We use this notation to avoid convolutions, especially deconvolutions in the spectral domain.
This enables us to show clearly the shift of a function, which is a convolution with $(z-m.),
instead of multiplying the spectrum with ezp(-j9muz,).Substituting convolutions by matrix multi-
plications immediately yields the discrete formulas. In this case the inverse of a matrix has to
be replaced by its pseudo inverse.
Stochastical variables, vectors or functions are underscored; Z is the true value of the. variable
E(+) and V(-) denote the expectation and the Variance operators. For notational convenience
giz) often is replaced by g. The signal to noise ratio (SNR) is defined by 9/0, sx" is the
transposed vector of x.
In
2. Filters for Object Location and Point Transfer
2.1 Least squares filters
Let the template be given as a continuous greylevel function g(x). According to fig. !a the
signal g,(z) is observed which results from g(z) by
=?
shifting the template by Z,, i.e. by convolving it with ó(z "2,4
1
2. possibly convolving the result with the point spread function A(z) and
3. adding noise n(x) with autocovariance function R (2); thus
Q
,Q0) = R(z) = giz) = §(z—=x.) + niz) G3
It is assumed that gí(z) and A(x)
zv
ave zero mean.
Fig. 1 Object location: given g, possibly A^; observed £j unknown i. , noise n
a.) matched filter for estimating Z
3
+
n, BE
— - E71 = n
T — 0 (5-Z.) A at £1 m,
= à =: =
x,
A; = À + iz) *g +n
sh +9 + zZ +
=
= ~ : j= ER
$30—[ ah A Tg S Ma 2.4
+ oe T E 2,035
n, 8
— Um
b.) filters for restoring 8(z-%_)
‘+
In general z and Z, are unknown. The task is to find a filter m(z) such that the maximum of c(z) =
m(z) = g, (=), the search function as we will call it, yields an estimate 2. for Z.. We will com-
pare the result of four different optimization criteria:
1. R, known, maximize the signal to noise ratio of the filtered signal and the filtered noise
at E. SNR? = V(gwm) / V(nwm).
2. 2, known, maximize the ratio of the expected maximum and the average standard deviation of
e(z) at all other points: æ = = E(g,=m) /VV( gU.
3. The expectation of the search function should be a óé-function with peak value at Z
|
Elclz)) $ 6(z-X
4. The maximum of E /2) should be at Z, and the autocovriance function of the search function
1 1 ^ = m; , t !
should be a S-function: E/a/
' ; 3 ELLAS Ci Xm 7. 53
Oí(r)xcí-ri)rz B fam} & Sf)
Ler {oa J T ws J tg nM - V m/s.
— —- e
he filter m(z), the expectation of the searen function and its autocovariance function are given
n le 1 for the case where no filter h(z) is applied.
