMRI maps from random sampling scheme using k-t FOCUSS

with KLT. Original fMRI maps from fully-sampled k-space data (a), fMRI maps with only

low frequency information (b), reconstructed fMRI maps from random down-sampling

scheme (c) are shown for significance level of α = 0.05. Down-sampling pattern and

acquisition type or phase-cycling angle are shown on the top and left-hand side of the

images, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.20 Comparison of mean ROI time course plot of original image and reconstructed images

using k-t FOCUSS with KLT. Mean ROI time course plot of reconstructed images from

Gaussian-weighted random down-sampling pattern with full sampling of k-space center 8

lines (a), random down-sampling pattern with full sampling of k-space center 8 lines (b),

Gaussian-weighted random down-sampling pattern with full sampling of k-space center

1 line (c), Gaussian-weighted random down-sampling pattern (d), and random down-

sampling pattern (e) are shown. Time course plots of bSSFP were obtained with phase-

cycling angle of 180◦ of a representative rat. . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.21 Comparison of FDR of reconstructed fMRI map using k-t FOCUSS with KLT. FDR of

bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat are shown. 59

4.22 Comparison of Type II error of reconstructed fMRI map using k-t FOCUSS with KLT.

Type II error of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a represen-

tative rat are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.23 Comparison of ROC curve of reconstructed fMRI map using k-t FOCUSS with KLT. ROC

curves of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat

are shown. The ROC curve with the highest area under curve (AUC) value is indicated

by the red-dashed box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.24 Implementation of paired random sampling pattern for suppression of Eddy current effects

in bSSFP with phase-cycling angle of 180◦. Paired random sampling trajectory (a) and

paired Gaussian-weighted random sampling scheme with full sampling of k-space center 1

line over time with a down-sampling factor of 4 (b) are shown. . . . . . . . . . . . . . . . 66

– viii –

4.25 Comparison of reconstructed fMRI maps from paired Gaussian-weighted random sampling

scheme with full sampling of k-space center 1 line using k-t FOCUSS with temporal FT.

Original fMRI maps from fully-sampled k-space (a), fMRI maps with only low frequency in-

formation (b), reconstructed fMRI maps from Gaussian-weighted random down-sampling

pattern with full sampling of k-space center 1 line (c), and reconstructed fMRI maps from

paired Gaussian-weighted random down-sampling pattern with full sampling of k-space

center 1 line (d) are shown for significance level of α = 0.05. Down-sampling pattern and

acquisition type or phase-cycling angle are shown on the top and left-hand side of the

images, respectively. Notice the shift of activation foci (white arrow) in the phase-cycled

bSSFP data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.26 Comparison of reconstructed fMRI maps from paired Gaussian-weighted random sampling

scheme with full sampling of k-space center 1 line using k-t FOCUSS with KLT. Original

fMRI maps from fully-sampled k-space (a), fMRI maps with only low frequency infor-

mation (b), reconstructed fMRI maps from Gaussian-weighted random down-sampling

pattern with full sampling of k-space center 1 line (c), and reconstructed fMRI maps from

paired Gaussian-weighted random down-sampling pattern with full sampling of k-space

center 1 line (d) are shown for significance level of α = 0.05. Down-sampling pattern and

acquisition type or phase-cycling angle are shown on the top and left-hand side of the

images, respectively. Notice the shift of activation foci (white arrow) in the phase-cycled

bSSFP data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.27 Comparison of FDR of reconstructed fMRI map from paired Gaussian-weighted random

sampling scheme using k-t FOCUSS. FDR obtained from fMRI map with only low fre-

quency information, reconstructed fMRI map using k-t FOCUSS with temporal FT and

Gaussian-weighted random sampling with full sampling of k-space center 1 line, recon-

structed fMRI map using k-t FOCUSS with temporal FT and paired Gaussian-weighted

random sampling with full sampling of k-space center 1 line, reconstructed fMRI map

using k-t FOCUSS with KLT and Gaussian-weighted random sampling with full sam-

pling of k-space center 1 line, and reconstructed fMRI map using k-t FOCUSS with KLT

and paired Gaussian-weighted random sampling with full sampling of k-space center 1

line are shown. FDR of bSSFP with phase-cycling angle of 180◦ (a), and GRE (e) of a

representative rat are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.28 Comparison of Type II error of reconstructed fMRI map from paired Gaussian-weighted

random sampling scheme using k-t FOCUSS. Type II error obtained from fMRI map

with only low frequency information, reconstructed fMRI map using k-t FOCUSS with

temporal FT and Gaussian-weighted random sampling with full sampling of k-space center

1 line, reconstructed fMRI map using k-t FOCUSS with temporal FT and paired Gaussian-

weighted random sampling with full sampling of k-space center 1 line, reconstructed fMRI

map using k-t FOCUSS with KLT and Gaussian-weighted random sampling with full

sampling of k-space center 1 line, and reconstructed fMRI map using k-t FOCUSS with

KLT and paired Gaussian-weighted random sampling with full sampling of k-space center

1 line are shown. Type II error of bSSFP with phase-cycling angle of 180◦ (a), and GRE

(e) of a representative rat are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

– ix –

4.29 Comparison of ROC curves of reconstructed fMRI map from paired Gaussian-weighted

random sampling scheme using k-t FOCUSS. ROC curves obtained from fMRI map with

only low frequency information, reconstructed fMRI map using k-t FOCUSS with temporal

FT and Gaussian-weighted random sampling with full sampling of k-space center 1 line,

reconstructed fMRI map using k-t FOCUSS with temporal FT and paired Gaussian-

weighted random sampling with full sampling of k-space center 1 line, reconstructed fMRI

map using k-t FOCUSS with KLT and Gaussian-weighted random sampling with full

sampling of k-space center 1 line, and reconstructed fMRI map using k-t FOCUSS with

KLT and paired Gaussian-weighted random sampling with full sampling of k-space center

1 line are shown. ROC curves of bSSFP with phase-cycling angle of 180◦ (a), and GRE

(e) of a representative rat are shown. The ROC curve with the highest area under curve

(AUC) value is indicated by the red-dashed box. . . . . . . . . . . . . . . . . . . . . . . . 71

– x –

Chapter 1. Introduction

Magnetic Resonance Imaging (MRI) is a modern medical imaging technique that allows for the

visualization of the internal human body non-invasively. MRI utilizes nuclear magnetic resonance phe-

nomenon to image atoms inside the body, and the most conventional measure is the most abundant atom,

the protons (1H nuclei). When the internal protons are subjected to an external magnetic field caused

by a powerful magnet, the magnetic moments of the protons align and precess at a specific frequency

depending on the strength of the magnet. The application of a radio frequency (RF) pulse alters the

alignment of the magnetization of these protons. This phenomenon, known as “magnetic resonance”,

causes a few precessing protons to absorb the energy of the RF pulse and flips the magnetization spin

of the protons to the opposite direction. The protons with the inverted spins slowly recovers its magne-

tization alignment back in time. Within this recovery process, magnetic gradients are applied in three

different spatial directions to retrieve the spatial information of the internal structures of the body.

Functional magnetic resonance imaging (fMRI) is a recent application of MRI that was developed to

measure brain activity. The conventional method use the positive blood oxygen level-dependent (BOLD)

response to map neural activity in the brain. As neuronal activity increases, metabolism also increases in

the region which leads to an increase in local blood flow [1]. The increased blood flow brings more oxygen

than is actually needed, and the content of oxygenated hemoglobin increases (also known as hemoglobin

saturation). Overall, this leads to an improved BOLD signal, and it has been verified that this indirect

measure of neural activity shows correlation with the true local neural activation [2].

The conventional MR imaging technique for acquiring BOLD fMRI images is gradient recalled echo

(GRE) echo planar imaging (EPI). The MR signals that are generated with GRE is governed by the T2*

free induction decay (FID) [3]. The technique is known to be sensitive to magnetic field susceptibility

and BOLD effect, and thus it is known as the best method for producing fMRI images [4]. Echo planar

imaging is one form of gradient recalled echo, which acquires an entire MR image in a single RF excita-

tion. As a consequence, a single image can be constructed in only a fraction of a second, however, with

susceptibility to image artifacts produced by magnetic field inhomogeneity.

Another MR imaging technique that can be used for the acquisition of fMRI images is balanced

steady-state free precession (bSSFP). Recently, pass-band balanced steady state free precession has

gained attention as a promising tool for high-resolution functional magnetic resonance imaging [5, 6, 7,

8, 9]. Specifically, at high magnetic fields, the conventional GRE-EPI is too sensitive to image distortion

and degradation caused by local magnetic field inhomogeneity, thus bSSFP has been proposed as an

alternative fMRI technique [9]. However, despite less image distortion and better signal to noise ratio

(SNR), the temporal resolution of bSSFP fMRI is lower than the typically used GRE-EPI fMRI [9, 10].

To overcome this problem, several methods can be applied to improve the temporal resolution of bSSFP

fMRI.

– 1 –

One solution to overcome the low temporal resolution of bSSFP fMRI is to use parallel imaging

technique. Parallel imaging is a widely used MR method that can improve the temporal resolution of

any dynamic MR images by the usage of multiple receiver coils for MR image acquisition. In princi-

ple, the data acquisition time for parallel imaging can be reduced by the number of receiver coils [3].

Various forms of parallel imaging reconstruction algorithms exist, depending on whether the reconstruc-

tion of the image occurs before or after the Fourier transformation (FT). For example, simultaneous

acquisition of spatial harmonics (SMASH)[11] and generalized autocalibrating partially parallel acquisi-

tion (GRAPPA)[12] recreates missing k-space lines in the frequency domain, while sensitivity encoding

(SENSE)[13] operates to “unfold” image aliasing artifacts in the image domain by using receiver coil sen-

sitivity information. However, although proven useful, the usage of parallel imaging results in reduction

of SNR and has several limitations due to the acceleration factor, the geometric factor of the different

coil elements, and the k-space filling trajectory [3].

Another solution to improve the temporal resolution of bSSFP fMRI is to use compressed sensing

(CS) [14, 15]. In MRI, the information received through the receiver coil is the spatial FT of the scanned

object image. Thus, in order to reconstruct an image without any aliasing artifacts, the sampling fre-

quency of echoes acquired through the receiver coil must satisfy the Nyquist sampling criterion. For

example, for an image with the highest temporal frequency of f , the temporal sampling rate needs to be

at least higher than 2f to reconstruct the image without any incurring aliasing effects. In contrast, CS

theory states that it is possible to reconstruct an aliasing free image even at sampling rates dramatically

smaller than the Nyquist sampling limit [14]. The CS theory operates by solving the lp minimization

problem (p ≤ 1); as long as the non-zero spectral signal is sparse and the samples are obtained with an

incoherent basis, the theory states that full reconstruction of the image is possible [14]. These require-

ments are well satisfied in dynamic MRI since arbitrary trajectories can be implemented for incoherent

sampling basis and dynamic MR images can be sparsified due to high temporal redundancy. In example,

a recent algorithm called k-t FOCUSS successfully applied CS theory to dynamic MRI by employing

random sampling pattern in k-t space and by using various sparsifying transforms such as temporal FT

and Karhunen-Loeve transform (KLT) to exploit the temporal redundancies [16, 17].

Though CS gained attraction for its vast potential for application [14], only a few studies have

successfuly applied CS theory to fMRI in the past. In addition, these studies have applied CS theory to

only GRE-EPI fMRI: ordinary GRE-EPI [18, 19] and spiral scan GRE-EPI [20]. Despite its application,

GRE-EPI is generally known to suffer from the contribution of magnetic field inhomogeneity, which can

degrade the performance of CS algorithms.

Although suffering from banding artifacts, bSSFP utilizes different RF excitations for each K-space

lines, thus may work better with CS algorithms than GRE-EPI. In this study, we test the feasibility of CS

for bSSFP fMRI at 9.4T using the model of rat somatosensory stimulation. The potential for improving

the temporal resolution of bSSFP fMRI using CS theory at high field without sacrificing image quality

and fMRI activation mapping results is discovered in this thesis.

The rest of the Chapters in this thesis are organized as follows: Chapter 2 will provide a brief review

of recent literatures related to the study; a description of fMRI, bSSFP, and kt-FOCUSS algorithm will

– 2 –

be provided in this section. Chapter 3 will decribe the materials and research methodology regarding the

experimental study. Data information, down-sampling patterns, and region of interest (ROI) selection

will be discussed in this chapter. Chapter 4 will show experimental results along with discussions. Finally,

Chapter 5 will provide a summary and future implications of this research.

– 3 –

Chapter 2. Literature Review

2.1 fMRI

Functional magnetic resonance imaging (fMRI) is a technique that utilizes MRI to measure neural

activations in the brain. To understand the motivation of this technique, it is necessary to exploit the

underlying physiology and mechanism of neuronal activation in the brain. Active neurons process and

transmit information through electrical and chemical signals. Energy is required for maintenance and

restoration of the neuronal membrane potentials, and is continuously supplied in the form of glucose

and oxygen via the vascular system. The key idea of fMRI is in the detection of a signal related to this

delivery process of glucose and oxygen, using MRI.

The conventional fMRI technique use positive BOLD response signal as a measure to map neural

activity in the brain. Oxygen is carried by hemoglobin molecules inside red blood cells, and BOLD

fMRI utilizes the magnetic property of hemoglobin for the detection of neural activation. It is discovered

that hemoglobin achieves different magnetic properties depending on its binding condition to oxygen

[21]. The oxygenated hemoglobin (Hb) is diamagnetic since it has no unpaired electrons and has zero

magnetic moment. In contrast, the deoxygenated hemoglobin (dHb) is paramagnetic since it has four

unpaired electrons and thus has significant magnetic moment. Within a strong uniform magnetic field, the

deoxygenated hemoglobin distorts the magnetic field locally and increases magnetic field inhomogeneity

in that region [21]. As neuronal activity increases, metabolism also increases in the region which leads

to an increase in local blood flow [1]. This increased blood flow brings more oxygen than is actually

needed through Hb in that region (also known as hemoglobin saturation). Thus, increase in neural

activation leads to an improved MR signal in that local region since the increase in diamagnetic Hb leads

to less interference with the external magnetic field [4]. This indirect measure of neural activity has been

discovered to correlate with the true neural activity [2].

– 4 –

2.2 bSSFP

Balanced steady-state free precession (bSSFP) is a MR pulse sequence that was first discovered by

Carr and others in 1958 [22]. The sequence consists of a consecutive train of identical excitation pulses

in the presence of “balanced” gradients, as shown in the pulse sequence diagram of bSSFP sequence

(Fig. 2.1). The RF excitation pulses are applied once in every repetition time (TR), and the gradients

are made to equally cancel out in each of the three spatial directions (i.e. slice-selection, phase-encoding,

and frequency-encoding direction) so that no net phase accumulates in the magnetization by the end of

each TR. The application of rapid train of identical RF excitation pulses results in an initial wild oscilla-

tion of the transverse magnetization signal, which eventually reaches a steady-state with a characteristic

magnetization spin precession (Fig. 2.2) and signal profile (Fig. 2.3) [22]. Once the magnetization signal

reaches steady-state, the magnitude of the transverse magnetization is maintained after every TR with

the application of identical RF excitation pulses (Fig. 2.2). Thus, with the choice of an optimal flip

angle (α), this sequence is known to offer the highest possible signal-to-noise rato (SNR) per unit time

of all known sequences. In addition, under common conditions the maximum refocusing of bSSFP signal

occurs at TE = TR/2 which results in echo signal similar to spin-echo formation [23] with unique char-

acteristic signal contrast of T2/T1 [24]. This steady-state magnetization signal of bSSFP is dependent

on the off-resonance precession angle, or in other words, off-resonance frequency of the magnetization.

The signal profile of bSSFP sequence mainly consists of two parts: the “pass-band” region and

the “transition-band” region (Fig. 2.3 a). The shape of the signal profile indicates that the pass-band

region is relatively insensitive to changes in off-resonance frequencies which are related to magnetic field

inhomogeneity, while the transition-band region is sensitive to changes in off-resonance precession angles

[22]. These two regions are also shown in both the magnitude and phase components of the signal. The

magnitude of the bSSFP signal (black-solid line in Fig. 2.3) is roughly constant in the pass-band region,

while the magnitude of the signal varies and decreases abruptly in the transition-band region. Similarly,

the phase of the bSSFP signal (red-dashed line in Fig. 2.3) is roughly constant in the pass-band region,

while the phase abruptly shifts in the transition-band region. Thus, in order to obtain a bSSFP image

with a fairly uniform signal and contrast, the spatial regions of bSSFP image requires the range of off-

resonance frequencies to be contained within the pass-band region. Because of this, recent applications

of bSSFP sequence focus on the pass-band region, which allows for achieving high SNR per unit time

while being relatively insensitive to changes in the off-resonance frequencies.

The signal magnitude profile of bSSFP is not fixed and can change relative to TR and phase of the

RF pulse (Fig. 2.3). The bSSFP signal magnitude profile is periodic, and repeats every TR−1 Hz. Thus,

as the TR value increases, the signal magnitude profile becomes broader with increase in the absolute

signal magnitude value [10, 9]. Also, the signal magnitude profile can be shifted along the off-resonance

frequency axis by incrementing the RF pulse phase in a constant amount for each TR period (Fig. 2.3

a, b, c, d) [10, 9]. This process, also known as phase-cycling, occurs when RF pulse phase is manually

incremented via changing the phase-cycling angle of the RF pulse.

– 5 –

The signal phase profile of bSSFP also changes relative to TR and phase of the RF pulse (Fig. 2.3).

The bSSFP signal phase evolves every TR−1 Hz, incrementing in value depending on the off-resonance

frequency. Thus, with the combination of phase-cycling technique, the total phase of the transverse mag-

netization signal in bSSFP (φ) is determined by the phase shift due to magnetic field inhomogeneity (ψ)

and phase-cycling angle (θ) (Fig. 2.4 a). This resultant total phase information is transmitted in k-space

during acquisition, resulting in a phase difference of φ between each consecutive k-space lines (Fig. 2.4

b). Therefore, manual control over phase-cycling angles lead to manipulation of the phase information in

each of the k-space lines, and thus is critical in order to maximize refocusing for acquisition considering

Eddy current effects.

One potential drawback of bSSFP sequence is that the signal suffers from banding artifacts due to

the disparity in signal between pass-band and transition-band regions (Fig. 2.3). The bSSFP signal pro-

file shows that the spatial regions with off-resonance frequencies in the range of transition band appears

at a lower MR signal (i.e. dark) compared to the spatial regions with off-resonance frequencies in the

range of pass-band regions in bSSFP image. This banding artifact can be controlled by decreasing the

TR (i.e. broadening the pass-band region) and changing the phase-cycling angle of RF pulse, or can be

removed completely by acquiring multiple datasets with multiple phase-cycling angles to suppress the

variations in pass-band and transition-band regions induced by the magnetic field inhomogeneity.

Recently, the bSSFP technique has been re-discovered in the context of fMRI [10]. Initial bSSFP

fMRI studies utilized transition-bands where both magnitude and phase signals change significantly

with small resonance frequency shifts to improve BOLD sensitivity[25, 26, 27, 28, 29], however, due to

limited spatial coverage (i.e. narrow range of off-resonance frequencies) and sensitivity to magnetic field

fluctuation, pass-band bSSFP has been adopted instead for recent bSSFP fMRI studies [5, 6, 7, 8, 9]. In

this study, CS is applied to pass-band bSSFP fMRI data acquired at high field.

– 6 –

Figure 2.1: Pulse sequence diagram of balanced steady-state free precession. The sum of all gradients in

each of the three directions (slice-selection, phase-encoding, and frequency-encoding) is zero. The phase

of the radio frequency pulse, denoted as the phase-cycling angle (θ), is incremented after every repetition

time.

– 7 –

Figure 2.2: Magnetization spin precession of bSSFP sequence. Magnetization spin precession of bSSFP

as viewed from the laboratory reference frame (a) and during steady-state for bSSFP with phase-cycling

angle θ = 180◦ (b) is shown. The z axis denotes the B0-field direction and the x and y axes denote the

direction of a pair of orthogonal vectors in a plane normal to the B0-field. M denotes the magnetization

spin, Mz denotes the longitudinal component of the magnetization spin M, Mxy denotes the transverse

component of the magnetization spin M, and α denotes the flip angle of RF pulse. Notice that the

magnitude of the transverse magnetization component is maintained once bSSFP signal reaches steady-

state with RF flip angles of alternating sign.

– 8 –

Figure 2.3: The balanced steady-state signal profile as a function of off-resonance frequency in one

repetition time. The bSSFP magnetization signal magnitude and phase responses are shown in response

to off-resonance precession frequency for one repetition time. Balanced steady-state signal profile for

phase-cycling angle of 0◦ (a), 90◦ (b), 180◦ (c), and 270◦ (d) are shown. Notice that signal magnitude

profile is periodic. Also notice the shift of signal magnitude profile as phase-cycling angle changes.

– 9 –

Figure 2.4: Phase evolution of bSSFP transverse magnetization. The transverse magnetization off-

resonance phase shift due to magnetic field inhomogeneity (ψ) and phase-cycling angle (θ) (a) and the

effect of total phase shift (φ) in the acquisition of cartesian k-space lines (b) are shown. Notice the

increment of phase shift (φ) in the acquisition of each consecutive k-space lines.

– 10 –

2.3 Compressed Sensing

Compressed sensing (CS) is a signal processing technique that efficiently reconstructs a signal from

measurements less than the signal dimension. The key idea is that it allows for reconstruction of an

aliasing free image even at sampling rates dramatically less than the Nyquist sampling limit, as long as

the non-zero spectral signal is sparse and the samples are obtained with an incoherent basis [14]. The

derivation of CS problem through mathematical expressions is shown to clearly understand CS.

In the context of CS, the basic problem is formulated by stating the relationship between the signal

and measurements, which is given as:

y = Ax, (2.1)

where y ∈ CM , x ∈ CN , A is a M ×N matrix, and N �M . Here y denotes the measurement vector, x

denotes the signal vector and A denotes the transformation through which the measurement and signal

are related. Notice that the N � M term indicates less number of measurements than the number of

elements of the signal.

CS theory states that the most direct method to solve this problem is by finding the “sparse” sig-

nal, which is found by solving the l0-norm minimization problem:

min||x||0 subject to y = Ax, (2.2)

where ||x||0 represents the total number of nonzero elements in vector x. However, solving this problem

is not practical for large N (i.e. NP-hard). Instead, CS theory solves the l1-norm minimization problem:

min||x||1 subject to y = Ax (2.3)

where l1-norm is defined as:

||x||1 =

N∑i=1

|xi| (2.4)

Although Equation (2.2) and Equation (2.3) are practically different, the solution to both problems

become identical when the transformation matrix A satisfies the restricted isometry property (RIP) [30],

which is shown below:

(1− δS)||x||22 ≤ ||Ax||22 ≤ (1 + δS)||x||22 (2.5)

where δS denotes the isometry constant. Then, the NP-hard problem of Equation (2.2) can be solved

indirectly by solving Equation (2.3).

– 11 –

In practical cases, additive noise is considered and Equation (2.1) becomes:

y = Ax + ε, (2.6)

where ε represents additive noise.

Then the minimization problem in Equation (2.3) becomes:

min||x||1 subject to ||y −Ax||2 ≤ ε. (2.7)

– 12 –

2.4 FOCUSS

FOCUSS algorithm was originally designed to find sparse solutions by iteratively solving quadratic

optimization problems, and its main application has been in EEG source localization[31, 32]. To under-

stand FOCUSS algorithm, it is important to know the basic structure of the algorithm and its relation to

CS theory. The initial step of FOCUSS begins by obtaining a low resolution estimate image of the object,

and this primary solution is then pruned to a sparse signal representation. The key of the pruning process

is the update of the current solution entries via a scaling process based on the entries of the previous solu-

tion. Thus, obtaining a good initial estimate is a crucial factor to guarantee the performance of FOCUSS.

Assuming the measurement-signal relationship shown in Equation (2.1), FOCUSS further considers

the signal (x) as:

x = Wq (2.8)

where W denotes weighting matrix and q denotes the solution to the following constrained minimization

problem:

min||q||22 s.t. AWq = y. (2.9)

The least squares solution of Equation (2.9) is found as:

q = (AW)†y. (2.10)

where (AW)† = (AW)H((AW)(AW)H)−1 denotes the pseudo-inverse.

The key idea of FOCUSS is motivated from Equations (2.8) and (2.10). The weighting matrix W

is iteratively updated using the previous solution x.

Specifically, if the (n− 1)th solution is given as:

xn−1 = [xn−1;1, xn−1;2, · · · , xn−1;N ]T

(2.11)

– 13 –

then the solution to the problem is found by using the following FOCUSS iteration steps:

Step 1 : Calculate Wn =

|xn−1;1|p 0 · · · 0

0 |xn−1;2|p · · · 0...

.... . .

...

0 0 · · · |xn−1;N |p

,1

2≤ p ≤ 1 (2.12)

Step 2 : Calculate qn = (AWn)†y (2.13)

Step 3 : Calculate xn = Wnqn (2.14)

Step 4 : Repeat Step 1-3 with nth solution (xn) until convergence.

The solution of FOCUSS is related to the preferred optimal solution of CS theory, which is the

solution to l1-norm minimization problem [16].

Using qn = Wn−1xn, Equation (2.9) can be rewritten as:

min||Wn−1x||22, subject to Ax = y, (2.15)

and the following asymptotic relationship is formulated for p = 0.5:

||Wn−1x||22 = xT (Wn

−1)H

Wn−1x

= xT

|xn−1;1|−1

0 · · · 0

0 |xn−1;2|−1 · · · 0...

.... . .

...

0 0 · · · |xn−1;N |−1

x

=N∑i=1

|xn−1, i|

= ||x||1 as n→∞. (2.16)

Thus, at p = 0.5, the solution of FOCUSS approximately equivalates the solution of l1-norm minimiza-

tion problem.

– 14 –

2.5 k-t FOCUSS

2.5.1 k-t FOCUSS with Temporal FT

k-t FOCUSS is a recent CS algorithm developed for the reconstruction of sparse-sampled dynamic

image data [17, 16]. As the name indicates, it is based on FOCUSS algorithm and utilizes random

sampling in the k-t domain [16]. Here, the basic structure of the algorithm will be speculated. For

simplicity, only the case of cartesian k-space trajectory will be discussed, with down-sampling only in

the phase-encoding direction and full sampling in the frequency-encoding direction and time.

The motivations of k-t FOCUSS are clearly described through its mathematical formulations. Prob-

lem is formulated by considering a 2D dynamic MR image data acquired over time. Recall that in MRI,

the information received through the receiver coil is the spatial Fourier transform (FT) of the scanned

object image. Thus, the acquisition of dynamic MR image data can be represented mathematically as:

υ(k, t) =

∫σ(x, t)e−j2πkxdx (2.17)

where υ(k, t) denotes k-space measurement at time t and σ(x, t) denotes the unknown image content x

at time t.

Then, the measurement-signal relationship equivalent in form to Equation (2.1) is:

υ = Fxσ (2.18)

where Fx denotes FT along content x, υ denotes the k-t space measurement vectors and σ denotes the

image vector.

If we assume less number of measurement vector elements than the number of image vector ele-

ments as in CS point of view (Equation (2.1)), Equation (2.18) results in a highly underdetermined

inverse problem. Thus, if the signal σ(x, t) can be effectively sparsified, full reconstruction of the image

is possible even at down-sampling rates lower than the Nyquist sampling limit.

By applying FT along the temporal-dimension assuming periodic motion over time, Equation (2.17)

is converted to:

υ(k, t) =

∫ ∫ρ(x, f)e−j2π(kx+ft)dxdf (2.19)

where ρ(x, f) denote the 2-D spectral signal in the x-f domain.

Then, similar to Equation (2.18), the measurement-signal relationship becomes:

υ = FxFtρ (2.20)

– 15 –

where Ft denotes the FT along temporal-dimension.

The signal solution of the sparse measurement is found by solving the following l1-norm minimization

problem:

min||ρ||1 s.t. ||υ − Fρ||2 ≤ ε (2.21)

where F = FxFt and ε denote additive noise.

If the unknown signal ρ is further decomposed as:

ρ = ρ0 + ∆ρ (2.22)

where ρ0 denotes predicted signal and ∆ρ residual signal,

then the sparsity is imposed on the residual signal ∆ρ rather than the total signal ρ. As a result, the

CS problem in Equation (2.21) changes to:

min||∆ρ||1 s.t. ||υ − Fρ0 − F∆ρ||2 ≤ ε (2.23)

In order to implement FOCUSS to solve the minimization problem above, ∆ρ is defined as:

∆ρ = Wq (2.24)

and Equation (2.23) changes to:

min||q||1 s.t. ||υ − Fρ0 − FWq||2 ≤ ε (2.25)

By converting the constrained optimization problem above into the un-constrained optimization problem

using Lagrangian multiplier, the cost function of Equation (2.25) becomes:

C(q) = ||υ − Fρ0 − FWq||22 + λ||q||22 (2.26)

Thus, the optimal solution of the problem is found by calculating the following:

ρ = ρ0 + ΘFH(FΘFH + λI)−1(υ − Fρ0) (2.27)

where Θ = WWH .

– 16 –

In the case of dynamic MR images, the matrix inversion expression in Equation (2.27) is practically

impossible to calculate. According to [16] the matrix inversion can be substituted with conjugate gradi-

ent (CG) method in order to reduce the computational burden.

In summary, the nth iteration of k-t FOCUSS is calculated by following the steps listed below:

Step 1: Compute the weighting matrix Wn using Equation (2.12)

Step 2: Compute Θn = WnWnH

Step 3: Compute the nth FOCUSS estimate: ρn = ρ0 + ΘnFH(FΘnFH + λI)−1(υ - Fρ0)

Step 4: Repeat Step 1-3 with increased iteration number (n) until convergence.

– 17 –

2.5.2 k-t FOCUSS with KLT

Even though k-t FOCUSS has been developed as above using the temporal FT as in Equation (2.20),

other general transformations could also be used to efficiently sparsify the signal [16]. One example is the

utilization of Karhunen-Loeve transform (KLT), which is also known as principal component analysis

(PCA) [33].

KLT or PCA is a data-dependent mathematical procedure that uses an orthogonal transformation

to convert possibly correlated elements of the data into a set of linearly uncorrelated components called

“principal components”. The transformation leads to a result in which the first principal component

accounts for the largest possible variance of the data, and each succeeding components have the next

largest variance possible under the restriction that it is orthogonal (i.e. uncorrelated) to the preceding

components. The transform is known to compact most of the energy into a small number of expansion

coefficients [33] and thus is ideal in CS perspective [16].

Similar to the notations in Equation (2.17), the MR image vector can be defined as:

σx = [σ(x, ∆t), σ(x, 2∆t), ..., σ(x,Nt ∆t)]T ∈ CNt (2.28)

where σx denote the unknown image vector of content x at time t.

According to [33], the covariance matrix Cx of σx is defined as:

Cx = E[σxσHx ] =

Nt∑i=1

λiψiψiH (2.29)

where λi and ψi denote the eignevalues and the corresponding eigenvectors (i.e. principal components)

of Cx, respectively.

Then, using Equation (2.29), the following compact form of the signal is deduced:

σx =

Nt∑i=1

ρxi ψi (2.30)

for some expansion coefficients ρx.

Using Equation (2.30), we can define ρ as the unknown vector to reconstruct using FOCUSS:

ρ = [ρ∆x, ρ2∆x, ..., ρNx∆x]T ∈ CNxNt×1 (2.31)

where ρ denote the KLT coefficients.

The main difference with the kt-FOCUSS algorithm that use temporal FT is the definition of the

basis expression to calculate the weighting matrix (W) and subsequently undergo FOCUSS iteration.

The kt-FOCUSS algorithm that use temporal FT defines the basis as ρ in Equation (2.20), while the

– 18 –

kt-FOCUSS algorithm that use KLT defines the basis as ρ in Equation (2.30). All subsequent calculation

steps are equivalent between the two methods.

As shown above, the principal components ψi are data dependent and thus in the case of CS this

leads to a problem since there are limited number of measurements and the estimation of covariance

matrix Cx becomes an underdetermined problem. To overcome this problem, the covariance matrix is

obtained from the initial k-t FOCUSS reconstruction using temporal FT.

– 19 –

Chapter 3. Methodology

3.1 Data Information

The acquisition of the fMRI data is the same as in [9]. Three male Sprague-Dawley rats weighing

250 ˜450g were used with approval from the Institutional Animal Care and Use Committe (IACUC) at

the University of Pittsburgh. For the activation studies, electrical stimulation was applied to either the

right or left forelimb using two needle electrodes with stimulation parameters of: current = 1.2 ˜1.6 mA,

pulse duration = 3 ms, repetition rate = 6 Hz, stimulation duration = 15 s, and inter-stimulation period

= 3 min. Animal handling and experiment procedure are detailed in [9].

All experiments were carried out on a Varian 9.4 T / 31-cm MRI system with an actively-shielded

gradient coil of 12-cm inner diameter. A homogeneous coil and a surface coil were used for RF exci-

tation and reception, respectively. For fMRI studies, four pass-band bSSFP and one GRE study were

performed. Four bSSFP studies were performed with TR / TE = 10 / 5 ms and with four different

phase-cycling angles (θ) of 0◦, 90◦, 180◦, and 270◦, which will be denoted as PC 0, PC 1, PC 2, PC

3, respectively. The GRE fMRI study was performed with TR / TE = 20 / 10 ms. The resolution

parameters were the same for all studies: matrix size = 256 × 192, FOV = 2.4 × 2.4 cm2, number of

slice = 1, and slice thickness = 2 mm. Flip angles for all the bSSFP studies and the GRE study were 16◦

and 8◦, respectively. Forty eight images were acquired for bSSFP fMRI studies; 16 during prestimulus

baseline, 8 during stimulation, and 24 during the poststimulus period. Number of slices for each epoch

was reduced by half for GRE fMRI study to match the total scan time of bSSFP fMRI study. Resonance

frequency was recalibrated before each fMRI study to minimize B0 drifting effects. Five fMRI studies of

bSSFP and GRE composed one full set and each full set was repeated 15 to 25 times for averaging.

3.2 Sampling Patterns

For all phase-cycled bSSFP and GRE data, down-sampling was applied to generate a sparse dataset

and undergo reconstruction with CS. In this study, down-sampling factor of 4 was applied to all dataset:

only 1/4th of the original k-space data of each data set was used for sparse sampling and reconstruction

with k-t FOCUSS. In CS, defining the sparse dataset is important to guarantee the performance of re-

construction. In order to determine the optimal sampling pattern for the reconstruction of fMRI data at

high field using k-t FOCUSS, five different sampling masks were considered (Fig. 3.1): Gaussian-weighted

random sampling mask with full sampling of k-space center 8 lines (Fig. 3.1 a), random sampling mask

with full sampling of k-space center 8 lines (Fig. 3.1 b), Gaussian-weighted random sampling mask with

full sampling of k-space center 1 line (Fig. 3.1 c), Gaussian-weighted random sampling mask (Fig. 3.1

d), and random sampling mask (Fig. 3.1 e) were generated and applied to each dataset before the k-t

– 20 –

FOCUSS reconstruction procedure. Masks with full sampling of k-space center lines were used to test the

effect of center k-space information (i.e. image contrast), by ensuring that all down-sampled time-series

slices had k-space center information preserved regardless of the down-sampling procedure. Gaussian-

weighting was considered as a derivative form of k-space center-weighted down-sampling pattern, in order

to generate down-sampled time-series slices with maintained but dispersed concentration of k-space cen-

ter information. A completely random pattern was considered to generate down-sampled time-series

slices without any prior manipulation in the down-sampling process. For comparison analysis, original

data (i.e. fully sampled k-space data) of each dataset was used as a control for comparing the effect of

the different sampling masks. All sampling patterns were used to test the effect of reconstruction using

k-t FOCUSS with temporal FT and k-t FOCUSS with KLT.

3.3 k-t FOCUSS Parameters

The choice of k-t FOCUSS parameters is important to ensure reconstruction of the down-sampled

fMRI data. The following kt-FOCUSS parameters were used for reconstruction using k-t FOCUSS with

temporal FT: weighting matrix power factor (p) of 0.5, FOCUSS iteration number of 2, Conjugate Gra-

dient iteration number of 20, regularization factor (λ) of 0, and no prediction. As for the reconstruction

using k-t FOCUSS with KLT, the following kt-FOCUSS parameters were used: weighting matrix power

factor (p) of 0.5, FOCUSS iteration number of 1, Conjugate Gradient iteration number of 100, regular-

ization factor (λ) of 0, and no prediction. Weighting matrix power factor (p) of 0.5 was chosen to find the

sparse solution approximately equivalent to the optimal solution of CS (shown in Equation (2.16)) [16].

FOCUSS iteration number and Conjugate Gradient iteration number were chosen through experiments

to ensure convergence and reconstruction of the down-sampled fMRI data. The fMRI data was acquired

at a relatively noiseless environment (i.e. high resolution data obtained at high field of 9.4 T), thus no

regularization factor (λ) was incorporated into the reconstruction process. No prediction was assumed

for the reconstruction of the down-sampled data in this study.

3.4 Region of Interest Selection

The selection of region of interest (ROI) is important and thus needs to be carefully selected since

the main goal of this study is to compare the effect of different sampling patterns on reconstruction of

fMRI activation maps under the same down-sampling factor, in order to determine the optimal setting

for reduced sampling and reconstruction of fMRI data with CS. The regions determined to be func-

tionally active (i.e. rejecting the null hypothesis H0) according to the t-statistics map of the original

(i.e. fully sampled k-space) data was chosen as the ROI for further analysis. Example images for ROI

selection is shown in Fig. 3.2 for GRE data: the t-statistics map (Fig. 3.2 a) was used to manually deter-

mine the boundaries of ROI mask for selection (Fig. 3.2 b) which was then applied to generate the ROI

(Fig. 3.2 c). New ROIs were defined for each different rat and data to perform ROI quantitative analysis

such as calculation of T value, number of activation voxels, percent signal change and roi time course plot.

– 21 –

3.5 Quantitative Analysis

Mean square error (MSE), T-statistics functional map, ROI quantitative analysis, false discovery

rate (FDR), Type II error, and receiver operating characteristics (ROC) curve were calculated to further

investigate the effect of CS with fMRI data at high field.

The MSE was calculated to quantify the improvement of baseline images, by comparing the baseline

images of the reconstructed time-series slices to the original (i.e. fully sampled k-space) data. The

normalized MSE calculation was performed using the following equation:

normalized MSE =||ρ− ρtrue||22||ρtrue||22

(3.1)

The Student’s t-test was performed for each dataset and sampling patterns to statistically analyse

fMRI data and generate the T-statistics functional map. The T-score is calculated on a pixel by pixel

basis over time as below:

T =x− y√s2xn +

s2ym

(3.2)

where x and y denote the mean value, s2x and s2

y denote the variation, and n and m denote the length of

the baseline and activation time series, respectively. The t-statistics functional map was generated for a

significance level of α = 0.05, excluding clusters of activation pixels less than 6.

For ROI quantitative analysis, mean T value, number of activation pixels, and percent signal changes

were quantified within the ROI for each different rat and data. ROI time course plot was generated by

viewing the time course of mean ROI value.

FDR was calculated to observe the proportion of false-positives from all statistically significant

findings (i.e. studies where the null-hypothesis is rejected). Assuming ground truth is given as the

original (i.e. fully sampled k-space) t-statistics map with significance level of α = 0.05, FDR is calculated

as below:

FDR =Number of True-Positive Activation Voxels

Number of Significant Activation Voxels(3.3)

Type II error was calculated to investigate the failure in rejecting the condition of functional ac-

tivation. Type II error is found by calculating the number of false-negative activation voxels assuming

ground truth is given as the original (i.e. fully sampled k-space) t-statistics map with significance level

of α = 0.05.

The ROC curve was generated to provide a standardized and statistically meaningful means for

comparing fMRI signal-detection accuracy [34]. Assuming ground truth is shown in the original (i.e.

– 22 –

fully sampled k-space) t-statistics map with significance level of α = 0.05, the relationship between true

positive fraction (TPF) and false positive fraction (FPF) are calculated over various significance levels

to generate the ROC curve. The performance is measured by the area under the ROC curve and ranges

from 0 to 1, with 1 representing better performance. The TPF and FPF are calculated using the follow-

ing equations:

TPF =Number of True-Positive Activation Voxels

Number of Truly Activated Voxels from Ground Truth(3.4)

and

FPF =Number of False-Positive Activation Voxels

Number of Truly Non-Activated Voxels from Ground Truth(3.5)

– 23 –

Figure 3.1: Masks of five different sampling patterns with down-sampling factor of 4. Patterns of

Gaussian-weighted random sampling with full sampling of center 8 lines (a), random sampling with full

sampling of center 8 lines (b), Gaussian-weighted random sampling with full sampling of center 1 line

(c), Gaussian-weighted random sampling (d), and random sampling (e) are shown.

– 24 –

Figure 3.2: Selection of ROI. Functionally active region is selected as the ROI from the t-statistics map

of original images acquired using GRE-EPI. Images of t-statistics map before ROI selection (a), ROI

mask (b), and selected ROI region (c) obtained for a representative animal is shown. New ROIs were

defined for each different rats. This ROI is used for further quantitative analysis such as calculation of

T values, calculation of percent signal changes and roi time course plot.

– 25 –

Chapter 4. Results and Discussion

4.1 k-t FOCUSS with Temporal FT

Visually the original baseline images became blurred with artifacts after sparse sampling was ap-

plied (Fig. 4.1). Down-sampling only the k-space center low frequency information Fig. 4.2 b), Gaussian-

weighted random sampling with full acquisition of k-space center 8 lines (Fig. 4.1 a), random sampling

with full acquisition of k-space center 8 lines (Fig. 4.1 b), Gaussian-weighted random sampling with full

acquisition of k-space center 1 line (Fig. 4.1 c), Gaussian-weighted random sampling (Fig. 4.1 d), random

sampling (Fig. 4.1 e) distorted the baseline images with the distortion of the image contrast increasing

in listed order, respectively. As expected, the baseline images with preserved k-space center informa-

tion even after down-sampling (i.e. down-sampling with only k-space center low frequency information,

down-sampling with Gaussian-weighted random sampling pattern with full acquisition of k-space center

8 lines, down-sampling with random sampling pattern with full acquisition of k-space center 8 lines,

and down-sampling with Gaussian-weighted random sampling pattern with full acquisition of k-space

center 1 line) maintained image contrast up to a certain degree, although the sampling pattern that pre-

served the most k-space center information (i.e. down-sampling with only k-space center low frequency

information) showed the image contrast closest to the original image even with only 1/4th of the whole

data. Similarly, the baseline images after down-sampling with concentrated but dispersed k-space center

information (i.e. Gaussian-weighted random sampling pattern) showed maintenance of the overall shape

of image, however, suffered from distortion and degradation of image artifacts. On the other hand, the

baseline images acquired after down-sampling with random sampling pattern were affected the most

by artifacts, since the k-space center information related to image contrast was mostly lost during the

down-sampling procedure.

Despite the distortion and degradation after down-sampling, the baseline images were well recon-

structed using k-t FOCUSS with temporal FT regardless of sampling pattern (Fig. 4.2 c and d, Fig. 4.3

c and d, and Fig. 4.4 c). Visually the image contrast was reconstructed well for all sampling patterns

compared to the original (i.e. fully sampled k-space) baseline image (Fig. 4.2 a or Fig. 4.3 a or Fig. 4.4

a) and the down-sampled baseline image with only k-space center low frequency information (Fig. 4.2

b or Fig. 4.3 b or Fig. 4.4 b). The MSE values of the reconstructed baseline images from all sampling

patterns indicate good reconstruction with mse values less than 0.06 and 0.08 for bSSFP and GRE time-

series data, while the down-sampled baseline image with only k-space center low frequency information

resulted in MSE values around 0.07 and 0.08 for the same bSSFP and GRE time-series data, respectively

(Fig. 4.5). Especially, the Gaussian-weighted random sampling scheme with full sampling of k-space cen-

ter 1 line reconstructed baseline images for both bSSFP with PC 2 and GRE very well, with the lowest

MSE values (i.e. lower than 0.01). Also, the reconstructed baseline images from the four center-weighted

sampling patterns (i.e. Gaussian-weighted random sampling pattern with full acquisition of k-space cen-

– 26 –

ter 8 lines, random sampling pattern with full acquisition of k-space center 8 lines, Gaussian-weighted

random sampling pattern with full acquisition of k-space center 1 line, and Gaussian-weighted random

sampling pattern) showed increased spatial resolution and signal-to-noise ratio (SNR) after reconstruc-

tion (Fig. 4.2 c and d, and Fig. 4.3 c and d). In addition, for phase-cycled bSSFP data, the reconstructed

baseline images from these center-weighted sampling schemes showed reduction of direct current (DC)

artifact (indicated by white arrow in Fig. 4.2 a or Fig. 4.3 a or Fig. 4.4 a) after reconstruction. This

shows that obtaining a good initial estimate through sparse sampling is important, and that certain

sampling patterns (e.g. k-space center-weighted sampling patterns) for the down-sampling process may

not guarantee full reconstruction of true image. This is in conjunction with previous literatures, since

obtaining a good initial estimate is a crucial factor to guarantee the performance of FOCUSS as explained

in [31, 32, 16].

The fMRI maps of both GRE and bSSFP were also reconstructed very well using from all k-space

center-weighted random sampling patterns using k-t FOCUSS with temporal FT (Fig. 4.6 c and d, and

Fig. 4.7 c and d), while the reconstructed fMRI maps of both GRE and bSSFP from random sampling

pattern did not show functional activation (Fig. 4.8 c). The down-sampled data from all four center-

weighted sampling patterns (i.e. Gaussian-weighted random sampling pattern with full acquisition of

k-space center 8 lines, random sampling pattern with full acquisition of k-space center 8 lines, Gaussian-

weighted random sampling pattern with full acquisition of k-space center 1 line, and Gaussian-weighted

random sampling pattern) resulted in reconstruction of fMRI maps that showed local functional activa-

tion in the contralateral somatocortical areas (Fig. 4.6 c and d, and Fig. 4.7 c and d). The down-sampling

patterns with full acquisition of k-space center 8 lines resulted in similar reconstruction of fMRI maps:

the reconstructed fMRI maps showed enlargement and dispersion of activation foci (red pixels), repre-

senting larger area of activation than the true original fMRI map for both bSSFP and GRE compared

to both the original fMRI maps and down-sampled fMRI maps with only k-space center low frequency

information (Fig. 4.6). The Gaussian-weighted random sampling pattern with full acquisition of k-space

center 1 line and Gaussian-weighted random sampling pattern resulted in the reconstruction of fMRI map

that resembled the original(Fig. 4.7). Especially, the fMRI map reconstructed from Gaussian-weighted

random sampling pattern with full acquisition of k-space center 1 line showed localization of the func-

tional activation area most similar to the original fMRI map (Fig. 4.7 c). The random sampling pattern

resulted in no reconstruction of fMRI map (Fig. 4.8 c).

In addition, visually the shifting of activation foci is observed in the fMRI map of original data: the

activation foci is located around the cortical surface area for PC 1 and 2, while it is located in the middle

cortical regions for PC 0 and 3 (indicated by white arrow in Fig. 4.6 a or Fig. 4.7 a or Fig. 4.8 a). This

is in conjunction with the findings from previous literature ([9]), and is presumed to be due to magnetic

field inhomogeneity in the brain resulting in the variation of on-resonance area for different PC angles.

Among all sampling patterns, only the Gaussian-weighted random sampling pattern with full sampling

of k-space center 1 line and Gaussian-weighted random sampling pattern resulted in reconstruction of

fMRI map that showed this shifting phenomenon of activation foci (indicated by white arrow in Fig. 4.7

c and d).

The time course of the mean ROI value was also relatively well preserved in reconstructed images

of bSSFP with PC 2 from all four k-space center-weighted sampling patterns (i.e. Gaussian-weighted

– 27 –

random sampling pattern with full acquisition of k-space center 8 lines, random sampling pattern with

full acquisition of k-space center 8 lines, Gaussian-weighted random sampling pattern with full acqui-

sition of k-space center 1 line, and Gaussian-weighted random sampling pattern) (Fig. 4.9 a, b, c and

d), while the time course of the mean ROI value in reconstructed images of bSSFP with PC 2 from

random sampling pattern differed from the original with significantly increased fluctuation (Fig. 4.9 e).

Reconstruction from the sampling patterns with full sampling of k-space center 8 lines (i.e. Gaussian-

weighted random sampling pattern with full sampling of k-space center 8 lines and random sampling

pattern with full sampling of k-space center 8 lines) resulted in similar mean ROI time course plots:

the mean amplitude difference and percent signal change between baseline and activation decreased,

however, with decreased signal fluctuation compared to the original (Fig. 4.9 a and b). Reconstruction

from the Gaussian-weighted random sampling pattern with full sampling of k-space center 1 line and

Gaussian-weighted random sampling pattern resulted in mean ROI time course plots similar to the orig-

inal data; Even with only 1/4th of the whole data, the mean ROI time course plot resembled the original

plot, with slightly reduced mean amplitude difference, percent signal change and slightly decreased signal

fluctuation (Fig. 4.9 c and d). Overall, Gaussian-weighted random sampling pattern with full sampling

of k-space center 1 line resulted in mean ROI time course plot most similar to the original. These im-

provements were applicable regardless of the phase-cycling angles.

The FDR calculations for both bSSFP data with PC 2 and GRE data over the significance level

interval of α = [5e-8, 5e-2] is shown in Fig. 4.10. Since FDR represents the fraction of false discov-

eries over total discoveries, higher values of FDR refer to higher rates of false detection of functional

activations. As for the bSSFP data with PC 2, overall the FDR of the reconstructed images from

Gaussian-weighted random sampling pattern with full sampling of k-space center 8 lines was the highest,

followed by the reconstructed images from random sampling pattern with full sampling of k-space center

8 lines, down-sampled image with only low frequency (i.e. k-space center) information, reconstructed

images from Gaussian-weighted random sampling pattern with full sampling of k-space center 1 line,

and reconstructed images from Gaussian-weighted random sampling pattern (Fig. 4.10 a). For the re-

constructed images from random sampling pattern, no detection of functional activations was found and

the FDR could not be calculated. As for the GRE data, overall the FDR of the reconstructed images

from random sampling pattern with full sampling of k-space center 8 lines was the highest, followed

by the reconstructed images from Gaussian-weighted random sampling pattern with full sampling of k-

space center 8 lines and down-sampled image with only low frequency (i.e. k-space center) information,

reconstructed images from Gaussian-weighted random sampling pattern, and reconstructed images from

Gaussian-weighted random sampling pattern with full sampling of k-space center 1 line (Fig. 4.10 b).

Similar to the case of bSSFP data with PC 2, the reconstructed images from random sampling pattern

showed no detection of functional activations and the FDR could not be calculated.

The Type II error calculations for both bSSFP data with PC 2 and GRE data over the significance

level interval of α = [5e-8, 5e-2] is shown in Fig. 4.11. Since Type II error represents false-negative,

higher values of Type II error indicates higher failure in rejecting the condition of functional activation.

As for the bSSFP data with PC 2, overall the Type II error of the reconstructed images from random

sampling pattern was the highest, followed by the reconstructed images from Gaussian-weighted ran-

dom sampling pattern, reconstructed images from Gaussian-weighted random sampling pattern with full

sampling of k-space center 1 line, down-sampled image with only low frequency (i.e. k-space center)

– 28 –

information, reconstructed images from random sampling pattern with full sampling of k-space center 8

lines, and reconstructed images from Gaussian-weighted random sampling pattern with full sampling of

k-space center 8 lines (Fig. 4.11 a). The reconstructed images from random sampling pattern showed no

detection of functional activations and thus a constant number of Type II error equivalent to the number

of true activation voxels were shown over the significance level interval. As for the GRE data, overall the

Type II error of the reconstructed images from random sampling pattern was the highest, followed by the

reconstructed images from Gaussian-weighted random sampling pattern and Gaussian-weighted random

sampling pattern with full sampling of k-space center 1 line, reconstructed images from random sampling

pattern with full sampling of k-space center 8 lines, reconstructed images from Gaussian-weighted ran-

dom sampling pattern with full sampling of k-space center 8 lines, and down-sampled image with only

low frequency (i.e. k-space center) information (Fig. 4.11 b). Similar to the case of bSSFP data with PC

2, the reconstructed images from random sampling pattern showed no detection of functional activations

and thus a constant number of Type II error equivalent to the number of true activation voxels were

shown over the significance level interval.

The ROC curve represents overall good performances of reconstruction using k-t FOCUSS with

temporal FT for all sampling patterns (Fig. 4.12). Since the ROC curve represents the degree of true-

positive activations with respect to the false-positive activations, ROC curves with area under curve

(AUC) closest to 1 represent better detection of truly activated voxels with respect to false-positive

activation voxels considering all significance levels. Although the reconstructed fMRI map from random

sampling pattern did not show functional activation for significance level of α = 0.05 as in Fig. 4.8 c,

the overall performance is good as indicated by the ROC curve generated over all significance levels.

As for the bSSFP data with PC 2, the ROC performance of the reconstructed images from Gaussian-

weighted random sampling pattern with full sampling of k-space center 1 line was the highest with AUC

of 0.9638, followed by the reconstructed images from Gaussian-weighted random sampling pattern with

full sampling of k-space center 8 lines, down-sampled image with only low frequency (i.e. k-space center)

information, reconstructed images from random sampling pattern with full sampling of k-space center

8 lines, reconstructed images from Gaussian-weighted random sampling pattern, and reconstructed im-

ages from random sampling pattern, in descending order of ROC performance over all significance levels

(Fig. 4.12 a). The AUC values were close to 1 with values above 0.95 for all images except reconstructed

image from random sampling pattern, which indicates overall very good performance of reconstruction

using k-t FOCUSS with temporal FT and k-space center-weighted random sampling schemes over all

significance levels. As for the GRE data, the ROC performance of the down-sampled images with only k-

space center low information were the highest with AUC of 0.9840, followed by the reconstructed images

from random sampling pattern with full sampling of k-space center 8 lines, reconstructed images from

Gaussian-weighted random sampling pattern with full sampling of k-space center 8 lines, reconstructed

images from Gaussian-weighted random sampling pattern with full sampling of k-space center 1 line,

reconstructed images from Gaussian-weighted random sampling pattern, and reconstructed images from

random sampling pattern, in descending order of ROC performance over all significance levels (Fig. 4.12

b). The AUC values were close to 1 with values above 0.93 for all images except reconstructed image

from random sampling pattern, which indicates overall very good performance of reconstruction using k-t

FOCUSS with temporal FT and k-space center-weighted random sampling schemes over all significance

levels.

– 29 –

The results of FDR, Type II error, and ROC curves represent the effect of different random sam-

pling patterns in terms of statistical testing. The reconstructed images from sampling patterns with full

sampling of k-space center 8 lines (i.e. Gaussian-weighted random sampling pattern with full sampling

of k-space center 8 lines and random sampling pattern with full sampling of k-space center 8 lines) have

overall high FDR, low Type II error, and high detection rate of true activation voxels with respect to

false-positive activations, which indicates enlargement of functional activation area compared to the true

original fMRI map for both bSSFP and GRE. The reconstructed images from Gaussian-weighted random

sampling pattern with full acquisition of k-space center 1 line and Gaussian-weighted random sampling

pattern have overall low FDR, some degree of Type II error, and high detection rate of true activation

voxels with respect to false-positive activations, which indicates functional activation area close to the

true original fMRI map for both bSSFP and GRE. Variations in the results of FDR, Type II error,

and ROC curves over different significance levels indicate that the performance of any reconstruction

algorithm with a given sampling scheme may not be guaranteed for certain significance levels depending

on the data. Thus, the statistical significance level must also be considered to verify the effectiveness

of the reconstruction algorithm and to perform correct analysis using reconstructed fMRI data. Other

informations, such as fMRI map, mean ROI time course and the quantitative ROI analysis methods for

a given significance level are also required to correctly perform fMRI analysis from reconstruction and

to determine optimal CS algorithm.

Quantitative ROI analysis results for reconstructed images using k-t FOCUSS with temporal FT are

shown in Table. 4.3, Table. 4.4, Table. 4.5, Table. 4.6 and Table. 4.7. Sampling patterns with full acqui-

sition of k-space center (i.e. Gaussian-weighted random sampling pattern with full sampling of k-space

center 8 lines, random sampling pattern with full sampling of k-space center 8 lines, and Gaussian-

weighted random sampling pattern with full sampling of k-space center 1 line) resulted in reconstruction

of image with increased ROI mean T values and number of activation pixels for all dataset, indicating

increase in functional activation sensitivity and specificity after reconstruction (Table. 4.3, Table. 4.4

and Table. 4.5). On the other hand, sampling patterns of Gaussian-weighted random sampling and ran-

dom sampling pattern resulted in reconstruction of image data with decreased ROI mean T values and

number of activation pixels for all dataset, indicating decrease in functional activation sensitivity and

specificity after reconstruction (Table. 4.6 and Table. 4.7). The percent signal change decreased after re-

construction regardless of sampling pattern, with the Gaussian-weighted random sampling pattern with

full sampling of k-space center 1 line resulting in reconstruction of image data with the highest percent

signal change after reconstruction (Table. 4.3, Table. 4.4, Table. 4.5, Table. 4.6 and Table. 4.7). Recon-

structed image data from Gaussian-weighted random sampling pattern resulted in mean ROI T values

closest to the original image data, while the reconstructed image data from Gaussian-weighted random

sampling pattern with full sampling of center 1 line resulted in number of activation pixels closest to

the original image data (Table. 4.6 and Table. 4.5). Low frequency image data resulted in percent signal

changes between baseline and activation closest to the original data (Table. 4.2). These quantitative

parameters indicate that low frequency image data, reconstructed image data from Gaussian-weighted

random sampling pattern with full sampling of k-space center 1 line, and reconstructed image data from

Gaussian-weighted random sampling pattern show closest resemblance to true functional activation and

sensitivity, depending on the quantitative analysis parameter.

Overall, the reconstruction results using k-t FOCUSS with temporal FT varies greatly with sampling

– 30 –

scheme, and thus the sampling scheme must be chosen wisely depending on the fMRI study. Random

sampling schemes with full sampling of k-space center 8 lines resulted in reconstruction of fMRI data

with enlarged functional activations, and seems to be most optimal for performing statistical tests at very

low significance levels. Gaussian-weighted random sampling scheme with full sampling of k-space center

1 line seems to be most optimal for correct reconstruction of fMRI data acquired at high field considering

significance level of α = 0.05. The results from random sampling scheme suggest that loss of k-space

central information can introduce significant transient artifacts to the reconstruction process, which is

in conjunction with the previous findings ([35] and [16]). These results indicate that both k-space center

information and edge information are important, and implies that the sampling scheme for correct CS

reconstruction of fMRI data acquired at high field must have a certain balance between the acquisition

of k-space center and edge information.

– 31 –

Figure 4.1: Comparison of baseline images after down-sampling. Baseline images after down-sampling

with Gaussian-weighted random down-sampling pattern with full sampling of k-space center 8 lines

(a), random down-sampling pattern with full sampling of k-space center 8 lines (b), Gaussian-weighted

random down-sampling pattern with full sampling of k-space center 1 line (c), Gaussian-weighted random

down-sampling pattern (d), and random down-sampling pattern (e) are shown. The 25th and 12th image

slice of bSSFP and GRE is shown, respectively. Down-sampling pattern and acquisition type or phase-

cycling angle are shown on the top and left-hand side of the images, respectively.

– 32 –

Figure 4.2: Comparison of reconstructed baseline images from sampling schemes with full acquisition of

k-space center 8 lines using k-t FOCUSS with temporal FT. Original baseline images from fully-sampled

k-space data (a), baseline images with only low frequency information (b), reconstructed baseline images

from Gaussian-weighted random down-sampling scheme with full sampling of k-space center 8 lines (c),

and reconstructed baseline images from random down-sampling scheme with full sampling of k-space

center 8 lines (d) are shown. The 25th and 12th image slice of bSSFP and GRE is shown, respectively.

Down-sampling pattern and acquisition type or phase-cycling angle are shown on the top and left-hand

side of the images, respectively. Notice the disappearance of DC artifact (white arrow) in all of the

reconstructed phase-cycled bSSFP images.

– 33 –

Figure 4.3: Comparison of reconstructed baseline images from Gaussian-weighted random sampling

scheme with full acquisition of k-space center 1 line and Gaussian-weighted random sampling scheme

using k-t FOCUSS with temporal FT. Original baseline images from fully-sampled k-space data (a),

baseline images with only low frequency information (b), reconstructed baseline images from Gaussian-

weighted random down-sampling scheme with full sampling of k-space center 1 line (c), and reconstructed

baseline images from Gaussian-weighted random down-sampling scheme (d) are shown. The 25th and

12th image slice of bSSFP and GRE is shown, respectively. Down-sampling pattern and acquisition type

or phase-cycling angle are shown on the top and left-hand side of the images, respectively. Notice the

disappearance of DC artifact (white arrow) in all of the reconstructed phase-cycled bSSFP images.

– 34 –

Figure 4.4: Comparison of reconstructed baseline images from random sampling scheme using k-t FO-

CUSS with temporal FT. Original baseline images from fully-sampled k-space data (a), baseline images

with only low frequency information (b), reconstructed baseline images from random down-sampling

scheme (c) are shown. The 25th and 12th image slice of bSSFP and GRE is shown, respectively. Down-

sampling pattern and acquisition type or phase-cycling angle are shown on the top and left-hand side of

the images, respectively. Notice the disappearance of DC artifact (white arrow) in all of the reconstructed

phase-cycled bSSFP images.

– 35 –

Figure 4.5: Comparison of MSE plots of reconstructed image using k-t FOCUSS with temporal FT. MSE

plots of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat are shown.

– 36 –

Figure 4.6: Comparison of reconstructed fMRI maps from sampling schemes with full acquisition of

k-space center 8 lines using k-t FOCUSS with temporal FT. Original fMRI maps from fully-sampled

k-space data (a), fMRI maps with only low frequency information (b), reconstructed fMRI maps from

Gaussian-weighted random down-sampling scheme with full sampling of k-space center 8 lines (c), and

reconstructed fMRI maps from random down-sampling scheme with full sampling of k-space center 8

lines (d) are shown for significance level of α = 0.05. Down-sampling pattern and acquisition type or

phase-cycling angle are shown on the top and left-hand side of the images, respectively.

– 37 –

Figure 4.7: Comparison of reconstructed fMRI maps from Gaussian-weighted random sampling scheme

with full acquisition of k-space center 1 line and Gaussian-weighted random sampling scheme using k-

t FOCUSS with temporal FT. Original fMRI maps from fully-sampled k-space data (a), fMRI maps

with only low frequency information (b), reconstructed fMRI maps from Gaussian-weighted random

down-sampling scheme with full sampling of k-space center 1 line (c), and reconstructed fMRI maps

from Gaussian-weighted random down-sampling scheme (d) are shown for significance level of α = 0.05.

Down-sampling pattern and acquisition type or phase-cycling angle are shown on the top and left-hand

side of the images, respectively. Notice the reconstruction of activation foci shift (white arrow) in the

phase-cycled bSSFP data.

– 38 –

Figure 4.8: Comparison of reconstructed fMRI maps from random sampling scheme using k-t FOCUSS

with temporal FT. Original fMRI maps from fully-sampled k-space data (a), fMRI maps with only low

frequency information (b), reconstructed fMRI maps from random down-sampling scheme (c) are shown

for significance level of α = 0.05. Down-sampling pattern and acquisition type or phase-cycling angle

are shown on the top and left-hand side of the images, respectively.

– 39 –

Figure 4.9: Comparison of mean ROI time course plot of original image and reconstructed images using

k-t FOCUSS with temporal FT. Mean ROI time course plot of reconstructed images from Gaussian-

weighted random down-sampling pattern with full sampling of k-space center 8 lines (a), random down-

sampling pattern with full sampling of k-space center 8 lines (b), Gaussian-weighted random down-

sampling pattern with full sampling of k-space center 1 line (c), Gaussian-weighted random down-

sampling pattern (d), and random down-sampling pattern (e) are shown. Time course plots of bSSFP

were obtained with phase-cycling angle of 180◦ of a representative rat.

– 40 –

Figure 4.10: Comparison of FDR of reconstructed fMRI map using k-t FOCUSS with temporal FT.

FDR of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat are shown.

– 41 –

Figure 4.11: Comparison of Type II error of reconstructed fMRI map using k-t FOCUSS with temporal

FT. Type II error of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat

are shown.

– 42 –

Figure 4.12: Comparison of ROC curve of reconstructed fMRI map using k-t FOCUSS with temporal

FT. ROC curves of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat are

shown. The ROC curve with the highest area under curve (AUC) value is indicated by the red-dashed

box.

– 43 –

Table 4.1: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

original image data.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 2.0 ± 0.6 1.7 ± 0.6 1.9 ± 0.7 1.7 ± 0.5 1.8 ± 0.5

Number of Active Pixels 723 ± 127 700 ± 100 713 ± 95 727 ± 113 697 ± 59

Percent Signal Changes 2.1 ± 0.2 1.5 ± 0.4 1.6 ± 0.7 1.8 ± 0.6 1.5 ± 0.3

Table 4.2: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data with only low frequency components.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 3.2 ± 1.4 2.9 ± 1.1 3.1 ± 1.3 2.7 ± 1.2 2.8 ± 0.9

Number of Active Pixels 842 ± 132 799 ± 116 813 ± 69 840 ± 211 810 ± 39

Percent Signal Changes 2.1 ± 0.2 1.5 ± 0.5 1.6 ± 0.7 1.8 ± 0.7 1.5 ± 0.3

Table 4.3: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from Gaussian-weighted random down-sampling scheme with full sampling of

k-space center 8 lines using k-t FOCUSS with temporal FT.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 4.3 ± 2.0 4.4 ± 2.0 4.5 ± 2.0 3.1 ± 2.3 2.9 ± 1.7

Number of Active Pixels 1055 ± 166 1075 ± 319 1059 ± 38 1158 ± 484 973 ± 255

Percent Signal Changes 0.9 ± 0.5 0.4 ± 0.2 0.4 ± 0.3 0.6 ± 0.4 0.3 ± 0.1

Table 4.4: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from random down-sampling scheme with full sampling of k-space center 8

lines using k-t FOCUSS with temporal FT.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 3.8 ± 1.7 4.0 ± 2.0 4.1 ± 1.8 3.1 ± 2.2 2.6 ± 1.3

Number of Active Pixels 1042 ± 120 1026 ± 327 1008 ± 40 1140 ± 467 896 ± 225

Percent Signal Changes 1.0 ± 0.5 0.5 ± 0.2 0.5 ± 0.3 0.7 ± 0.5 0.4 ± 0.2

– 44 –

Table 4.5: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from Gaussian-weighted random down-sampling scheme with full sampling of

k-space center 1 line using k-t FOCUSS with temporal FT.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 2.5 ± 0.9 2.4 ± 0.7 2.4 ± 0.8 1.9 ± 0.6 1.3 ± 0.3

Number of Active Pixels 888 ± 219 738 ± 188 808 ± 177 735 ± 185 504 ± 56

Percent Signal Changes 1.3 ± 0.0 1.0 ± 0.2 1.0 ± 0.3 1.0 ± 0.4 0.8 ± 0.3

Table 4.6: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from Gaussian-weighted random down-sampling scheme using k-t FOCUSS

with temporal FT.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 2.0 ± 0.6 1.8 ± 0.7 1.8 ± 0.8 1.6 ± 0.7 1.4 ± 0.3

Number of Active Pixels 571 ± 11 543 ± 73 556 ± 67 648 ± 123 1339 ± 134

Percent Signal Changes 1.0 ± 0.2 0.8 ± 0.3 0.6 ± 0.4 0.8 ± 0.4 0.3 ± 0.4

Table 4.7: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from random down-sampling scheme using k-t FOCUSS with temporal FT.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 1.0 ± 0.1 0.7 ± 0.1 0.7 ± 0.2 0.8 ± 0.1 0.6 ± 0.0

Number of Active Pixels 619 ± 384 521 ± 203 578 ± 183 646 ± 178 129 ± 48

Percent Signal Changes 1.6 ± 0.6 1.0 ± 0.2 1.0 ± 0.5 1.2 ± 0.3 1.4 ± 0.4

– 45 –

4.2 k-t FOCUSS with KLT

Similar to the results obtained through k-t FOCUSS with temporal FT, the baseline images were

well reconstructed using k-t FOCUSS with KLT regardless of sampling pattern, despite the distortion and

degradation after down-sampling (Fig. 4.13 c and d, Fig. 4.14 c and d, and Fig. 4.15 c). Visually the im-

age contrast was reconstructed well for all sampling patterns compared to the original (i.e. fully sampled

k-space) baseline image (Fig. 4.13 a or Fig. 4.14 a or Fig. 4.15 a) and the down-sampled baseline image

with only k-space center low frequency information (Fig. 4.13 b or Fig. 4.14 b or Fig. 4.15 b). The MSE

values of the reconstructed baseline images from all sampling patterns were very similar to the resulted

obtained through k-t FOCUSS with temporal FT and indicates good reconstruction: the mse values were

less than 0.06 and 0.08 for bSSFP and GRE time-series data, while the down-sampled baseline image

with only k-space center low frequency information resulted in MSE values around 0.07 and 0.08 for

the same bSSFP and GRE time-series data, respectively (Fig. 4.16). Especially, the Gaussian-weighted

random sampling scheme with full sampling of k-space center 1 line reconstructed baseline images for

both bSSFP with PC 2 and GRE very well, with the lowest MSE values (i.e. lower than 0.01). Also, the

reconstructed baseline images from the four center-weighted sampling patterns (i.e. Gaussian-weighted

random sampling pattern with full acquisition of k-space center 8 lines, random sampling pattern with

full acquisition of k-space center 8 lines, Gaussian-weighted random sampling pattern with full acquisi-

tion of k-space center 1 line, and Gaussian-weighted random sampling pattern) showed increased spatial

resolution and signal-to-noise ratio (SNR) after reconstruction (Fig. 4.13 c and d, and Fig. 4.14 c and

d), with reduction of direct current (DC) artifact (indicated by white arrow in Fig. 4.13 a or Fig. 4.14 a

or Fig. 4.15 a) after reconstruction. These results also suggest the importance of obtaining a good initial

estimate through sparse sampling to guarantee the performance of CS reconstruction algorithms.

The fMRI maps of both GRE and bSSFP were also reconstructed very well using k-t FOCUSS with

KLT from all k-space center-weighted random sampling patterns (Fig. 4.17 c and d, and Fig. 4.18 c

and d), while the reconstructed fMRI maps of both GRE and bSSFP from random sampling pattern

did not show functional activation (Fig. 4.19 c). The down-sampled data from all four center-weighted

sampling patterns (i.e. Gaussian-weighted random sampling pattern with full acquisition of k-space cen-

ter 8 lines, random sampling pattern with full acquisition of k-space center 8 lines, Gaussian-weighted

random sampling pattern with full acquisition of k-space center 1 line, and Gaussian-weighted random

sampling pattern) resulted in reconstruction of fMRI maps that showed local functional activation in

the contralateral somatocortical areas (Fig. 4.17 c and d, and Fig. 4.18 c and d). The down-sampling

patterns with full acquisition of k-space center 8 lines resulted in similar reconstruction of fMRI maps:

the reconstructed fMRI maps showed enlargement and dispersion of activation foci (red pixels), repre-

senting larger area of activation than the true original fMRI map for both bSSFP and GRE compared

to both the original fMRI maps and down-sampled fMRI maps with only k-space center low frequency

information (Fig. 4.17). The Gaussian-weighted random sampling pattern with full acquisition of k-space

center 1 line and Gaussian-weighted random sampling pattern resulted in the reconstruction of fMRI

map that resembled the original in terms of functional sensitivity and the presence of activation foci

shifting phenomenon in the bSSFP data (i.e. activation foci located in the cortical surface area for PC 1

and 2, and located in the middle cortical regions for PC 0 and 3) (Fig. 4.18). Especially, the fMRI map

reconstructed from Gaussian-weighted random sampling pattern with full acquisition of k-space center

1 line showed localization of the functional activation area most similar to the original fMRI map with

– 46 –

the shifting phenomenon of activation foci present (Fig. 4.18 c). The random sampling pattern resulted

in no reconstruction of fMRI map (Fig. 4.19 c).

The time course of the mean ROI value in the reconstructed images of bSSFP with PC 2 from

all four center-weighted sampling patterns (i.e. Gaussian-weighted random sampling pattern with full

acquisition of k-space center 8 lines, random sampling pattern with full acquisition of k-space center

8 lines, Gaussian-weighted random sampling pattern with full acquisition of k-space center 1 line, and

Gaussian-weighted random sampling pattern) indicated difference between baseline and activation times

(Fig. 4.20 a, b, c and d), while the time course of the mean ROI value in reconstructed images of bSSFP

with PC 2 from random sampling pattern showed no difference between baseline and activation times

with significantly increased fluctuation (Fig. 4.20 e). Reconstruction from the two sampling patterns

with full sampling of k-space center 8 lines (i.e. Gaussian-weighted random sampling pattern with full

sampling of center 8 lines and random sampling pattern with full sampling of center 8 lines) resulted

in similar mean ROI time course plots, showing decrease in the mean amplitude difference and percent

signal change between baseline and activation but also with decrease in the signal fluctuation compared

to the original (Fig. 4.20 a and b). Reconstruction from the Gaussian-weighted random sampling pat-

tern with full sampling of k-space center 1 line and Gaussian-weighted random sampling pattern resulted

in mean ROI time course plots resembling the original plot, showing slightly reduced mean amplitude

difference, percent signal change and also slight decrease in signal fluctuation (Fig. 4.20 c and d). These

improvements were applicable regardless of the phase-cycling angles for bSSFP data.

The FDR calculations for both bSSFP data with PC 2 and GRE data over the significance level

interval of α = [5e-8, 5e-2] is shown in Fig. 4.21. Recall that FDR represents the fraction of false

discoveries over total discoveries, and higher values of FDR refer to higher rates of false detection of

functional activations. As for the bSSFP data with PC 2, overall the FDR of the reconstructed im-

ages from Gaussian-weighted random sampling pattern with full sampling of k-space center 8 lines was

the highest, followed by the reconstructed images from random sampling pattern with full sampling of

k-space center 8 lines, down-sampled image with only low frequency (i.e. k-space center) information,

reconstructed images from Gaussian-weighted random sampling pattern with full sampling of k-space

center 1 line, and reconstructed images from Gaussian-weighted random sampling pattern (Fig. 4.21 a).

For the reconstructed images from random sampling pattern, no detection of functional activations was

found and the FDR could not be calculated. As for the GRE data, overall the FDR of the reconstructed

images from Gaussian-weighted random sampling pattern with full sampling of k-space center 8 lines

was the highest, followed by the reconstructed images from random sampling pattern with full sampling

of k-space center 8 lines, down-sampled image with only low frequency (i.e. k-space center) information,

reconstructed images from Gaussian-weighted random sampling pattern, and reconstructed images from

Gaussian-weighted random sampling pattern with full sampling of k-space center 1 line (Fig. 4.21 b).

Similar to the case of bSSFP data with PC 2, the reconstructed images from random sampling pattern

showed no detection of functional activations and the FDR could not be calculated.

The Type II error calculations for both bSSFP data with PC 2 and GRE data over the significance

level interval of α = [5e-8, 5e-2] is shown in Fig. 4.22. Recall that Type II error represents false-negative,

and higher values of Type II error indicates higher failure in rejecting the condition of functional acti-

vation. As for the bSSFP data with PC 2, overall the Type II error of the reconstructed images from

– 47 –

random sampling pattern was the highest, followed by the reconstructed images from Gaussian-weighted

random sampling pattern, reconstructed images from Gaussian-weighted random sampling pattern with

full sampling of k-space center 1 line, down-sampled image with only low frequency (i.e. k-space center)

information, reconstructed images from random sampling pattern with full sampling of k-space center 8

lines, and reconstructed images from Gaussian-weighted random sampling pattern with full sampling of

k-space center 8 lines (Fig. 4.22 a). The reconstructed images from random sampling pattern showed no

detection of functional activations and thus a constant number of Type II error equivalent to the number

of true activation voxels were shown over the significance level interval. As for the GRE data, overall the

Type II error of the reconstructed images from random sampling pattern was the highest, followed by the

reconstructed images from Gaussian-weighted random sampling pattern and Gaussian-weighted random

sampling pattern with full sampling of k-space center 1 line, reconstructed images from random sampling

pattern with full sampling of k-space center 8 lines and down-sampled image with only low frequency

(i.e. k-space center) information, and reconstructed images from Gaussian-weighted random sampling

pattern with full sampling of k-space center 8 lines (Fig. 4.22 b). Similar to the case of bSSFP data

with PC 2, the reconstructed images from random sampling pattern showed no detection of functional

activations and thus a constant number of Type II error equivalent to the number of true activation

voxels were shown over the significance level interval.

The ROC curve represents overall good performances of reconstruction using k-t FOCUSS with

KLT for all sampling patterns (Fig. 4.23). Although the reconstructed fMRI map from random sam-

pling pattern did not show functional activation for significance level of α = 0.05 as in Fig. 4.19 c,

the overall performance is good as indicated by the ROC curve generated over all significance levels.

As for the bSSFP data with PC 2, the ROC performance of the reconstructed images from Gaussian-

weighted random sampling pattern with full sampling of k-space center 1 line was the highest with

AUC of 0.9660, followed by the down-sampled image with only low frequency (i.e. k-space center)

information, reconstructed images from Gaussian-weighted random sampling pattern, reconstructed im-

ages from random sampling pattern with full sampling of k-space center 8 lines, reconstructed images

from Gaussian-weighted random sampling pattern with full sampling of k-space center 8 lines, and re-

constructed images from random sampling pattern, in descending order of ROC performance over all

significance levels (Fig. 4.23 a). The AUC values were close to 1 with values above 0.93 for all images ex-

cept reconstructed image from random sampling pattern, which indicates overall very good performance

of reconstruction using k-t FOCUSS with temporal FT and k-space center-weighted random sampling

schemes over all significance levels. As for the GRE data, the ROC performance of the reconstructed

images from random sampling pattern with full sampling of k-space center 8 lines was the highest with

AUC of 0.9857, followed by the down-sampled image with only low frequency (i.e. k-space center) in-

formation, reconstructed images from Gaussian-weighted random sampling pattern with full sampling

of k-space center 8 lines, reconstructed images from Gaussian-weighted random sampling pattern with

full sampling of k-space center 1 line, reconstructed images from Gaussian-weighted random sampling

pattern, and reconstructed images from random sampling pattern, in descending order of ROC perfor-

mance over all significance levels (Fig. 4.23 b). The AUC values were close to 1 with values above 0.93

for all images except reconstructed image from random sampling pattern, which indicates overall very

good performance of reconstruction using k-t FOCUSS with temporal FT and k-space center-weighted

random sampling schemes over all significance levels.

– 48 –

The results of FDR, Type II error, and ROC curves represent the effect of different random sam-

pling patterns in terms of statistical testing. The reconstructed images from sampling patterns with full

sampling of k-space center 8 lines (i.e. Gaussian-weighted random sampling pattern with full sampling

of k-space center 8 lines and random sampling pattern with full sampling of k-space center 8 lines) have

overall high FDR, low Type II error, and high detection rate of true activation voxels with respect to

false-positive activations, which indicates enlargement of functional activation area compared to the true

original fMRI map for both bSSFP and GRE. The reconstructed images from Gaussian-weighted random

sampling pattern with full acquisition of k-space center 1 line and Gaussian-weighted random sampling

pattern have overall low FDR, some degree of Type II error, and high detection rate of true activation

voxels with respect to false-positive activations, which indicates functional activation area close to the

true original fMRI map for both bSSFP and GRE. Variations in the results of FDR, Type II error, and

ROC curves over different significance levels also indicate that the performance of any reconstruction

algorithm may not be guaranteed for certain significance levels depending on the data, and thus the

significance level must be considered for correct fMRI analysis in case using CS for improving temporal

resolution of bSSFP data acquired at high field. Other informations, such as fMRI map, mean ROI

time course and the quantitative ROI analysis methods for a given significance level are also needed to

correctly perform fMRI analysis from reconstruction and to determine optimal CS algorithm.

Quantitative ROI analysis results for reconstructed images using k-t FOCUSS with KLT are shown

in Table. 4.8, Table. 4.9, Table. 4.10, Table. 4.11 and Table. 4.12. Reconstructed images from sam-

pling schemes with full acquisition of k-space center (i.e. Gaussian-weighted random sampling pattern

with full sampling of k-space center 8 lines, random sampling pattern with full sampling of k-space

center 8 lines, and Gaussian-weighted random sampling pattern with full sampling of k-space center 1

line) had increased ROI mean T values and number of activation pixels for all dataset compared to

original, indicating increase in functional activation sensitivity and specificity after reconstruction (Ta-

ble. 4.8, Table. 4.9 and Table. 4.10). On the other hand, reconstructed images from sampling patterns of

Gaussian-weighted random sampling and random sampling pattern had decreased ROI mean T values

and number of activation pixels for all dataset compared to original, indicating decrease in functional

activation sensitivity and specificity after reconstruction (Table. 4.11 and Table. 4.12). The percent

signal change decreased after reconstruction regardless of sampling pattern, with the Gaussian-weighted

random sampling pattern with full sampling of k-space center 1 line resulting in reconstruction of image

data with the highest percent signal change after reconstruction (Table. 4.8, Table. 4.9, Table. 4.10, Ta-

ble. 4.11 and Table. 4.12). Reconstructed image data from Gaussian-weighted random sampling pattern

resulted in mean ROI T values closest to the original image data, while the reconstructed image data

from Gaussian-weighted random sampling pattern with full sampling of center 1 line resulted in number

of activation pixels closest to the original image data (Table. 4.11 and Table. 4.10). Low frequency

image data resulted in percent signal changes between baseline and activation closest to the original

data (Table. 4.2). These quantitative parameters indicate that low frequency image data, reconstructed

image data from Gaussian-weighted random sampling pattern with full sampling of k-space center 1

line, and reconstructed image data from Gaussian-weighted random sampling pattern show closest re-

semblance to true functional activation and sensitivity, depending on the quantitative analysis parameter.

Overall, similar to the reconstruction using k-t FOCUSS with temporal FT, the reconstruction of

fMRI data acquired at high field using k-t FOCUSS with KLT varies greatly with sampling scheme,

– 49 –

and thus the sampling scheme must be chosen wisely depending on the fMRI study. Random sampling

schemes with full sampling of k-space center 8 lines resulted in reconstruction of fMRI data with en-

larged functional activations, and seems to be most optimal for performing statistical tests at very low

significance levels. Gaussian-weighted random sampling scheme with full sampling of k-space center 1

line seems to be most optimal for correct reconstruction of fMRI data acquired at high field considering

significance level of α = 0.05. The results from random sampling scheme also suggest that loss of k-space

central information can introduce significant transient artifacts to the reconstruction process. These

results indicate that both k-space center information and edge information are important, and implies

that the sampling scheme for correct CS reconstruction of fMRI data acquired at high field must have a

certain balance between the acquisition of k-space center and edge information.

– 50 –

Figure 4.13: Comparison of reconstructed baseline images from sampling schemes with full acquisition

of k-space center 8 lines using k-t FOCUSS with KLT. Original baseline images from fully-sampled k-

space data (a), baseline images with only low frequency information (b), reconstructed baseline images

from Gaussian-weighted random down-sampling scheme with full sampling of k-space center 8 lines (c),

and reconstructed baseline images from random down-sampling scheme with full sampling of k-space

center 8 lines (d) are shown. The 25th and 12th image slice of bSSFP and GRE is shown, respectively.

Down-sampling pattern and acquisition type or phase-cycling angle are shown on the top and left-hand

side of the images, respectively. Notice the disappearance of DC artifact (white arrow) in all of the

reconstructed phase-cycled bSSFP images.

– 51 –

Figure 4.14: Comparison of reconstructed baseline images from Gaussian-weighted random sampling

scheme with full acquisition of k-space center 1 line and Gaussian-weighted random sampling scheme

using k-t FOCUSS with KLT. Original baseline images from fully-sampled k-space data (a), baseline

images with only low frequency information (b), reconstructed baseline images from Gaussian-weighted

random down-sampling scheme with full sampling of k-space center 1 line (c), and reconstructed baseline

images from Gaussian-weighted random down-sampling scheme (d) are shown. The 25th and 12th

image slice of bSSFP and GRE is shown, respectively. Down-sampling pattern and acquisition type

or phase-cycling angle are shown on the top and left-hand side of the images, respectively. Notice the

disappearance of DC artifact (white arrow) in all of the reconstructed phase-cycled bSSFP images.

– 52 –

Figure 4.15: Comparison of reconstructed baseline images from random sampling scheme using k-t

FOCUSS with KLT. Original baseline images from fully-sampled k-space data (a), baseline images with

only low frequency information (b), reconstructed baseline images from random down-sampling scheme

(c) are shown. The 25th and 12th image slice of bSSFP and GRE is shown, respectively. Down-sampling

pattern and acquisition type or phase-cycling angle are shown on the top and left-hand side of the

images, respectively. Notice the disappearance of DC artifact (white arrow) in all of the reconstructed

phase-cycled bSSFP images.

– 53 –

Figure 4.16: Comparison of MSE plots of reconstructed image using k-t FOCUSS with KLT. MSE plots

of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat are shown.

– 54 –

Figure 4.17: Comparison of reconstructed fMRI maps from sampling schemes with full acquisition of

k-space center 8 lines using k-t FOCUSS with KLT. Original fMRI maps from fully-sampled k-space

data (a), fMRI maps with only low frequency information (b), reconstructed fMRI maps from Gaussian-

weighted random down-sampling scheme with full sampling of k-space center 8 lines (c), and reconstructed

fMRI maps from random down-sampling scheme with full sampling of k-space center 8 lines (d) are shown

for significance level of α = 0.05. Down-sampling pattern and acquisition type or phase-cycling angle

are shown on the top and left-hand side of the images, respectively.

– 55 –

Figure 4.18: Comparison of reconstructed fMRI maps from Gaussian-weighted random sampling scheme

with full acquisition of k-space center 1 line and Gaussian-weighted random sampling scheme using k-t

FOCUSS with KLT. Original fMRI maps from fully-sampled k-space data (a), fMRI maps with only low

frequency information (b), reconstructed fMRI maps from Gaussian-weighted random down-sampling

scheme with full sampling of k-space center 1 line (c), and reconstructed fMRI maps from Gaussian-

weighted random down-sampling scheme (d) are shown for significance level of α = 0.05. Down-sampling

pattern and acquisition type or phase-cycling angle are shown on the top and left-hand side of the images,

respectively. Notice the reconstruction of activation foci shift (white arrow) in the phase-cycled bSSFP

data.

– 56 –

Figure 4.19: Comparison of reconstructed fMRI maps from random sampling scheme using k-t FOCUSS

with KLT. Original fMRI maps from fully-sampled k-space data (a), fMRI maps with only low frequency

information (b), reconstructed fMRI maps from random down-sampling scheme (c) are shown for signif-

icance level of α = 0.05. Down-sampling pattern and acquisition type or phase-cycling angle are shown

on the top and left-hand side of the images, respectively.

– 57 –

Figure 4.20: Comparison of mean ROI time course plot of original image and reconstructed images using

k-t FOCUSS with KLT. Mean ROI time course plot of reconstructed images from Gaussian-weighted

random down-sampling pattern with full sampling of k-space center 8 lines (a), random down-sampling

pattern with full sampling of k-space center 8 lines (b), Gaussian-weighted random down-sampling pat-

tern with full sampling of k-space center 1 line (c), Gaussian-weighted random down-sampling pattern

(d), and random down-sampling pattern (e) are shown. Time course plots of bSSFP were obtained with

phase-cycling angle of 180◦ of a representative rat.

– 58 –

Figure 4.21: Comparison of FDR of reconstructed fMRI map using k-t FOCUSS with KLT. FDR of

bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat are shown.

– 59 –

Figure 4.22: Comparison of Type II error of reconstructed fMRI map using k-t FOCUSS with KLT.

Type II error of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat are

shown.

– 60 –

Figure 4.23: Comparison of ROC curve of reconstructed fMRI map using k-t FOCUSS with KLT. ROC

curves of bSSFP with phase-cycling angle of 180◦ (a) and GRE (b) of a representative rat are shown.

The ROC curve with the highest area under curve (AUC) value is indicated by the red-dashed box.

– 61 –

Table 4.8: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from Gaussian-weighted random down-sampling scheme with full sampling of

k-space center 8 lines using k-t FOCUSS with KLT.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 5.0 ± 3.4 5.2 ± 3.3 5.0 ± 3.5 3.4 ± 2.8 3.7 ± 3.1

Number of Active Pixels 1119 ± 328 1180 ± 784 1161 ± 345 1264 ± 580 1028 ± 480

Percent Signal Changes 1.0 ± 0.5 0.4 ± 0.2 0.4 ± 0.3 0.7 ± 0.5 0.4 ± 0.2

Table 4.9: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from random down-sampling scheme with full sampling of k-space center 8

lines using k-t FOCUSS with KLT.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 4.2 ± 2.4 4.2 ± 2.7 4.2 ± 2.7 3.1 ± 2.2 3.1 ± 2.1

Number of Active Pixels 1017 ± 178 939 ± 366 992 ± 262 1112 ± 508 925 ± 356

Percent Signal Changes 1.0 ± 0.4 0.5 ± 0.2 0.5 ± 0.3 0.7 ± 0.5 0.4 ± 0.2

Table 4.10: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from Gaussian-weighted random down-sampling scheme with full sampling of

k-space center 1 line using k-t FOCUSS with KLT.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 2.5 ± 0.9 2.4 ± 0.8 2.5 ± 0.8 1.9 ± 0.6 1.3 ± 0.3

Number of Active Pixels 878 ± 208 704 ± 198 788 ± 148 725 ± 175 489 ± 60

Percent Signal Changes 1.3 ± 0.0 1.1 ± 0.2 1.0 ± 0.3 1.0 ± 0.4 0.8 ± 0.3

Table 4.11: Comparison of T values, number of activation pixels, and percent signal changes of ROI

in image data reconstructed from Gaussian-weighted random down-sampling scheme using k-t FOCUSS

with KLT.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 2.0 ± 0.6 1.8 ± 0.7 1.8 ± 0.8 1.6 ± 0.7 1.4 ± 0.3

Number of Active Pixels 557 ± 17 514 ± 82 561 ± 70 647 ± 142 1339 ± 134

Percent Signal Changes 1.0 ± 0.2 0.8 ± 0.3 0.7 ± 0.4 0.7 ± 0.4 0.3 ± 0.4

– 62 –

Table 4.12: Comparison of T values, number of activation pixels, and percent signal changes of ROI in

image data reconstructed from random down-sampling scheme using k-t FOCUSS with KLT.

bSSFP GRE

PC0 PC1 PC2 PC3

T Values 1.0 ± 0.1 0.7 ± 0.1 0.7 ± 0.2 0.7 ± 0.1 0.6 ± 0.0

Number of Active Pixels 621 ± 385 525 ± 207 573 ± 182 651 ± 178 129 ± 48

Percent Signal Changes 1.6 ± 0.6 1.0 ± 0.2 1.0 ± 0.5 1.2 ± 0.3 1.4 ± 0.4

– 63 –

4.3 Practical Implications

Experimental results suggests that the sampling scheme must be considered wisely for the CS al-

gorithms to correctly reconstruct the image, since defining the sparse dataset is an important step to

guarantee the performance of reconstruction for any CS algorithm. As for the reconstruction using both

k-t FOCUSS algorithms (i.e. k-t FOCUSS with temporal FT and k-t FOCUSS with KLT), the sampling

scheme of Gaussian-weighted random sampling with full sampling of k-space center 1 line reconstructs

both baseline images and fMRI maps well and seems to be the most optimal sampling scheme for fMRI

data acquired at high field to apply CS. This indicates that both k-space center information and edge

information is required to guarantee the performance of CS algorithm. Further tests with other sampling

schemes that incorporates both k-space center and edge information is necessary to ensure correct recon-

struction and to determine the best sampling scheme that maximizes the performance of CS algorithms

on fMRI data acquired at high field.

Considering practical implementation of sampling scheme framework, pair-wise random sampling

scheme of k-space lines is recommended for the acquisition of bSSFP data with PC 2 at high fields to

suppress the Eddy current effects [35]. It is discovered that Eddy currents generated by consecutive

phase-encoding gradients due to magnetic field inhomogeneity can lead to loss of net magnetization in

the refocused echo signal during MRI acquisition. This can be a major problem especially when random

sampling scheme is incorporated in the phase-encoding direction. If the bSSFP data is obtained with PC

2, this Eddy current effect due to random sampling in the phase-encoding direction can be suppressed

by acquiring the consecutive k-space lines in pairs of even numbers (Fig. 4.24 a) [35].

Incorporating this idea to our experimental data, a simulation study was performed to see the ef-

fect of pair-wise downsampling scheme. A paired Gaussian-weighted random sampling pattern with full

sampling of k-space center 1 line was generated, with a down-sampling factor of 4 (Fig. 4.24 b). This

down-sampling pattern was applied to original k-space data and the down-sampled data underwent re-

construction using both k-t FOCUSS with temporal FT and k-t FOCUSS with KLT. The reconstructed

fMRI maps using k-t FOCUSS with temporal FT and k-t FOCUSS with KLT are shown in Fig. 4.25 and

Fig. 4.26, respectively. The effect of pair-wise sampling in Gaussian-weighted random sampling scheme

with full sampling of k-space center 1 line with both k-t FOCUSS reconstruction algorithms showed

maintenance of activation foci shift but decrease in functional activation specificity for all phase-cycled

bSSFP data (Fig. 4.25 and Fig. 4.26). This indicates that the pair-wise sampling scheme should be im-

plemented wisely, considering the trade-off between the suppression of Eddy current effects and reduced

specificity of fMRI activation.

The FDR, Type II error, and ROC curves were calculated to determine the effect of pair-wise ran-

dom sampling scheme in terms of statistical testing. The FDR of the down-sampled image with only

low frequency (i.e. k-space center) information was the highest, followed by the reconstructed images

from the Gaussian-weighted random sampling pattern with full sampling of k-space center 1 line using

k-t FOCUSS algorithm with temporal FT and KLT, and the reconstructed images from the pair-wise

Gaussian-weighted random sampling pattern with full sampling of k-space center 1 line using k-t FOCUSS

algorithm with temporal FT and KLT (Fig. 4.27). The Type II error of the reconstructed images from the

pair-wise Gaussian-weighted random sampling pattern with full sampling of k-space center 1 line using

– 64 –

k-t FOCUSS algorithm with temporal FT and KLT was the highest, followed by the reconstructed images

from the Gaussian-weighted random sampling pattern with full sampling of k-space center 1 line using

k-t FOCUSS algorithm with temporal FT and KLT, and down-sampled image with only low frequency

(i.e. k-space center) information (Fig. 4.28). The ROC performance of the reconstructed images using

k-t FOCUSS algorithm with KLT and Gaussian-weighted random sampling pattern with full sampling of

k-space center 1 line was the highest with AUC of 0.9660, followed the reconstructed images using k-t FO-

CUSS algorithm with temporal FT and Gaussian-weighted random sampling pattern with full sampling

of k-space center 1 line, down-sampled image with only low frequency (i.e. k-space center) information,

reconstructed images using k-t FOCUSS algorithm with temporal FT and pair-wise Gaussian-weighted

random sampling pattern with full sampling of k-space center 1 line, and reconstructed images using

k-t FOCUSS algorithm with KLT and pair-wise Gaussian-weighted random sampling pattern with full

sampling of k-space center 1 line, in descending order of ROC performance over all significance levels

(Fig. 4.29). The AUC values were close to 1 with values above 0.94 for all images, which indicates overall

very good performance of reconstruction using k-t FOCUSS algorithm with any of the listed sampling

schemes. Overall, the incorporation of pair-wise sampling scheme resulted in reduction in FDR and

ROC performance while increasing Type II error, regardless of k-t FOCUSS reconstruction algorithm.

The trade-off between the suppression of Eddy current effects and reduction in ROC performance with

increase in Type II error must be considered when incorporating pair-wise sampling scheme in bSSFP

data with PC 2.

In summary, the reconstruction of fMRI data acquired at high field using k-t FOCUSS varies greatly

with sampling scheme and thus the sampling scheme must be chosen wisely depending on the fMRI study.

Random sampling schemes with full sampling of k-space center 8 lines resulted in reconstruction of fMRI

data with enlarged functional activations, and seems to be most optimal for performing statistical tests at

very low significance levels. For fMRI study analysis at significance level of α = 0.05, Gaussian-weighted

random sampling scheme with full sampling of k-space center 1 line resulted in reconstruction of fMRI

data showing closest resemblance to the original image in terms of fMRI maps, mean ROI time course

plot, number of activation pixels within ROI, FDR, Type II error and ROC performance, and thus seems

to be the most optimal sampling scheme. Both k-space center information and edge information seems

to be important, and thus the sampling scheme for correct CS reconstruction of fMRI data acquired at

high field must have a certain balance between the acquisition of k-space center and edge information.

Further studies need to be conducted for understanding the sources of the fMRI signal improvements,

effects of other CS algorithms, optimization of parameters, and changes in the sparse sampling scheme

to deduce practical implications regarding fMRI with CS at high field.

– 65 –

Figure 4.24: Implementation of paired random sampling pattern for suppression of Eddy current effects

in bSSFP with phase-cycling angle of 180◦. Paired random sampling trajectory (a) and paired Gaussian-

weighted random sampling scheme with full sampling of k-space center 1 line over time with a down-

sampling factor of 4 (b) are shown.

– 66 –

Figure 4.25: Comparison of reconstructed fMRI maps from paired Gaussian-weighted random sampling

scheme with full sampling of k-space center 1 line using k-t FOCUSS with temporal FT. Original fMRI

maps from fully-sampled k-space (a), fMRI maps with only low frequency information (b), reconstructed

fMRI maps from Gaussian-weighted random down-sampling pattern with full sampling of k-space center

1 line (c), and reconstructed fMRI maps from paired Gaussian-weighted random down-sampling pattern

with full sampling of k-space center 1 line (d) are shown for significance level of α = 0.05. Down-sampling

pattern and acquisition type or phase-cycling angle are shown on the top and left-hand side of the images,

respectively. Notice the shift of activation foci (white arrow) in the phase-cycled bSSFP data.

– 67 –

Figure 4.26: Comparison of reconstructed fMRI maps from paired Gaussian-weighted random sampling

scheme with full sampling of k-space center 1 line using k-t FOCUSS with KLT. Original fMRI maps

from fully-sampled k-space (a), fMRI maps with only low frequency information (b), reconstructed fMRI

maps from Gaussian-weighted random down-sampling pattern with full sampling of k-space center 1 line

(c), and reconstructed fMRI maps from paired Gaussian-weighted random down-sampling pattern with

full sampling of k-space center 1 line (d) are shown for significance level of α = 0.05. Down-sampling

pattern and acquisition type or phase-cycling angle are shown on the top and left-hand side of the images,

respectively. Notice the shift of activation foci (white arrow) in the phase-cycled bSSFP data.

– 68 –

Figure 4.27: Comparison of FDR of reconstructed fMRI map from paired Gaussian-weighted random

sampling scheme using k-t FOCUSS. FDR obtained from fMRI map with only low frequency information,

reconstructed fMRI map using k-t FOCUSS with temporal FT and Gaussian-weighted random sampling

with full sampling of k-space center 1 line, reconstructed fMRI map using k-t FOCUSS with temporal FT

and paired Gaussian-weighted random sampling with full sampling of k-space center 1 line, reconstructed

fMRI map using k-t FOCUSS with KLT and Gaussian-weighted random sampling with full sampling of

k-space center 1 line, and reconstructed fMRI map using k-t FOCUSS with KLT and paired Gaussian-

weighted random sampling with full sampling of k-space center 1 line are shown. FDR of bSSFP with

phase-cycling angle of 180◦ (a), and GRE (e) of a representative rat are shown.

– 69 –

Figure 4.28: Comparison of Type II error of reconstructed fMRI map from paired Gaussian-weighted

random sampling scheme using k-t FOCUSS. Type II error obtained from fMRI map with only low

frequency information, reconstructed fMRI map using k-t FOCUSS with temporal FT and Gaussian-

weighted random sampling with full sampling of k-space center 1 line, reconstructed fMRI map using

k-t FOCUSS with temporal FT and paired Gaussian-weighted random sampling with full sampling of

k-space center 1 line, reconstructed fMRI map using k-t FOCUSS with KLT and Gaussian-weighted

random sampling with full sampling of k-space center 1 line, and reconstructed fMRI map using k-t

FOCUSS with KLT and paired Gaussian-weighted random sampling with full sampling of k-space center

1 line are shown. Type II error of bSSFP with phase-cycling angle of 180◦ (a), and GRE (e) of a

representative rat are shown.

– 70 –

Figure 4.29: Comparison of ROC curves of reconstructed fMRI map from paired Gaussian-weighted

random sampling scheme using k-t FOCUSS. ROC curves obtained from fMRI map with only low

frequency information, reconstructed fMRI map using k-t FOCUSS with temporal FT and Gaussian-

weighted random sampling with full sampling of k-space center 1 line, reconstructed fMRI map using

k-t FOCUSS with temporal FT and paired Gaussian-weighted random sampling with full sampling of

k-space center 1 line, reconstructed fMRI map using k-t FOCUSS with KLT and Gaussian-weighted

random sampling with full sampling of k-space center 1 line, and reconstructed fMRI map using k-

t FOCUSS with KLT and paired Gaussian-weighted random sampling with full sampling of k-space

center 1 line are shown. ROC curves of bSSFP with phase-cycling angle of 180◦ (a), and GRE (e) of a

representative rat are shown. The ROC curve with the highest area under curve (AUC) value is indicated

by the red-dashed box.

– 71 –

Chapter 5. Conclusion

In conclusion, CS with sampling reduction by a factor of 4 works well for both GRE and bSSFP

fMRI at high field. The k-t FOCUSS algorithms with temporal FT and KLT reconstruct both baseline

images and fMRI maps well for fMRI data acquired at high field. The reconstruction of fMRI data

acquired at high field using k-t FOCUSS varies greatly with sampling scheme, and thus the sampling

scheme must be chosen wisely depending on the aim of the fMRI study. Random sampling schemes with

full sampling of k-space center 8 lines resulted in reconstruction of fMRI data with enlarged functional

activations, and seem to be most optimal for performing statistical tests at very low significance levels.

Gaussian-weighted random sampling scheme with full sampling of k-space center 1 line seems to be the

most optimal sampling scheme for fMRI study analysis at significance level of α = 0.05. The results

from the study suggest that combination of CS with both GRE and bSSFP data acquired at high field

could significantly improve the temporal resolution while maintaining image contrast and SNR. Further

studies are required for understanding the sources of the fMRI signal improvements, effects of other CS

algorithms, optimization of parameters, and changes in the sparse sampling scheme to determine the

optimal settings for the application of CS in fMRI data acquired at high fields.

– 72 –

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Summary

Compressed Sensing for fMRI at High Field

일반적인뇌기능자기공명영상기법으로잘알려진 gradient recalled echo (GRE) echo-planar imag-

ing (EPI) 는 고자장에서 국소 자기장 변화에 매우 민감하다. 최근 고자장에서 고해상도 뇌기능 자

기공명영상을 얻기 위해 Pass-band balanced steady state free precession (bSSFP) 이 제안되었지만,

bSSFP 기법의 주기해상도는 기존의 뇌기능 자기공명영상법인 GRE-EPI 보다 낮다는 문제가 존재한

다. Nyquist 샘플링 한계보다 적은 샘플 수로도 선명한 영상을 얻을수 있음이 밝혀진 압축 센싱 이론의

적용이 bSSFP기법의주기해상도를높이기위한한방안이될수있다. 최근몇몇뇌기능자기공명영상

연구에서 GRE-EPI기법에압축센싱이론을적용해본사례들이있지만, bSSFP기법에적용한사례는

없다. 이론상으로는 하나의 자기공명영상을 얻기 위해 단일 Radio frequency (RF) pulse 를 이용하고

자기장 변화에 민감한 GRE-EPI 기법 보다, 여러 다른 Radio frequency (RF) pulse 를 이용하는 bSSFP

기법이 압축 센싱 이론과 잘 접목될수 있는 가능성이 보인다. 본 학위 논문에서는 9.4T 에서 얻은 쥐

체감각신경의고해상도뇌기능자기공명영상에압축센싱이론을적용해보았다. 다양한샘플링패턴

및 통계 분석 방법등을 통해 적은 샘플 수로도 기본 영상의 복원 뿐만 아니라 뇌기능 자기공명영상의

분석 또한 할 수 있어서, 압축 센싱 이론으로 bSSFP 기법의 주기해상도를 높이고 고자장에서 고해상도

뇌기능 자기공명영상을 얻을수 있는 가능성이 보인다.

– 76 –