Magnetic Resonance Im

Improved optimization strategies for autofocusing motion compensation in

MRI via the analysis of image metric maps

Wei Lin, Hee Kwon Song4

Department of Radiology, University of Pennsylvania Medical Center, Philadelphia, PA 19104, USA

Received 11 May 2005; accepted 14 February 2006

Abstract

Autofocusing is a postprocessing technique for motion correction, which optimizes an image quality metric against various trial motions.

In this work, image metric maps, which are measures of image quality plotted as a function of in-plane 2-D trial translations, are

systematically studied to develop improved autofocusing motion correction algorithms. It is shown that determining object motion with

autofocusing is equivalent to an image metric map optimization problem. These maps provide insights into the motion compensation

process and help improve several aspects of the correction algorithm, including the selection of the image metric and motion search

strategy. A highly efficient and robust 2-D global optimization method is devised, exploiting the properties of the metric map pattern. The

improved algorithm is used to correct phantom and clinical MR images with in-plane 2-D translational motion and is shown to be more

effective than existing methods.

D 2006 Elsevier Inc. All rights reserved.

Keywords: Motion correction; Autofocusing; Autocorrection; Image metric

1. Introduction

From the early years of MR imaging, the acquisition of

high-quality MR images in vivo was often hampered by

motion-related artifacts, and various prospective and retro-

spective techniques have been developed for their compen-

sation. Among these are physiologic gating, data reordering

and spatial presaturation [1–3]. Although effective, these

techniques are limited to known periodic physiologic

movements and cannot be used for arbitrary motion. One

of the more effective means of compensation for transla-

tional motion in use today is the method of bnavigatorechoesQ [4], which requires the acquisition of additional data

during the scan to extract the motion information. In some

high-resolution imaging sequences, however, it may not be

desirable or feasible to obtain additional data since the

minimum sequence repetition time (TR) or the total scan

time can become prolonged. The use of navigator echoes

may also undesirably affect the steady state.

Several postprocessing techniques have also been pro-

posed for correction of general translational motion. The

0730-725X/$ – see front matter D 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.mri.2006.02.003

4 Corresponding author. Tel.: +1 215 662 6265; fax: +1 215 349 5925.

E-mail address: hsong@uphs.upenn.edu (H.K. Song).

method of generalized projections attempts to correct for

motion by estimating the phase errors in the acquired data

[5], while the method of Zoroofi et al. [6] uses a

combination of both edge detection and phase error

estimation. Although shown to be successful in simulation

studies and simple phantom experiments, these techniques

have not been tested successfully in in vivo imaging. They

are dependent on the existence of significant ghosting

artifacts or bright object boundaries, which are often not

present in high-resolution images of objects undergoing

subtle, nonperiodic motion, and it is likely that these

methods will fail in typical in vivo situations.

One of the postprocessing techniques devised to correct

motion artifacts is autofocusing (also known as autocorrec-

tion), which was first introduced for application in MR

imaging by Atkinson et al. [7] and has been improved since

[8–11]. In autofocusing, motion is estimated by trial and

error but in a systematic fashion. For each k-space segment,

which consists of one or more contiguous phase-encoding

views, a range of possible object displacements is assumed

and the corresponding phase shifts in k-space are applied.

The data set that yields a minimum (or a maximum) in an

image metric, a measure of image sharpness and quality, is

retained and assumed to be motion compensated. The

aging 24 (2006) 751–760

Fig. 1. k-Space regions used for metric map computation. The segment S, to

which the trial motions are applied, and the background B do not overlap,

and the combined area does not necessarily cover the entire k-space.

W. Lin, H.K. Song / Magnetic Resonance Imaging 24 (2006) 751–760752

method assumes that there is little or no motion during

acquisition of each segment and that those data are acquired

contiguously in time. Autofocusing does not require

additional data and has been shown to successfully

compensate for motion, often performing as well as the

navigator echo technique in cooperative volunteers [7,8].

Both 1-D and 2-D autofocusing have also been demonstrat-

ed in several clinical examples [10,11]. However, high

computational costs and the lack of robustness still impede

wide clinical applicability of the method.

In this work, we utilize the patterns observed in the

metric maps, which are 2-D images whose individual pixel

intensity corresponds to the metric value for a given 2-D

trial translation, for developing improved autofocusing

algorithms. From a systematic analysis of the metric map

patterns, improvements in both speed and accuracy of the

motion correction process are developed. Both the choice of

an image metric and an efficient search strategy could be

determined based on the behavior of these maps. It should

be emphasized here that these maps are used solely to

develop improved autofocusing strategies; they are not

computed during actual motion compensation. The im-

proved algorithm is used to correct for translational motion

in phantom and clinical MR images of the wrist, and the

results compare with previous autofocusing schemes, as

well as with navigator correction techniques.

2. Theory

A bulk rigid-body in-plane object translation of (Dx, Dy)

causes a complex phase factor to be multiplied to the

measured MR signal:

F Dx;Dyð Þ kx; ky� �

¼ F 0;0ð Þ kx; ky� �

e�j2p kxDxþkyDyð Þ: ð1Þ

Here, superscripts indicate the amount of the translation.

Throughout this article, it is assumed that frequency

encoding is along the x-axis and phase encoding is along

y. In a typical spin- or gradient-echo sequence, the duration

of the phase encoding and readout gradients are much

shorter than the TR period between successive views (i.e.,

different ky values). Therefore, intra-view motion can be

considered negligible compared with inter-view motion, and

both Dx and Dy can be taken as only dependent on ky. As a

result, translational motion correction is essentially estimat-

ing a total of NPE 2-D vectors T, where NPE is the number of

phase-encoding views:

T ky� �

¼ Dx ky� �

;Dy ky� �� �

;

kya � NPE=2; N ; 0; N ;NPE=2� 1f g: ð2Þ

The autofocusing technique attempts to successively recover

each component of T by searching for the trial translation

(�Dx, �Dy) that would minimize (or maximize) the metric

for the reconstructed image I.

Let us define S, the k-space data segment in which the

trial motion is applied, and B, the botherQ ky lines includedin the current analysis, which may contain only a portion of

the remaining data (Fig. 1). We define the metric map for

translational motion MS |B(dx,dy) as the computed image

metric versus translation (dx,dy) applied to the k-space

segment S, while the image is reconstructed from the

combined k-space data of S and B. Formally,

MSjB dx; dyð ÞuRðFT�1ðF dx;dyð ÞSjB ÞÞ

¼ RðFT�1ðF dx;dyð ÞS þ F

0;0ð ÞB ÞÞ: ð3Þ

Here, R is the metric operation, the subscript denotes the

k-space coverage of the data F and the superscript is the

translation applied.

If autofocusing were to estimate motion correctly, a

global extremum should appear at the origin of the metric

map computed on a motion-free image. Assuming that the

metric minimum is sought, we have

arg mindx;dyð Þ

Mf

SjB dx; dyð Þ ¼ 0; 0ð Þ: ð4Þ

Here, the superscript f indicates that metric map is computed

over a motion-free image. For a motion-corrupted image

where a translation of (Dx,Dy) occurs when acquiring data

contained in segment S, the metric map is (assuming that

background B is already corrected and therefore motion

free):

McSjB dx; dyð Þ ¼ M

f

SjB dxþ Dx; dyþ Dyð Þ: ð5Þ

Here, the superscript c indicates that metric map is

computed over a motion-corrupted image. From Eqs. (4)

and (5), we have

arg mindx;dyð Þ

McSjB dx; dyð Þ ¼ � Dx; � Dyð Þ: ð6Þ

Fundamentally, autofocusing is a global optimization

problem [12] that searches the variable (dx ,dy) that

W. Lin, H.K. Song / Magnetic Resonance Imaging 24 (2006) 751–760 753

minimizes (or maximizes) the metric map function

MS |Bc(dx,dy) for various S and B configurations. The

characteristic of the metric map is therefore of great interest

since it dictates how to perform the optimization process.

Metric maps computed from a motion-free brain MR data

set are shown in Fig. 2. As expected, a minimum always

occurs at the center of the maps, which corresponds to a

motion-free state. The maps also show an oscillation along

the y-direction (phase encoding), confined in a narrow band

along the dx=0 line, with multiple minima at

arg mindx;dyð Þ

MSjB dx; dyð Þi 0; nDy

� �; Dy ¼ 1=kSy : ð7Þ

Here, kyS is the average spatial frequency of segment

S. Segments close to the k-space center have smaller kyS

values and larger oscillation periods. The periodicity

along the phase-encoding direction on the metric map is

expected: a shift of Dy while acquiring one view with

a spatial frequency of ky causes a phase factor of

Uy=exp[�j2pkyDy(ky)] to be multiplied to the acquired

data. One cannot distinguish between shifts of Dy and

Dy+nDy, where n is an arbitrary integer. Eq. (7) is exact

when S is a single phase-encoding line. When segment S

contains more lines, however, only the minimum at (0,0) is

global while the other minima become local, as shown in

Fig. 2B. This is due to the existence of a distribution of Dy

values within segment S; that is, each line within S has

Fig. 2. The metric maps. (A) The motionless image used for the metric comp

consecutive phase-encoding lines S ={(kx,ky)|kya[k1,k1+n�1]}. First row: n =1 (o

locations: k1=8, 16, 32, 64 and 96 are shown from left to right. The NGS metric (E

in number of pixel shifts, and the intensity of each map is scaled individually fo

slightly different periodicity, resulting in diminished oscil-

lation amplitudes away from the true object position.

3. Methods

3.1. Image metrics

Two types of image metrics were primarily studied based

on their successful outcomes in previous reports: entropy

[7,8] and normalized gradient squared (NGS) [9]. Entropy is

an image metric used for autofocusing in inverse synthetic

aperture radar imaging [13] from which the idea of

autofocusing MR images was derived and was shown to

successfully compensate for motion in MR studies of

volunteers [7,8]. It is defined as

E ¼ �Xi;j

Bi;j=Bmax

� �ln Bi;j=Bmax

� �;

Bmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXi;j

B2i;j

sð8Þ

where Bi,j is the pixel intensity at coordinate (i,j). The term

Bmax can be interpreted as the pixel brightness if all the

energy was contained in a single pixel. Since translational

motion only introduces a phase factor to each k-space

data point, the total image energy and Bmax are invariant.

NGS was among the best-performing metrics in a study

of 24 different metrics (including entropy) [9] and was

utation, SNR=25. (B) Metric maps computed for segment S containing

ne phase-encoding line), second row: n =32. Maps for segments at different

q. (9)) and Background Selection Method 1 (Fig. 3) were used. The axes are

r display purposes.

W. Lin, H.K. Song / Magnetic Resonance Imaging 24 (2006) 751–760754

therefore also studied. Although another metric, the gradient

entropy, slightly outperformed NGS, the latter was used in

this study due to its lower computation cost. It is defined as

NGS ¼Xi;j

jGi;jj2� �

=� X

i;j

jGi;jj�2

: ð9Þ

|Gi,j|, the gradient magnitude at pixel coordinate (i,j), was

computed by taking the square root of the sum of the

squares of x and y gradients. Both directional gradients were

computed using a simple filter of [�1,1]. Other filters such

as the steerable Gaussian first-derivative basis filters [14],

which has a proven optimality for edge detection and allows

trade-off between noise filtering and fine-scale structure

detection, can be used, if desired. However, the computation

cost by using such a kernel would be significantly higher,

and it is therefore not considered in this study. For a sharper

image, entropy is minimized while NGS is maximized. To

simplify the discussion, we always minimized metrics in

this work. Therefore, for NGS, the metric used was:

M ¼ � NGS: ð10Þ3.2. k-Space segment and background selection

Different k-space segment S and background B selection

methods have been used in previous autofocusing algo-

rithms [7–11]. However, to the best of the authors’

knowledge, no methodical comparison of the various

possible techniques has been reported. Manduca et al. [10]

proposed a bmultiresolutionQ scheme that optimizes the

k-space in several iterations, first using large segment sizes

(e.g., 64 lines) followed by successively smaller ones. In

their scheme, as well as in the originally proposed method

[7], the k-space data were always used in their entirety

during motion compensation. In contrast, Atkinson et al. [8]

proposed excluding k-space lines that have not yet been

corrected, proceeding in a center-out fashion in k-space.

This method has the advantage in that all data in

background B is already motion compensated, improving

the likelihood of correctly determining the motion present in

the uncorrected segment S.

Fig. 3. Four different k-space background selection methods. The black

area indicates segment S, and the gray area indicates background k-space

region B.

To evaluate the performance of these differing strategies,

we studied the metric maps of four k-space background

selection methods, as shown in Fig. 3. In Method 1, only the

k-space regions more central to segment S are included. The

ky lines that are conjugate of S are not included even if those

lines have been previously motion compensated. In Method

2, background B contains not only the more central regions

as in Method 1 but also the region that is the conjugate of S.

Atkinson et al.’s [8] center-out method, including only

motion-compensated background region, essentially alter-

nates between Methods 1 and 2, with the exception that their

processing is performed on a reduced matrix that only

includes the corrected data and current segment (B and S

shown in Fig. 3), without zero filling to the original matrix

size. Method 3 is the conventional ball otherQ k-space data

technique. Finally, in Method 4, the conjugate of segment S

is always excluded from the background used in Method 3.

k-space regions not included in S or B are zero filled to the

original size to maintain consistent image resolution.

For each of the background selection schemes, the effects

of different locations and sizes of the segment S were

investigated systematically by analyzing the metric maps.

Several motion-free phantom and clinical MR images were

used to compute the metric map MS |Bf(dx,dy). For each

configuration of k-space segment S and background B, the

metric map was computed for

dx; dyð Þj dxa � Nx;Nx½ ; dya � Ny;Ny

� �: ð11Þ

Here, Nx and Ny set the limits of the trial motion. The

segment S contains data in consecutive ky lines:

S ¼ kx; ky� �

jkya k1; k1F n� 1ð Þ½ �

: ð12Þ

The positive sign was used for k1N0, and the negative sign

was used for k1b0. The segments studied contain n=1, 2, 4,

8, 16, 32 and 64 ky lines. The locations of the segments,

measured by the distance from the k-space center, were

|k1|=8, 16, 32, 64 and 96. To test the metric map behavior at

different SNRs, we added Gaussian noise with zero mean

and various standard deviations to the k-space data prior to

the metric map computation.

3.3. Application of trial motions in k-space and image space

Trial motions can be applied either in k-space [7–10] or

in image space [11] to compute the image metrics. In the

original k-space approach, trial motions involve a phase

factor to be multiplied to the k-space segment S:

Fdx;dyð Þ

S ¼ F0;0ð ÞS e�j2p kxdxþkydyð Þ: ð13Þ

Then, an inverse FFT is taken over the combined k-space

data S+B to obtain the image:

ISjB dx; dyð Þ ¼ FT�1 Fdx;dyð Þ

S þ F0;0ð ÞB

�:

�ð14Þ

Autofocusing can also be performed in image space. Due

to the linearity of the Fourier transform, the reconstructed

W. Lin, H.K. Song / Magnetic Resonance Imaging 24 (2006) 751–760 755

MR image I is a summation of many bcomponent imagesQIi, each of which contains contribution from a spatial

frequency component Fi:

I ¼ FT�1ðXi

FiÞ ¼Xi

Ii; Ii ¼ FT�1 Fið Þ: ð15Þ

From this viewpoint, a translation occurring while acquiring

one phase-encoding line simply causes the same amount of

the translation to be applied to the corresponding component

image Ii. In the image-space autofocusing approach, two

complex images are first obtained by separate inverse FFTs,

the bsegment imageQ IS for S and the bbackground imageQ IBfor B. Then, translational shifts are applied to IS before it is

added to IB:

ISjB dx; dyð Þ ¼ IS dx; dyð Þ þ IB 0; 0ð Þ: ð16Þ

The k-space data have to be zero filled to the full prescribed

resolution before transforming to image space to maintain

consistent image sizes and to permit single pixel shifts.

In this study, translations were applied in the image space

for its speed advantage. In the image-space approach, only

two FFTs need to be performed for each k-space segment,

while Ntrial FFTs are required for the k-space approach

where Ntrial is the number of trial motions. Since, typically,

NtrialH2 and image shifting and summation involve less

computation than FFT, image-space approach has consider-

able speed advantage, between 30% and 50% [11].

Furthermore, if the image metric is computed only within

a small user-defined region of interest, pixels only in that

region need to be shifted and summed, further reducing the

computational cost. Fractional pixel shifts can be performed,

by first zero filling the segment S to a larger matrix before

the FFT. For example, if the segment S contains 8 phase-

encoding lines in a 256�256 original data set, the 8�256

data can be zero filled to a matrix size of 512�512 before

the FFT, which subsequently allows a translational precision

of 1/2 pixel. This method of subpixel shifting is mathemat-

ically equivalent to applying a 1/2 pixel shift in k-space

using the appropriate phase.

3.4. Optimization strategy

The task of motion detection for a given configuration of

segment S and background B by a measure of image metric is

a 2-D global optimization problem. In the presence of 2-D

translation, the position of the global minimum (motion-free

state) is shifted accordingly. Various searching strategies to

locate this point have been used in the past, including

exhaustive 2-D [7], successive 1-D [8], golden section [10]

and simplex [11]. An exhaustive 2-D search is equivalent to

computing the entire metric map. Although robust, its

computational cost becomes prohibitively high when number

of trial motions becomes large. Successive 1-D, golden

section and simplex are local optimization techniques [12],

whose success relies on the assumption that the initial choice

of the translation vector lie in the local convergence region of

the global minimum. Since there could be multiple local

minima on the metric map, the algorithm may not always

converge to the true global location.

The patterns that are observed in the metric maps allow

us to design a more efficient optimization strategy. The new

strategy proposed here is composed of two distinct steps. In

the first step, the displacement of the metric minimum

along the x-direction is first located. Since the periodic

pattern that appear along dx=0 occurs throughout the entire

length of the metric map, the search could be facilitated by

first locating this periodic band. In the NGS metric map,

the vertical band is approximately 1 pixel wide. Therefore,

a small step size of 1 pixel should be used along x. Along

the y-axis, a search range of one oscillation period (Dy in

Eq. (7)) and a step size of a quarter of this period are

sufficient in locating this band. Therefore, the number of

metric evaluations performed in this step is 4Nx, where Nx is

the search range along x. To reiterate, the goal of this first

step is not to locate the absolute minimum but to locate the

x-axis shift.

In the second step, various local optimization techniques

previously proposed can be applied. In this work, a

successive 1-D searching strategy is used. Starting from

the minimum detected in Step 1, a y-axis search is first

performed, alternating with an x-axis search until the

location of the minimum no longer changes in two

consecutive searches. However, because of the existing

periodicity of the local minima along the y-axis, the

alternating 1-D local search is performed not only from

the minimum detected in Step 1 but also in the regions of

periodically occurring local minima, whose positions are

solely determined by the location of segment S, ensuring

that the global minimum could be found.

Our proposed strategy is different from previous techni-

ques in that a coarse 2-D search is used first to detect x-axis

shift, followed by successive 1-D iterations. Compared to

just a successive 1-D approach, the technique is more robust

in locating the desired minimum. It is worth repeating here

that the entire metric map does not need to be computed

during actual motion compensation; the patterns observed in

these metric maps were only used to develop the best means

to locate the global minimum.

3.5. Image acquisition and analysis

Motion-corrupted phantom and high-resolution in vivo

wrist data were acquired. Translational motion was manu-

ally applied during the phantom scan, while in one of the in

vivo scans, the subject was informed to move voluntarily

several times during the imaging session. Two other in vivo

data sets were from a high-resolution wrist imaging study in

which the subjects are advised to lie still [15]. In the

phantom experiment, a 2-D spin-echo sequence with the

following parameters was used: 24 cm field of view (FOV),

5 mm thickness, 256�256 matrix, 500 ms TR. A piecewise

constant translational motion was applied manually eight

times during the scan. For the in vivo experiments, a 3-D

Fig. 4. The NGS metric maps at low SNR. Background Selection Method 1 was used. Noise was added to the data for the image shown in Fig. 2A and the

resulting SNR=4. NGS maps were computed for segments S ={(kx,ky)|kya[k1,k1+n�1]}. First row: n = 1 (one phase-encoding line), second row: n = 32.

W. Lin, H.K. Song / Magnetic Resonance Imaging 24 (2006) 751–760756

FLASE sequence [15] with the following parameters was

used: TR/TE, 80/7.8 ms; 7.0�3.5 cm FOV; 0.5 mm section

thickness; 512�256�32 matrix (137�137�500 Am3 reso-

lution); 1208 excitation flip angle; F16 kHz bandwidth;

10.9 min scan time. An alternating navigator echo was

incorporated at the end of each readout to recover both

readout and phase-encoding direction translations [16],

allowing the comparison of our autofocusing techniques

with motion compensation using the navigators. For auto-

Fig. 5. The effects of background and metric selections on the metric map pattern

ky =k1 line. Two step shifts were applied on each half of the k-space for data

(Dx,Dy)= (�3,0) pixels for kya[�128,�k1] and kya[k1,127], respectively. Meth

focusing, an inverse FFT along the z-direction was first

performed to separate the slices, and data from 8 central

(out of 32) slices were used for 2-D in-plane motion

compensation by summing up their NGS values. Since

TR=80 ms and since 32 slices were acquired, the temporal

resolution of motion detection was effectively 2.5 s. In

addition to these acquisitions, several additional motion-free

scans in vivo were collected to help evaluate the properties

of the metric maps.

. The image used is the one shown in Fig. 2A. The segment S contains the

that have not been corrected. The resulting motion is (Dx,Dy)= (4,4) and

ods 1–4 are those shown in Fig. 3. (A) Entropy maps. (B) NGS maps.

Fig. 6. Comparison of image quality of different autofocusing schemes in a

phantom experiment with manually applied translations. (A) Motion-free

control image. (B) Motion-corrupted image. (C) Entropy, BG1 (Back-

ground Selection Method 1), proposed two-step search. (D) NGS, BG1,

1-D successive search. (E) NGS, BG3, two-step search. (F) NGS, BG1,

two-step search. (G) Detected displacements for Panel F. In Panel G, ad-

justments (additional shifts) were made to the detected Dy at some kylocations, such that while the phases applied to those lines were unchanged,

the detected displacements were more consistent with the expected

piecewise motion. This causes Dy to appear sloped at those locations.

W. Lin, H.K. Song / Magnetic Resonance Imaging 24 (2006) 751–760 757

On a 3.0-GHz Pentium computer (with 1 GB RAM), the

total processing time for the phantom experiment was about

100 s using the following parameters: 4 phase encoding lines

per segment near k-space center, 8 lines per segment near

k-space edge, (dx,dy) search range of (F10,F10) pixels and

precision of (1/4,1/4) pixel. Larger segment sizes were used

in the outer k-space regions due to their lower signal content

to improve the robustness. For the in vivo experiments, the

total processing times were about 5 min using these

parameters: 4 or 8 lines per segment near k-space center,

16 lines per segment near k-space edge, (dx,dy) search rangeof (F4,F4) pixel and precision of (1/4,1/4) pixel. NGS was

used as the image metric.

The effectiveness of the autofocusing algorithm was

evaluated in three ways. First, the motion-corrected images

were inspected visually and compared with the motion-

corrupted images and either a motionless image (phantom)

or images corrected using navigator echoes (in vivo).

Second, a measure of image improvement (relative to the

navigator technique) for the in vivo study was computed

according to the following equation:

Maf �Mcorrupted

Mnavigator �Mcorrupted

: ð17Þ

Here, Mnavigator, Mcorrupted and Maf are the metric values for

navigator-corrected, corrupted and autofocusing-corrected

images, respectively, evaluated over the eight central slices

where autofocusing was carried out. Finally, the derived

motion trajectory was compared with the navigator-

detected motion.

4. Results

4.1. The metric map pattern

Metric maps were computed for several motion-free

phantom and in vivo MR images. Despite variations in the

image features and metrics used, all metric maps share

similar patterns as in Fig. 2. It was found that metric maps of

segments that are near the k-space center or which consists

of more views, both of which have more image energy, have

a higher absolute range of signal intensity (peak minus

valley) and are, therefore, less susceptible to noise. The

latter phenomenon can be seen in Fig. 2, in which the

regions outside the dx=0 line are more homogeneous (less

noisy) for larger n and smaller k1.

One unexpected observation in the metric maps is the

presence of high-intensity maxima between the local

minima. This phenomenon could be explained by consid-

ering the interaction between the segment image IS and the

background image IB. Because IS contains finite spatial

frequencies, ringing will appear at the object’s edges along

the y-direction. The frequency of this modulation corre-

sponds to that of the missing spatial frequencies of the

background image IB. Local minima appear when the two

complex images are aligned such that the object’s edges are

sharpened, while, simultaneously, the ringing is reduced by

destructive interference. In between the minima, however,

the phases become aligned such that edges become blurred,

while ringing from two component images adds construc-

tively, resulting in local maxima.

At high SNR, a minimum occurs at the metric map’s

center, which corresponds to a motion-free state (Fig. 2,

SNR=25).When noise is added to reduce the image SNR and

when the segment used contains only a small number of

phase-encoding lines, one can no longer rely on ametricmini-

mum for proper motion compensation (Fig. 4A). Including

more phase-encoding lines to the segment S reduces the

undesired intensity fluctuations in the image metric map and

yields a more predictable pattern from which the proper

minimum can be extracted (Fig. 4B). This example visually

demonstrates the necessity of using larger segment sizes in

low SNR conditions, particularly in the outer k-space regions.

4.2. Background selection and image metric

Metric map patterns, as shown in Fig. 5, reveal that

the background selection method can have a significant

impact on the effectiveness of autofocusing algorithms. If

Table 1

Comparison of metric values for different autofocusing schemes in the phantom experiment

Metric used for autofocusing Optimization strategy Background selection method (Fig. 3) Image Quality measure

Entropy NGS

Motion free Fig. 6A 204.5 �19.72

Motion corrupted Fig. 6B 335.9 �7.37

Autofocused Entropy Proposed two-step search 1 Fig. 6C 327.4 �7.52

NGS 1-D successive search 1 Fig. 6D 332.6 �7.93

NGS Proposed two-step search 3 Fig. 6E 335.4 �7.86

NGS Proposed two-step search 1 Fig. 6F 323.0 �10.13

Lower values (more negative) indicate better motion compensation.

W. Lin, H.K. Song / Magnetic Resonance Imaging 24 (2006) 751–760758

autofocusing was to proceed in a center-out fashion in

k-space, among the four different background selection

methods, only Method 1 uses exclusively previously

motion-corrected data in background B. To simulate the

effects when B contains uncorrected motion, we applied

two step shifts on each half of the k-space for data that

have not been corrected. Since segment S contains motion

(Dx,Dy)=(�3,0), a distinct oscillating band should ideally

appear at dx=�Dx=3. However, for both entropy and

NGS, Methods 2–4 all generate additional oscillating

bands that may interfere with the correct motion estima-

tion. The problem is likely to worsen with reduced image

SNR. Overall, NGS maps with Method 1 produce the most

distinct oscillatory pattern along dx=3, as well as the

smallest intensity variations outside this band. Based on

these findings, the background selection scheme of Method

1 in conjunction with NGS image metric was chosen for

our experiments. This proposed scheme, proceeding in a

center-out fashion and alternating between positive and

negative ky, is similar to that proposed by Atkinson et al.

[8] for entropy, with the exception that the conjugate

region of S is always excluded and k-space is always zero

filled to the full matrix size. These results underscore the

potential usefulness of the metric maps in helping to

develop an improved combination of image metric and

motion search procedure.

Fig. 7. Motion-corrupted (A), navigator-corrected (B) and autofocusing-corrected

detail. (D–F) A portion of the image with �2 magnification. The arrows in Panels

D due to motion corruption.

4.3. Phantom and in vivo experiments

The results of the phantom experiment comparing our

proposed technique with others are shown in Fig. 6. Only

the image corrected with our proposed global optimiza-

tion procedure is of acceptable quality. The motion trajec-

tories determined from the proposed method are plotted in

Fig. 6G, showing several step motions along both x- and y-

directions, consistent with the actual motion applied during

the experiment. Table 1 shows the metric values of the

corrected images. Both entropy and NGS were computed

from the final corrected image, although only NGS was

used for the correction itself (except Fig. 6C).

The results from one of the three in vivo scans are shown

in Fig. 7. The image quality of the autofocused and

navigator-corrected images is similar and has significantly

improved compared with that of the motion-corrupted

images. The finer details of the trabecular structure are

more clearly visualized after motion correction. Such high

image quality is demanded in applications measuring bone

parameters, such as trabecular bone volume, thickness and

topology [15]. Table 2 shows the computed metric values of

the corrected images. Using the navigator correction as a

gold standard and based on the computed entropy and NGS

values, the relative image quality improvement after

autofocusing correction ranged from 0.72 to 0.90 in the

three subjects. There was no visually discernible difference

(C) images of the distal radius of Subject 1. Images were cropped to show

E and F show a trabecular element recovered, which was not visible in Panel

Table 2

Comparison of motion correction by autofocusing and navigator echoes for three high-resolution in vivo trabecular bone MR images

Subject Motion Image quality measure Motion corrupted Navigator corrected Autofocused Relative image quality improvement

1 Involuntary Entropy 1430.3 1418.2 1419.8 0.872

NGS �1.637 �1.671 �1.655 0.823

2 Involuntary Entropy 1613.9 1600.4 1602.9 0.818

NGS �1.521 �1.545 �1.539 0.756

3 Voluntary Entropy 1592.4 1584.0 1584.8 0.903

NGS �1.578 �1.596 �1.591 0.720

The image quality improvement relative to the navigator technique is defined by Eq. (17). Autofocusing was performed using the average NGS of eight slices

of the 3-D data set.

W. Lin, H.K. Song / Magnetic Resonance Imaging 24 (2006) 751–760 759

between the images corrected with either technique. These

results show that autofocusing is a very sensitive technique

able to compensate for motion in in vivo micro-MR

applications where SNR can be limited (8–10 in our

examples), yielding image quality improvements similar

to that of navigator echoes but without requiring addi-

tional data.

Motion trajectory recovered from the in vivo data is

compared with that determined from the navigator data in

Fig. 8. The trajectories from the two methods agree well,

although the errors tend to increase near the edges of

k-space due to lower SNR. It is also possible that the navi-

gator correction itself contains errors, as the acquired

navigator projections are gradient echoes with a long echo

time and are therefore noisy [16]. Potential through-plane

and rotational motion are additional sources of error not

addressed by the techniques implemented.

5. Discussion and conclusions

Although this work concentrates on entropy and NGS,

other metrics (such as correlation and gradient entropy [9])

were found to produce similar metric map patterns and,

therefore, could also be studied using the methodology

established here.

Fig. 8. Motion trajectories recovered from autofocusing applied on eight

slices of the data set and navigator echoes in Subject 1.

The computation time required by the proposed global

optimization strategy is highly predictable. In the first step

(the coarse 2-D search), the number of metric evaluations is

predetermined by the motion search range and the segment

position. The subsequent 1-D successive search typically

requires only two to three iterations since the global

minimum is already in proximity after the first step. In

contrast, a local searching approach might require a long

convergence time, depending on the position of the initial

trial motion vector on the entire metric map. In addition,

local searches may become trapped in a local minimum, as

evidenced by the presence of many local minima in the

metric maps, as shown in Figs. 2, 4 and 5. As previously

mentioned, high computational costs and the lack of

robustness are the two main drawbacks of current algo-

rithms. The improvements in both speed and robustness

presented in this work may help achieve the ultimate goal of

clinical applicability.

As discussed previously, it is necessary to use larger

segment sizes when SNR is low since otherwise noisy

metric maps would prevent reliable detection of the global

ig. 9. The effects of intrasegment motion on the metric map pattern. The

age used is the one shown in Fig. 2A. NGS maps (Background Selection

ethod 1) were computed for k -space segments S = {(k x ,k y )|

ya[k1,k1+7]}, with k1=8, 32 and 96. (A) A linear shift, with no motion

t ky =k1 and a maximal shift of (Dx,Dy)= (4,4) pixels at k =k1+7. (B) A

F

im

M

k

a

step shift of (Dx,Dy)= (4,4) pixels for kya[k1+4,k1+7].

W. Lin, H.K. Song / Magnetic Resonance Imaging 24 (2006) 751–760760

minimum (see Fig. 4). However, with larger segment size,

bintrasegmentQ motion may occur. Fig. 9 shows the effects

of such motion on the metric map pattern. In the first

example, a linear motion is present within segment S. The

maxima/minima oscillations (Fig. 9A) become less well

behaved, and the global minimum may potentially appear at

incorrect locations. If instead a sudden shift occurs in

segment S, a second set of local minima may appear

(Fig. 9B), particularly for large k1. These results show that it

is important to include a minimal number of ky lines in each

segment, as allowed by the SNR, in order to minimize the

influence of intrasegment motion.

Correction for combined in-plane rotation/translation has

been previously demonstrated [8], which involves applying

various trial rotations, in addition to translations. Generally

speaking, if motion other than in-plane translation occurs

during data acquisition and if only 2-D translation

correction was applied, then the resulting metric map

may not yield the expected patterns, leading to incorrect

results with the proposed strategies. However, if a

rotational motion is first compensated by applying a trial

rotation to the k-space segment, then it is anticipated that

the metric map would yield the expected pattern, permitting

the detection of the metric minimum that corresponds to the

motion-free state.

Like most existing techniques, the proposed autofocus-

ing motion compensation scheme does not address through-

plane motion or nonrigid deformations. Since through-plane

motion affects the excitation slice, it is not possible to fully

compensate retrospectively in a 2-D data set. Correction for

nonrigid deformations, on the other hand, may be possible,

but a more complex process beyond simple k-space phase

correction would be required [17]. For certain non-

Cartesian sampled data, it may be possible to mitigate the

effects of motion by considering data consistency [18].

Prospective gating techniques may also be used to mitigate

through-plane and nonrigid body motion [19], although

real-time signal processing and longer scan times are

often required.

This work introduces the concept of an image metric map

for the autofocusing optimization process. The metric map

pattern aids in the formulation of several significant

improvements to the existing autofocusing algorithm. The

maps were useful not only for determining the most relevant

image metric for motion correction and strategies to reduce

the overall processing time but also for developing a more

robust algorithm that is less susceptible to local minima.

Phantom experiments demonstrated that the proposed

algorithm is superior to previous methods. In clinical

high-resolution in vivo data corrupted with 2-D translational

motion, the improved autofocusing technique achieved

motion correction comparable with that of navigator echoes.

Acknowledgments

This work is supported by the NSF Grant BES-0302251.

The authors also wish to acknowledge Dr. Felix W. Wehrli

for useful discussions and for providing the in vivo images.

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