1. Decomposition of flow signals into basis functions: Performance advantages, disadvantages, and computational complexity
Hans Torp and Lasse Løvstakken Norwegian University of Science and Technology
Trondheim, Norway
Disposisjon:
Signal-model 2D -> 1D (?)
Hvorfor lineært filter? -> lineær transf., Matrise, Eks. FIR-filter
Shift invarianse
Frequency response. Advantage complex signals
Basis functions and transformation to obtain diagonal matrix
Regression filter. Linear regression (Hooks). Polynomial regression
Discrete Legendre polynomials. Show freq. Resp. basis and filter
Fourier basis, and regression filter. Fourier basis with zeropadding. How to obtain zero in freq. Response
Kahrunen-Loeven transform- generelt
Modellbasert: Gaussisk og rektang. Frekv. Respons. Adaptiv
Deteksjonsproblemet
Estimering av hastighet – bias . ML-estimering av hastighet
Computatonal complexity
General linear filter matrix. Regression filter, compare FIR
Compex/real coefficients. Adaptive filter

2. Acknowledgement Steinar Bjærum, GE Vingmed Ultrasound, Horten
Substantial part of the results in this presentation is taken from his phd work
Kjell Kristoffersen, GE Vingmed Ultrasound, Horten
Collaboration for 25 years in Doppler ultrasound

3. Outline Methods for clutter filtering in color flow imaging
IIR , FIR, regression filter
General linear filtering – basis functions
Optimum choice of basis functions
Best frequency response
Optimum detection
Disadvantages:
Bias in velocity estimation
Computational complexity
Comparison FIR filter and regression filter
Clutterfilter in Doppler/cfi, history (brief)
Signal model 2D -> 1D Doppler signal
Properties:
- linearity
-shift invariance
Performance criteria:
Frequency response
Detection
Velocity estimation
Basis types:
Fourier
Extended Fourier (spør Kjell!)
Legendre
Karuhen Leuwe
Performance advantages:
Best frequency response
Optimum detection
Disadvantages:
Bias in velocity estimation
Computational complexity
Max N FIR-filter av orden N+1
Ta med bilde av DSP-kort i System 5

4. Clutter filter in color flow imaging?
Beam k-1
Beam k
Beam k+1

5. Doppler Signal Model x = [x(1),…,x(N)]T x = c + n + b
Signal vector for each sample volume:
-0.6
-0.4
-0.2
0.2
0.4
0.6 20 40 60 80
100
Blood velocity [m/s]
Doppler spectrum [dB]
Clutter
Blood
x = [x(1),…,x(N)]T
Zero mean complex random process
Three independent signal components:
Signal = Clutter + White noise + Blood
x = c + n + b
Typical clutter/signal level: 30 – 80 dB
Clutter filter stopband suppression is critical!
Hans Torp
NTNU, Norway

6. IIR filter with initialization
Frequency response
-80
-60
-40
-20
Steady state
Step init.
Discard first samples
Power [dB]
Projection init*
0.1
0.2
0.3
0.4
0.5
Frequency
Chebyshev order 4, N=10
*Chornoboy: Initialization for improving IIR filter response, IEEE Trans. Signal processing, 1992

7. FIR Filters å = - M k z b H ) (
Discard the first M output samples, where M is equal to the filter order
Improved amplitude response when nonlinear phase is allowed
-20
Linear phase
Power [dB]
-40
Minimum phase
-60
-80
0.1
0.2
0.3
0.4
0.5
Frequency
Frequency response, order M= 5, packet size N=10

8. Regression Filters x y c y = x - c b3 Signal space x = [x(1),…,x(N)]
Clutter space b1 b2 b3 x y c
Subtraction of the signal component contained in a K-dimensional clutter space:
x = [x(1),…,x(N)]
y = x - c
Linear regression first proposed by Hooks & al. Ultrasonic imaging 1991

9. Why should clutter filters be linear?
No intermodulation between clutter and blood signal
Preservation of signal power from blood
Optimum detection (Neuman-Pearson test) includes a linear filter
Any linear filter can be performed by a matrix multiplication of the N - dimensional signal vector x *
Matrix A
Input vector x
Output vector y
y = Ax
This form includes all IIR filters with linear initialization, FIR filters, and regression filters

10. Frequency response Linear Filters
y = Ax
Definition of frequency response function
= power output for single frequency input signal 1 Ho 2 ( w ) = Ae
; -p < w < p w N [ ] T w e = 1 e i w e i ( N - 1 ) L w
Note 1. The output of the filter is not in general a single frequency signal
(This is only the case for FIR-filters)
Note 2. Frequency response only well defined for complex signals

11. FIR filter matrix structure
FIR filter order M=5
Packet size N=10
Output samples: N-M= 5
+ Improved clutter rejection
Increasing filter order
- Increased estimator variance
Hans Torp
NTNU, Norway

12. å Regression Filters x y c y = x - c x b I y ÷ ø ö ç è æ - = A b3
Signal space
Clutter space b1 b2 b3 x y c
Subtraction of the signal component contained in a K-dimensional clutter space:
y = x - c x b I y ÷ ø ö ç è æ - = å K i H 1 A
Choise of basis function is crucial for filter performance

13. Fourier basis functions
i*k*n/N
b (k)=1/sqrt(N)e n
n= 0,.., N-1
are orthonormal, and equally distributed in frequency

14. Fourier Regression Filters
Frequency response
DFT
Set low frequency
coefficients to zero
Inverse DFT
-20
Power [dB]
-40
-60
-80
0.1
0.2
0.3
0.4
0.5
Frequency
N=10, clutter dim.=3

15. Legendre polynom basis functions
Gram-Schmidt process to obtain
Orthonormal basis functions
-> Legendre polynomials

16. Polynomial Regression Filters
b0 =
Frequency responses, N=10
-80
-60
-40
-20
b1 =
b2 =
Power [dB]
b3 =
0.1
0.2
0.3
0.4
0.5
Frequency

17. Fourier-basis with extended period

18. Frequency Response comparison
-80
-60
-40
-20
Polynom regression
Power [dB]
IIR projection init.
FIR minimum phase
0.1
0.2
0.3
0.4
0.5
Frequency
Polynomial regression and IIR filter with projection initializarion
have almost identical performance, and are superior to FIR filters

19. Optimal Basis functions
Eigenvalue decomposition of the clutter correlation matrix: å = N i ‘ c 1 b R l
Use the eigenvectors bi as a basis for the clutter space (Karhunen-Loeve transform)
This basis provides maximum energy concentration of the clutter signal
Energy l1 lK lN

20. Adaptive basis functions
The correlation matrix may be estimated by spatial averaging in a region with uniform motion: å = M i H c 1 ˆ x R
Adapt to clutter velocity - skewed filter center freq.
May account for irregular wall motion, non-stationary clutter signal

21. Adaptive Regression Filter
Eigenvalue spectrum of clutter + blood
Clutter filter
Energy l1 lK lN
Eigenvectors

22. Adaptive Regression Filter Projection along each single basis function
Legendre polynomial basis functions
Eigenvector basis functions

23. Detection of Blood A rule for deciding between the two hypotheses:
H0: No blood is present
H1: 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)
Coronary artery

24. The Optimal Detector >  < >  < The Neyman-Pearson lemma:
PD is maximized under the constraint
PF   by a likelihood ratio test (LRT) ) ( H p 1 L x = 
>
< H0 H1
For Gaussian signals, the LRT can be simplified to: 2 ) ( Ax x = l 
>
< H0 H1
Matrix A is given by the signal covariance matrix

25. The Optimal Detector for Gaussian signals
||  ||2 x
Clutter filter
Power calc.
> g
< g H1 H0
Matrix A is a linear filter which suppress the clutter signal.

26. ROC for different clutter filters
Blood velocity = 10 cm/s 1
Optimal detector
Eigenvector reg. filter
Pol. reg. filter PF
IIR proj. init.
FIR min. phase
FIR linear phase
IIR step init. PD 1
Example from coronary artery flow in rapid moving myocard
Note that the eigenvector regression filter is close to optimal

27. Is the Gaussian assumption valid?
Non-Gaussian histogram due to
variation in signal power
Histogram from smaller region
shows Gaussian distribution

28. Basis functions for non-adaptive filters
Clutter signal with Gaussian
shaped power spectrum:
Eigenvectors of covariance matrix
~ Legendre polynomials
Covariance matrix estimated from
Color Doppler signals with moderate wall motion:
Eigenvectors of covariance matrix
~ Legendre polynomials

29. Summary Optimal choice of basis function for blood vessel detection
Doppler signals show Gaussian distribution locally in color flow images
Eigenvector basis functions give optimum separation of clutter and blood
Legendre polynomials are the best choice for non-adaptive clutter filtering, and give substantially better performance than FIR filters.
Adaptive eigenvector filters show significant improvements over polynomial regression filters for spatially uniform wall motion

30. Autocorrelation method for blood velocity estimationx y
Phase
angle
&
scaling
Clutter
Rejection
Filter
Auto
Correlation
Estimator
y =A x
Hans Torp
NTNU, Norway

31. Regression filter bias
Single frequency input may give
high frequency distortion
Severe bias in band width and
mean frequency estimator
below cutoff frequency.
Magnitude and phase frequency response Polynomial regression filter packet size = 10, polynom order 3.

32. Computer simulation of Doppler signal including clutter
Frequency 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 m/s
Angle blood flow deg.
Hans Torp
NTNU, Norway

33. Blood velocity estimator performance
0.15
0.15
Polyreg. filter
Polyreg filter
FIR filter
FIR filter
0.1
0.1
Bias [m/s]
0.05
Standard deviation [m/s]
0.05
-0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.2
0.4
0.6
Blood velocity [m/s]
Blood velocity [m/s]
Hans Torp
NTNU, Norway
FIR filter order 7

34. Optimal methods for velocity estimation Maximum Likelihood estimator
Probability density function given by the covariance matrix C(v)
l(v|x)
Likelihood function
vML v
l(v|x) = log p(x|v)
Hans Torp
NTNU, Norway

35. Log likelihood function and Cramer - Rao lower bound
Hans Torp
NTNU, Norway

36. Blood velocity estimator bias variance
ML-estimator is not minimum variance, but better than the two other approaches

37. Summary Blood velocity estimation in the presence of clutter signals
Polynomial Regression (PR) filters give substatial positive bias for low velocity blood flow
FIR filters give less bias, but much higher variance than PR-filters
ML-estimator has lowest bias and variance, but the algorithm is not suitable for practical use.
PR-filter approach the performance of ML-estimator when the Dopplershift is above the filter cutoff frequency

38. Computational complexity
# multipications
+ additions
per packet
M=5 *
Filter matrix
Input signal vector
Output signal vector
Packet size
N=8
Full matrix
multipication:
N*N * -
Basis vector
Projection
1 basis function
2*N
y = A* x
FIR-filter
Order M
(M+1)*(N-M) 
M+1
N-M

39. Real-time clutter filtering
Data rate in color flow imaging:
1 – 5 M samples/sec (complex samples)
Processing speed test: Pentium M 1.6 GHz, using Matlab R13, N=8, M=6
Matrix multipication: Msamples/sec
Projection filter, 3 basis functions: 17 Msamples/sec
FIR filter Msamples/sec
Adaptive filters is much more computer demanding
Double CPU-time with complex filter coefficients
CPU-time for filter coefficient calculation
Example: Adaptive Eigenvector filter 2.1 Msamples/sec

40. Summary Computational complexity of clutter filter algorithms
Regression filters have 1 – 2 times longer computation time than FIR-filters
A standard laptop computer is able to do real-time regression filtering using less than 10% of available cpu-time
Adaptive eigenvector filter requires ~ 10 times more computation power than the regression filter

41. Conclusions
Regression filters are superior to FIR and IIR filters in blood flow detection and velocity estimation
For non-adaptive clutter filtering, the optimum choice of basis functions are the Legendre polynomials
Regression filtering can be done in real-time with a standard PC. Adaptive algorithms are probably also possible to perform with current state-of-the-art PC technology

42. Future work
Algoritm improvements and real-time implementation of adaptive clutter filters
Closed form approximation for ML-estimator, and algorithm for real-time use
Multi-dimensional clutter filtering (space and time) e.g. by tracking material points in tissue
Algoritms optimized for blood motion imaging

43. Extras

44. Signal from moving scatterer
Pulse no
Doppler shift frequency [kHz]
Ultrasound pulse frequency [MHz] 1 … N
Fast time
Slow time
Thermal noise
Clutter
Signal
2D Fourier
transform
Signal from one range
Doppler shift frequency [kHz]
Power
Hans Torp
NTNU, Norway

45. RF versus baseband Remove negative ultrasound
Ultrasound frequency [MHz]
Remove negative ultrasound
Frequencies by Hilbert transform
or complex demodulation
Clutter
Blood signal
Skewed clutter filter (signal adaptive filter) can be implemented with 1D filtering
Doppler frequency [kHz]
Axial sampling frequency reduced by a factor > 4
Doppler shift frequency [kHz]

46. Doppler signal from one range
Pulse no 1 …
Doppler shift frequency [kHz]
Ultrasound pulse frequency [MHz] N
Signal from one range
Doppler shift frequency [kHz]
Power
FFT
Hans Torp
NTNU, Norway

47. Blood detection and velocity estimation from 2D signal
Pulse no
Doppler shift frequency [kHz]
Ultrasound pulse frequency [MHz] 1 … N
Doppler
Spectrum 3
Doppler
Spectrum 2
Range no 1 2 … M
Doppler
Spectrum 1
Hans Torp
NTNU, Norway

48. Blood detection and velocity estimation from 2D signal
Pulse no
Doppler shift frequency [kHz]
Ultrasound pulse frequency [MHz] 1 … N
Doppler
Spectrum 3
Doppler
Spectrum 2
Range no 1 2 … M
Doppler
Spectrum 1
Hans Torp
NTNU, Norway

49. Image Improvement Example of image improvement with adaptive
regression filter
Polynomial regression filter
Thyroid gland
Adaptive regression filter

50. Blood detection and velocity estimation from 2D signal
Increased number of range samples M give better performance but lower spatial resolution
Best spatial resolution with M=1
In this work optimum estimators for the case M=1 is treated
Extension to the case M > 1 is straight forward
Hans Torp
NTNU, Norway

51. Clutter suppression by high pass filtering
100
Before filtering
Packet size N=10
FIR order 6
FIR order 8
Order M=6: 4 samples left after
initialization 50
Order M=8: 2 samples left after
initialization
Doppler spectrum [dB]
-50 -2 -1 1 2
Doppler shift frequency [kHz]
FIR filter
Hans Torp
NTNU, Norway

52. Clutter suppression by high pass filtering
Polynom regression filter
FIR filter
Hans Torp
NTNU, Norway

53. Cramer - Rao lower bound Approximation
Hans Torp
NTNU, Norway

54. Cramer - Rao lower bound Approximation
Hans Torp
NTNU, Norway

55. Linear filters Argumenter for linearitet Lineærtransform - matrise
Eks. FIR-filter
Basis-functions y= A*X
A=V*D*V’
V’*y = D*V’*x
v = D*u
Change coord. System + gain adjustment (complex) of each component
Makes no sense in most cases, e.g. FIR filters
Fourier basis makes sense, but poor clutter filter
Fourier with expanded period better; -> poly regression

56. Computational complexity
# multipications
+ additions
per packet *
Filter matrix
Input signal vector
Output signal vector
Packet size
N=8
Full matrix
multipication:
N*N * -
Basis vector
Projection
1 basis function
2*N
y = A* x
FIR-filter
Order M
(M+1)*(N-M) 
M+1
N-M

57. Atlanta, GA april 1986 Horten, Norway september 2001
Vingmed, introduced CFM The first commercial colorflow imaging scanner with mechanical probe
Horten, Norway september 2001
GE-Vingmed closed down production line for CFM
after 15 years of continuous production

58. The role of basis functions for linear filters
y = A* x
u= B*x, v= B*y;
B is a matrix of orthonormal basis functions;
B’*B = I (identity matrix)
v = B’*A*B *u
If B is eigenvectors for A, the filter matrix B’*A*B will be diagonal, i.e.
v= diag (l1,..,lN)*u
Filter output is a weighted sum of the projections
along the basis functions