Page 124 - Introduction to Information Optics

2.6. Algorithms for Processing

logf
Logarithmic log f + log n Noise + white noise ^ Linear log f ® log f
Processor Pre- Optical
whitening Correlator
Fig. 2.32. A logarithmic processing system for multiplicative noise.

which were not exploited. In this section we discuss a type of composite filter
in which a number of filters can be encoded in a single spatial filter.
Nevertheless, composite filters can also be constructed by means of thick
photorefractive crystals (a three-dimensional volume filter discussed later). In
general, a composite filter can be constructed by means of a set of filter
function q = {q n}, which are used to correlate with a set of multivariant (or
multiclass) objects / = {/„}, n = 1,2, . . . , N, where N is the number of object
orientations. We note that the sets of / and g could be entirely different objects.
For simplicity, we assume the set {g n} represents different orientations of g and
that {/„} belongs to the set {#„}. We begin by expanding each orientation of
the input object and filter function f and g in sets of orthonormal sets, as given
by

g(x) = (2.86)

where a and b are coefficients, and

It is trivial that / and g can be represented by vectors in a vector space, such

= (a i,a 2,...,a n) (2.87)

and

(2.88)

In terms of these expansions, the correlation of / and g can be shown as

x
T
a
x
^( ) =/(*) ® d( ) — ZZ A I ^j( + ?)</>i(x)dx. (2.89)
' ./