Page 81 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

70 PARAMETER ESTIMATION

Assuming that some initial estimate x ref is available we expand (3.46)
in a Taylor series and neglect all terms of order higher than two:

z ¼ hðxÞþ v

qhðxÞ
hðx ref Þþ H ref x x ref þ v with: H ref ¼ ð3:52Þ
qx
x¼x ref
where H ref is the Jacobian matrix of h(:) evaluated at x ref , see Appendix
B.4. With such a linearization, (3.51) applies. Therefore, the following
approximate value of the LS estimate is obtained:

1

T
^ x x LS x ref þ H H ref H ref z hðx ref Þ ð3:53Þ
ref
A refinement of the estimate could be achieved by repeating the proce-
dure with the approximate value as reference. This suggests an iterative
x
approach. Starting with some initial guess ^ x(0), the procedure becomes
as follows:
1
T
x
x
ð
^ x xði þ 1Þ¼ ^ xðiÞþ H ðiÞHðiÞ HðiÞ z hð^ xðiÞÞÞ

ð3:54Þ
with: HðiÞ¼ qhðxÞ
qx
x
x¼^ xðiÞ
In each iteration, the variable i is incremented. The iterative process
stops if the difference between x(i þ 1) and x(i) is smaller than some
predefined threshold. The success of the method depends on whether the
first initial guess is already close enough to the global minimum. If not,
the process will either diverge, or get stuck in a local minimum.
Example 3.8 Estimation of the diameter of a blood vessel
In vascular X-ray imaging, one of the interesting parameters is the
diameter of blood vessels. This parameter provides information about
a possible constriction. As such, it is an important aspect in cardiolo-
gic diagnosis.
Figure 3.8(a) is a (simulated) X-ray image of a blood vessel of the
coronary circulation. The image quality depends on many factors.
Most important are the low-pass filtered noise (called quantum mot-
tle) and the image blurring due to the image intensifier.
Figure 3.8(b) shows the one-dimensional, vertical cross-section of
the image at a location as indicated by the two black arrows in
Figure 3.8(a). Suppose that our task is to estimate the diameter of the
imaged blood vessel from the given cross-section. Hence, we define