Page 125 - Introduction to Information Optics

1 1 0 2. Signal Processing with Optics

Thus, at t = 0, we have
(190)

which is the dot product of vectors / and g.
The set of vectors /, as well as g, forms a hypervector space that yields
discriminant surfaces which enable us to group the rnulticlass objects (i.e.. the
oriented vectors) apart from the false input objects. In other words, the shape
of the discriminant surface decides the kind of average matched filter to be
designed. For instance, the autocorrelation of an oriented input object / with
respect to a specific oriented filter function g should lie within the discriminant
surface.
For simple illustration, we assume only two object vectors are to be detected
and all the other object vectors are to be rejected. By representing these input
vectors as
/j = (a j j , a ! 2 ) and / 2 = (a 2 1 , a 2 2),

a hyperplane resulting from /, and f 2 can be described by

f- h - K,

where h represent the filter vector and K is a constant. In this illustration, h
represents a vector perpendicular to the hyperplane form by / t and / 2, by
112
which K = (g -g) . Thus, we see that the average filter function is a specific
linear combination of c l /, and c 2 f 2, as written by

h = GI /i + c 2 / 2 = (c ia u + c la l2)cf) l + (c 2a 2l + c 2 a 22 )0 2 .
A synthetic discriminant function (SDF) filter can be described as a linear
combination of a set of reference functions; that is,

h(x) = J]cj^(jc), (2.91)
./
for which the correlation output must be

b
0 = g- n = Y, n- (2.92)
The remaining task is to find {4>j}, n nj, then GJ, and finally h(x), under the
assumed acceptable correlation peaks R(0). However, in the construction of the
SDF filter, the shift invariance property is imposed; that is, all the autocorrela-
tion peaks must occur at i = 0. These requirements ensure that we shift each
g n or (j>j to the correct input object location when we synthesized the filter. Such
technique is acceptable, since the synthesis is an off-line algorithm. This off-line