Exercise 7

Topic: Estimation theory. ML estimator applied on motion-estimation in ultrasound images.

Literature: van Trees: Detection and estimation... and "Waves…" chapter 10.1 and 10.2

We are starting with the complex envelope of the received signal from two following images. The signal is complex Gaussian. The radial motion dz causes a phase shift from image 1 to image 2 which gives s2 = s1*exp(i*4*pi*f0/c*dz). f0 is the ultrasound center frequency and c is the propagation velocity. Thermal noise, which is assumed to be uncorrelated in two neighbor images, is added to the signal.
x1 = s1 + n1; x2 = s2 +n2; n1 and n2 are uncorrelated.

Make an expression for the covariance matrix for [x1,x2] (with amplitude respectively As and An) from a point in the image. Also find the simultaneous probability density for [x1,x2].

An example is given in the m-file Oppgave7, with simulation of signal + noise, and a numeric computation of the ML-estimate of the motion dz.

Make an analytic expression for the ML-estimate.
(Note: the inverse matrix of [1, a ; a’, 1] is 1/b*[1, -a ; -a’, 1], where b = 1-a*a’; a is an arbitrary complex number, a’=conj(a)).

Compare the analytic and the numerical ML-estimate (run for instance 100 simulations). Compute bias and variance. Compare with the Cramer-Rao limit.

We will now try to reduce the estimation variance by using more points in the image, which are assumed to have the same motion dz.
We assume the covariance of the signal and the noise to be 0 between different points in the image.
Make an expression for the ML-estimate based on N points.
Will this give the same result as using the mean value of the ML-estimate from each point? Illustrate this by computing the variance numerically (by simulation) for the two methods and/or by comparing the Cramer-Rao limit.