Page 180 - Introduction to Statistical Pattern Recognition

162 Introduction to Statistical Pattern Recognition

,.
(3) Average over i = 1,. . . ,n-1, to get p.

(4) Insert into (4.126) to form k.

Note that only (n+l) parameters, oi (i = 1,. . . ,n) and p, are used to approxi-
mate a covariance matrix.

Example 5: Figure 4-10 shows the correlation matrix for Camaro of
Data RADAR. The radar transmitted a left circular electro-magnetic wave, and
received left and right circular waves, depending on whether the wave bounced
an even or odd number of times off the target surface. With 33 time-sampled
values from each return, we form a vector with 66 variables. The first 33 are
from the left-left and the latter 33 are from the left-right. The two triangular
parts of Fig. 4-10 show the correlation matrices of the left-left and left-right.
In both matrices, adjacent time-sampled variables are seen to be highly corre-
lated, but the correlation disappears quickly as the intervals between two sam-
pling points increase. The ith sampling point of the left-left and the ith sam-
pling point of the left-right are the returns from the same target area. There-
fore, they are somewhat correlated, which is seen in the rectangular part of Fig.
4- 10.
The toeplitz form of (4.126) cannot approximate the rectangular part of
Fig. 4-10 properly. Therefore, we need to modify the form of the approxima-
tion. The structure of Fig. 4-10 is often seen in practice, whenever several sig-
nals are observed from the same source. In the radar system of Example 5, we
have two returns. In infrared sensors, it is common to observe several wave-
length components. Furthermore, if we extend our discussion to two-
dimensional images, the need for the form of Fig. 4-10 becomes more evident
as follows.

Block toeplitz: Let x(i,j) be a variable sampled at the i,j position in a
two-dimensional random field as illustrated in Fig. 4-1 1. Also, let us assume
toeplitz forms for correlation coefficients as