Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

Principal component analysis: The detail of the analysis [4] is beyond
the scope of the book. We just illustrate the concept from a practical
standpoint. We, in this study, considered 9 facial images of (32 x 32) size for
40 individuals. Thus we have 360 images each of (32 x 32) size. The average
gray value of each image matrix is then computed and subtracted from the
corresponding image matrix elements. One such image after subtraction of the
average value is presented in fig. 17.9. This is some form of a normalization
that keeps the image free from illumination bias of the light source. Now, we
construct a matrix X of ((32 x 32) x 360) = (1024 x 360) dimension with the
above data points. We also construct a covariance matrix by taking the
product X X T and evaluate the eigen vectors of X X T. . Since X is of dimension
(1024 x 360), X X T will have a dimension of (1024 x 1024). X X T thus will
have 1024 eigen values, out of which we select the first 12, on experimental
basis. The eigen vectors corresponding to these 12 eigen values are called the
first 12 principal components. Since the dimension of each principal
component is (1024 x 1), grouping these 12 principal components in columnwise
fashion, we construct a matrix of dimension (1024 x 12). We denote this
matrix by EV (for eigen vector).
To represent an image in the eigen space, we first represent that image
in (1 x 1024) format and then project it onto the face space by taking the dot
product of the image matrix IM, represented in (1 x 1024) format and the EV
and call it a point (PT) in the face space. Thus
( PT ) 1 x 12 = ( IM ) 1 x 1024 . ( EV) 1024 x 12. (17.11)
The projection of images onto face space as 12-dimensional points is
presented in fig. 17.10. Representing all 360 images by the above expression,
we thus get 360 points in 12-dimensional image space.
Now, suppose given a test image and we want to classify it to one of the
40 persons. One simple way to solve this problem is to determine the
corresponding image point (PT) in 12-dimensional space and then determine
the image point (out of 360 points) that has the least Euclidean distance w.r.t
the test image point. The test image thus can be classified to one of 360
images.
The principal component analysis (PCA) thus reduces the dimension of
matching from (32 x 32) to (1 x 12) but requires computing the distance of a
test point with all image points. An alternative scheme that reduces less
computing time is by a self-organizing neural net. The self-organizing scheme
inputs the (1 x 12) points for all of the 360 images, constructs a network and
searches a test point by the best first search paradigm in the search space. A