Clutter filtering in color flow imaging Hans Torp 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

Outline Methods for clutter filtering in color flow imaging IIR , FIR, regression filter General linear filtering – basis functions 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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: 13 Msamples/sec Projection filter, 3 basis functions: 17 Msamples/sec FIR filter 45 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

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