decomposition of flow signals into basis functions: performance advantages, disadvantages, and...

Decomposition of flow signals into basis functions: Decomposition of flow signals into basis functions: Performance advantages, disadvantages, and Performance advantages, disadvantages, and

computational complexitycomputational complexity

Hans Torp and Lasse LøvstakkenHans Torp and Lasse LøvstakkenNorwegian University of Science and TechnologyNorwegian University of Science and Technology

Trondheim, NorwayTrondheim, Norway

AcknowledgementAcknowledgement

• Steinar Bjærum, GE Vingmed Ultrasound, HortenSteinar Bjærum, GE Vingmed Ultrasound, Horten– Substantial part of the results in this presentation is taken from his Substantial part of the results in this presentation is taken from his

phd workphd work

• Kjell Kristoffersen, GE Vingmed Ultrasound, HortenKjell Kristoffersen, GE Vingmed Ultrasound, Horten– Collaboration for 25 years in Doppler ultrasoundCollaboration for 25 years in Doppler ultrasound

OutlineOutline

• Methods for clutter filtering in color flow imagingMethods for clutter filtering in color flow imaging– IIR , FIR, regression filterIIR , FIR, regression filter

• General linear filtering – basis functionsGeneral linear filtering – basis functions

• Optimum choice of basis functions Optimum choice of basis functions – Best frequency responseBest frequency response

– Optimum detectionOptimum detection

• Disadvantages:Disadvantages:– Bias in velocity estimationBias in velocity estimation

• Computational complexityComputational complexity– Comparison FIR filter and regression filterComparison FIR filter and regression filter

Clutter filter in color flow imaging?Clutter filter in color flow imaging?

Beam k-1 Beam k Beam k+1 Signal discontinuity causes ringing in high-pass filters

Raw signal from beam k

Signal after high-pass filter

Lost settling time

Doppler Signal ModelDoppler Signal Model

• Signal vector for each sample volume:Signal vector for each sample volume:

x = [x(1),…,x(N)]T

Signal = Clutter + White noise + Blood

x = c + n + b• Typical clutter/signal level: 30 – 80 dBTypical clutter/signal level: 30 – 80 dB

• Clutter filter stopband suppression is critical!Clutter filter stopband suppression is critical!

• Zero mean complex random processZero mean complex random process

• Three independent signal components:Three independent signal components:

Hans TorpNTNU, Norway

-0.6-0.4-0.2 0 0.2 0.4 0.6

0

20

40

60

80

100

Blood velocity [m/s]

Do

pp

ler

spe

ctru

m [

dB

]

Clutter

Blood

IIR filter with initializationIIR filter with initialization

Signal discontinuity causes ringing in high-pass filters

Raw signal from beam k

Signal after high-pass filter

Lost settling time

Chebyshev order 4, N=10

Pow

er [

dB]

Frequency

0 0.1 0.2 0.3 0.4 0.5-80

-60

-40

-20

0

Frequency response

Steady state

Step init.

Discard first samples

Projection init*

*Chornoboy: Initialization for improving IIR filter response, IEEE Trans. Signal processing, 1992

FIR FiltersFIR Filters

• Discard the first Discard the first MM output samples, where output samples, where MM is equal to the filter is equal to the filter orderorder

• Improved amplitude Improved amplitude response when nonlinear response when nonlinear phase is allowedphase is allowed

0 0.1 0.2 0.3 0.4 0.5-80

-60

-40

-20

0

Pow

er [

dB]

Frequency

Frequency response, order M= 5, packet size N=10

M

k

kkzbzH

0

)(

Linear phaseMinimum phase

Regression FiltersRegression Filters

Subtraction of the signal Subtraction of the signal component contained in a component contained in a KK-dimensional clutter -dimensional clutter space:space:

Signal space

Clutter spaceb1

b2

b3

xy

cy = x - c

Linear regression first proposed by Hooks & al. Ultrasonic imaging 1991

x = [x(1),…,x(N)]

Why should clutter filters be linear?Why should clutter filters be linear?• No intermodulation between clutter and blood signalNo intermodulation between clutter and blood signal

• Preservation of signal power from bloodPreservation of signal power from blood

• Optimum detection (Neuman-Pearson test) includes a linear Optimum detection (Neuman-Pearson test) includes a linear filter filter

y = Ax

• Any linear filter can be performed by a matrix multiplication Any linear filter can be performed by a matrix multiplication of the of the N N - dimensional signal vector - dimensional signal vector xx

• This form includes all IIR filters with linear initialization, FIR This form includes all IIR filters with linear initialization, FIR filters, and regression filtersfilters, and regression filters

*Matrix A

Input vector x Output vector y

Frequency response Linear FiltersFrequency response Linear Filters

y = Ax

• Definition of frequency response function Definition of frequency response function

= power output for single frequency input signal= power output for single frequency input signal

21)( Ae

NHo

TNii eee

)1(1

Note 1. The output of the filter is not in general a single frequency signalNote 1. The output of the filter is not in general a single frequency signal

(This is only the case for FIR-filters)(This is only the case for FIR-filters)

Note 2. Frequency response only well defined for complex signalsNote 2. Frequency response only well defined for complex signals

FIR filter matrix structureFIR filter matrix structure

654321

654321

654321

654321

654321

0..0

0.

..

.0

0..0

bbbbbb

bbbbbb

bbbbbb

bbbbbb

bbbbbb

AFIR filter order M=5Packet size N=10Output samples: N-M= 5

Increasing filter order + Improved clutter rejection

- Increased estimator variance


Regression FiltersRegression Filters

• Subtraction of the signal Subtraction of the signal component contained in a component contained in a KK-dimensional clutter -dimensional clutter space:space:

xbbIy

K

i

Hii

1

Signal space

Clutter spaceb1

b2

b3

xy

c

Choise of basis function is crucial for filter performance

y = x - c

A

Fourier basis functionsFourier basis functions

nb (k)=1/sqrt(N)ei*k*n/N

n= 0,.., N-1are orthonormal, and equally distributed in frequency

Fourier Regression FiltersFourier Regression Filters

DFT

Set low frequency coefficients to zero

Inverse DFT0 0.1 0.2 0.3 0.4 0.5

-80

-60

-40

-20

0

Frequency

Pow

er [

dB]

Frequency response

N=10, clutter dim.=3

Legendre polynom basis functionsLegendre polynom basis functions

b0 =

b1 =

b2 =

b3 =

Gram-Schmidt process to obtainOrthonormal basis functions -> Legendre polynomials

Polynomial Regression FiltersPolynomial Regression Filters

b0 =

b1 =

b2 =

b3 =

Pow

er [

dB]

Frequency

0 0.1 0.2 0.3 0.4 0.5-80

-60

-40

-20

0

Frequency responses, N=10

basepolynomialLegendrebbb

bbIA

N

P

k

TkkN

110

0

,..,,

Fourier-basis with extended periodFourier-basis with extended period1 N

Frequency Response comparisonFrequency Response comparison

0 0.1 0.2 0.3 0.4 0.5-80

-60

-40

-20

0

Frequency

Pow

er [

dB]

Polynom regression

IIR projection init.

FIR minimum phase

Polynomial regression and IIR filter with projection initializarionhave almost identical performance, and are superior to FIR filters

Optimal Basis functionsOptimal Basis functions

• Eigenvalue decomposition of the clutter correlation matrix:Eigenvalue decomposition of the clutter correlation matrix:

• Use the eigenvectors Use the eigenvectors bbi i as a basis for the clutter space (Karhunen-Loeve as a basis for the clutter space (Karhunen-Loeve transform)transform)

• This basis provides maximum energy concentration of the clutter signalThis basis provides maximum energy concentration of the clutter signal

N

i

‘iiic

1

bbR

1 NK

Ene

rgy

Adaptive basis functionsAdaptive basis functions• The correlation matrix may be estimated by spatial The correlation matrix may be estimated by spatial

averaging in a region with uniform motion:averaging in a region with uniform motion:

• Adapt to clutter velocity - skewed filter center freq.Adapt to clutter velocity - skewed filter center freq.

• May account for irregular wall motion, non-May account for irregular wall motion, non-stationary clutter signalstationary clutter signal

M

i

Hiic M 1

1ˆ xxR

Adaptive Regression FilterAdaptive Regression Filter

Eigenvalue spectrum of clutter + blood

Clutter filter

1 NK

Ene

rgy

Eigenvectors

Legendre polynomial basis functions

Eigenvector basis functions

Adaptive Regression FilterAdaptive Regression FilterProjection along each single basis functionProjection along each single basis function

Coronary artery

Detection of BloodDetection of Blood

A rule for deciding between the two hypotheses:

H0: No blood is presentH1: Blood is present

The detector is characterized by:

• Probability of false alarm PF = P(choose H1 | H0 is true)

• Probability of detection PD = P(choose H1 | H1 is true)

The Optimal DetectorThe Optimal DetectorThe Neyman-Pearson lemma:

PD is maximized under the constraintPF by a likelihood ratio test (LRT)

><H0

H1

)(

)()(

0Hp1Hp

L0H

1H

x

xx

x

x

For Gaussian signals, the LRT can be simplified to:

Matrix A is given by the signal covariance matrix

2)( Axx l ><

H0

H1

The Optimal DetectorThe Optimal Detectorfor Gaussian signalsfor Gaussian signals

A2

x

Clutter filter Power calc.

H1

H0

Matrix A is a linear filter which suppress the clutter signal.

ROC for different clutter filtersROC for different clutter filters

Blood velocity = 10 cm/s

IIR proj. init.

IIR step init.

FIR min. phase

FIR linear phase

0 10

1

PD

PF

Optimal detectorEigenvector reg. filterPol. reg. filter

Example from coronary artery flow in rapid moving myocard

Note that the eigenvector regression filter is close to optimal

Is the Gaussian Is the Gaussian assumption valid?assumption valid?

Non-Gaussian histogram due tovariation in signal powerHistogram from smaller regionshows Gaussian distribution

Basis functions for non-adaptive Basis functions for non-adaptive filtersfilters

Clutter signal with Gaussian Clutter signal with Gaussian

shaped power spectrum:shaped power spectrum:

Eigenvectors of covariance Eigenvectors of covariance matrix matrix

~ Legendre polynomials~ Legendre polynomials

Covariance matrix estimated fromCovariance matrix estimated from

Color Doppler signals with Color Doppler signals with moderate wall motion:moderate wall motion:

Eigenvectors of covariance matrix Eigenvectors of covariance matrix

~ Legendre polynomials~ Legendre polynomials

SummarySummaryOptimal choice of basis function Optimal choice of basis function

for blood vessel detectionfor blood vessel detection

• Doppler signals show Gaussian distribution locally in color Doppler signals show Gaussian distribution locally in color flow imagesflow images

• Eigenvector basis functions give optimum separation of Eigenvector basis functions give optimum separation of clutter and bloodclutter and blood

• Legendre polynomials are the best choice for non-adaptive Legendre polynomials are the best choice for non-adaptive clutter filtering, and give substantially better performance clutter filtering, and give substantially better performance than FIR filters.than FIR filters.

• Adaptive eigenvector filters show significant improvements Adaptive eigenvector filters show significant improvements over polynomial regression filters for spatially uniform wall over polynomial regression filters for spatially uniform wall motionmotion

Autocorrelation methodAutocorrelation methodfor blood velocity estimationfor blood velocity estimation

Clutter Rejection

Filter

Auto CorrelationEstimator

Phaseangle&scaling

x

y =A x n

nynyR )1()()1(ˆ * )1(ˆˆ02 Ranglev f

PRFc


y )1(R̂ v̂

Regression filter biasRegression filter bias

Magnitude and phase frequency response Polynomial regression filter packet size = 10, polynom order 3.

Severe bias in band width and mean frequency estimator below cutoff frequency.

Single frequency input may give high frequency distortion

Computer simulation of Doppler Computer simulation of Doppler signal including cluttersignal including clutter

-2 -1 0 1 2

0

20

40

60

80

100

Doppler shift frequency [kHz]

Dopp

ler sp

ectru

m [d

B]

Frequency 2.5 MHz

Beam width 3 mm

Pulse length 2 mm

PRF 5 kHz

packet size 10 samples

Signal level 20 dB

Clutter level 80 dB

Thermal noise level 0 dB

Blood velocity 0.2 -0.8 m/s

Angle blood flow 20 deg.


Blood velocity estimator performance Blood velocity estimator performance


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.05

0

0.05

0.1

0.15


Bia

s [m

/s]

Polyreg. filter

FIR filter

0 0.2 0.4 0.60

0.05

0.1

0.15


Sta

ndar

d de

viat

ion

[m/s

]

Polyreg filter

FIR filter

FIR filter order 7

Optimal methods for velocity estimation Optimal methods for velocity estimation Maximum Likelihood estimatorMaximum Likelihood estimator

vl(v|x) = log p(x|v) vML


xvCx

vCevxp N

)('

)(1

1

)(

Probability density function given by the covariance matrix C(v)

l(v|x)

Likelihood function

Log likelihood functionLog likelihood functionandand

Cramer - Rao lower bound Cramer - Rao lower bound

1

2

2

min

1

)(var

)()()(log)(

xvlv

vCxvCxvxpxvl T


Blood velocity estimator Blood velocity estimator bias variance bias variance

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.05

0

0.05

0.1

0.15


Bias

[m/s]

Polyreg. filterMax. LikelihoodFIR filter

0 0.2 0.4 0.60

0.05

0.1

0.15


Stan

dard

dev

iatio

n [m

/s]

Polyreg filterML Cramer-Rao FIR filter

ML-estimator is not minimum variance, but better than the two other approaches

SummarySummaryBlood velocity estimation in the Blood velocity estimation in the

presence of clutter signalspresence of clutter signals

• Polynomial Regression (PR) filters give substatial Polynomial Regression (PR) filters give substatial positive bias for low velocity blood flowpositive bias for low velocity blood flow

• FIR filters give less bias, but much higher variance FIR filters give less bias, but much higher variance than PR-filtersthan PR-filters

• ML-estimator has lowest bias and variance, but the ML-estimator has lowest bias and variance, but the algorithm is not suitable for practical use.algorithm is not suitable for practical use.

• PR-filter approach the performance of ML-estimator PR-filter approach the performance of ML-estimator when the Dopplershift is above the filter cutoff when the Dopplershift is above the filter cutoff frequencyfrequency

y = A* x

Computational complexityComputational complexity# multipications+ additions per packet

Full matrix multipication:

N*N

Projection1 basis function

2*N

FIR-filterOrder M

(M+1)*(N-M)(M+1)*(N-M)

*

* -Basis vector

*Filter matrix

Input signal vector Output signal vector

Packet sizeN=8

M+1 N-M

M=5

• Data rate in color flow imaging: Data rate in color flow imaging: – 1 – 5 M samples/sec (complex samples)1 – 5 M samples/sec (complex samples)

• Processing speed test: Processing speed test: Pentium M 1.6 GHz, using Matlab R13, N=8, M=6 Pentium M 1.6 GHz, using Matlab R13, N=8, M=6

– Matrix multipication: Matrix multipication: 13 Msamples/sec13 Msamples/sec

– Projection filter, 3 basis functions:Projection filter, 3 basis functions: 17 Msamples/sec17 Msamples/sec

– FIR filterFIR filter 45 Msamples/sec45 Msamples/sec

• Adaptive filters is much more computer demandingAdaptive filters is much more computer demanding– Double CPU-time with complex filter coefficientsDouble CPU-time with complex filter coefficients

– CPU-time for filter coefficient calculationCPU-time for filter coefficient calculation

– Example: Adaptive Eigenvector filterExample: Adaptive Eigenvector filter 2.1 Msamples/sec2.1 Msamples/sec

Real-time clutter filteringReal-time clutter filtering

SummarySummaryComputational complexity ofComputational complexity of

clutter filter algorithmsclutter filter algorithms

• Regression filters have 1 – 2 times longer computation Regression filters have 1 – 2 times longer computation time than FIR-filterstime than FIR-filters

• A standard laptop computer is able to do real-time A standard laptop computer is able to do real-time regression filtering using less than 10% of available regression filtering using less than 10% of available cpu-timecpu-time

• Adaptive eigenvector filter requires ~ 10 times more Adaptive eigenvector filter requires ~ 10 times more computation power than the regression filtercomputation power than the regression filter

ConclusionsConclusions

• Regression filters are superior to FIR and IIR filters Regression filters are superior to FIR and IIR filters in blood flow detection and velocity estimationin blood flow detection and velocity estimation

• For non-adaptive clutter filtering, the optimum choice For non-adaptive clutter filtering, the optimum choice of basis functions are the Legendre polynomialsof basis functions are the Legendre polynomials

• Regression filtering can be done in real-time with a Regression filtering can be done in real-time with a standard PC. Adaptive algorithms are probably also standard PC. Adaptive algorithms are probably also possible to perform with current state-of-the-art PC possible to perform with current state-of-the-art PC technologytechnology

Future workFuture work

• Algoritm improvements and real-time implementation Algoritm improvements and real-time implementation of adaptive clutter filtersof adaptive clutter filters

• Closed form approximation for ML-estimator, and Closed form approximation for ML-estimator, and algorithm for real-time usealgorithm for real-time use

• Multi-dimensional clutter filtering (space and time) e.g. Multi-dimensional clutter filtering (space and time) e.g. by tracking material points in tissueby tracking material points in tissue

• Algoritms optimized for blood motion imagingAlgoritms optimized for blood motion imaging

ExtrasExtras

Thermal noise

Signal from moving scattererSignal from moving scattererPulse no1 2 .. … N

2D Fouriertransform

Hans TorpNTNU, Norway Doppler shift frequency [kHz]

Pow

er

Signal from one range

Fas

t ti

me

Slow time


Ult

raso

und

puls

e fr

eque

ncy

[MH

z]

Signal

Clutter

RF versus basebandRF versus baseband


Blood signalClutter

Ultrasound frequency [MHz]

Doppler frequency [kHz]

Remove negative ultrasoundFrequencies by Hilbert transformor complex demodulation

• Skewed clutter filter (signal Skewed clutter filter (signal adaptive filter) can be adaptive filter) can be implemented with 1D filteringimplemented with 1D filtering

• Axial sampling frequency Axial sampling frequency reduced by a factor > 4reduced by a factor > 4

Doppler signal from one range Doppler signal from one range Pulse no1 2 .. … N



Ult

raso

und

puls

e fr

eque

ncy

[MH

z]

Signal from one range


Pow

er

FFT

Blood detection and Blood detection and velocity estimation from 2D signal velocity estimation from 2D signal Pulse no1 2 .. … N



Ult

raso

und

puls

e fr

eque

ncy

[MH

z]

Range no12…M

DopplerSpectrum 1

DopplerSpectrum 2

DopplerSpectrum 3

Image ImprovementImage Improvement

Polynomial regression filter

Adaptive regression filter

Example of image improvement with adaptive regression filter

Thyroid gland

• Increased number of range samples M give better Increased number of range samples M give better performance but lower spatial resolutionperformance but lower spatial resolution

• Best spatial resolution with M=1Best spatial resolution with M=1

• In this work optimum estimators for the case M=1 is In this work optimum estimators for the case M=1 is

treated treated

• Extension to the case M > 1 is straight forwardExtension to the case M > 1 is straight forward


Blood detection and Blood detection and velocity estimation from 2D velocity estimation from 2D

signal signal

Clutter suppression by high pass Clutter suppression by high pass filteringfiltering

-2 -1 0 1 2-50

0

50

100


Do

pp

ler

spe

ctru

m [

dB

]

Before filteringFIR order 6 FIR order 8

FIR filterHans TorpNTNU, Norway

Order M=8: 2 samples left afterinitialization

Order M=6: 4 samples left afterinitialization

Packet size N=10

Clutter suppression by high pass Clutter suppression by high pass filteringfiltering

-2 -1 0 1 2-50

0

50

100


Dop

pler

spe

ctru

m [d

B]

Before filteringFIR order 6 FIR order 8

FIR filter

-2 -1 0 1 2-50

0

50

100


Dop

pler

spe

ctru

m [d

B]

Before filteringPoly reg order 2Poly reg order 3

Polynom regression filter


Cramer - Rao lower boundCramer - Rao lower boundApproximation Approximation

0)(

)()(;0)(

)()(')'()(

2''

''12

2

mRv

emRmRvC

vCxvvCxxvlv

b

vmkTibb

T


Cramer - Rao lower boundCramer - Rao lower boundApproximation Approximation

1'''1'1

,

,

,2

2

2),(

),(

),(),()(

CCCCCCnmbB

CICnmcC

nmbnmcxvlv

bbbnm

bCnm

nm


Linear filtersLinear filters

y = A* x

Computational complexityComputational complexity# multipications+ additions per packet

Full matrix multipication:

N*N

Projection1 basis function

2*N

FIR-filterOrder M

(M+1)*(N-M)(M+1)*(N-M)

*

* -Basis vector

*Filter matrix

Input signal vector Output signal vector

Packet sizeN=8

M+1 N-M

Atlanta, GA april 1986Atlanta, GA april 1986

Vingmed, introduced Vingmed, introduced CFMCFM The first commercial The first commercial

colorflow imaging scanner colorflow imaging scanner with mechanical probewith mechanical probe

Horten, Norway Horten, Norway september 2001september 2001

GE-Vingmed closed down GE-Vingmed closed down production line for production line for CFMCFM

after 15 years of continuous after 15 years of continuous productionproduction

The role of basis functions for The role of basis functions for linear filterslinear filters

y = A* xy = A* x

u= B*x, v= B*y; u= B*x, v= B*y;

B is a matrix of orthonormal basis functions;B is a matrix of orthonormal basis functions;

B’*B = I (identity matrix)B’*B = I (identity matrix)

v = B’*A*B *uv = B’*A*B *u

If B is eigenvectors for A, the filter matrix B’*A*B will be If B is eigenvectors for A, the filter matrix B’*A*B will be diagonal, i.e. diagonal, i.e.

v= diag (l1,..,lN)*u v= diag (l1,..,lN)*u

Filter output is a weighted sum of the projections Filter output is a weighted sum of the projections

along the basis functionsalong the basis functions

decomposition of flow signals into basis functions: performance advantages, disadvantages, and...

Documents

iir filter response

filter performancey

linear filter y

axany linear filter

clutter filtering

signal processing

clutter dim

power output