Master Thesis - Department of Computer Science

ΩNull and ΩRange represents the class templates in null space and range space, re-
spectively. On arrival of a test sample, we project it onto W Null
opt
to compare with
the templates in ΩNull. Similarly, the projection of the test sample onto W Range
opt
be compared with the templates in ΩRange. So, on the presentation of a test sample
x test , the feature vectors are obtained as,
will
ω test
Null = W Null
opt x test , (4.22)
ω test
Range
= W Range
opt x test . (4.23)
Then ω test
Null is compared to the vectors in ΩNull using Euclidean distance to ob-
tain response vector DNull(x test ) = [d 1 Null(x test ), ..., d C Null(x test )]. The response vec-
tor DRange(x test ) = [d 1 Range (xtest ), ..., d C Range (xtest )] in range space can be obtained
by comparing ω test
Range with the vectors in ΩRange. The elements in DNull(x test ) and
DRange(x test ) represent the similarity measures or score values for different classes.
More specifically, d i Null(x test ) is the amount of evidence provided by null space clas-
sifier, DNull, to the fact that x test belongs to class i. Now the task of decision com-
bination rule is to combine DNull(x test ) and DRange(x test ) for obtaining a combined
decision vector, ˜ D(x test ). Detail discussion on the techniques for combining response
vectors (also known as soft class labels and decision vectors) from two classifiers is
provided in Section 4.4.
4.3 Feature Fusion
4.3.1 Techniques for Merging Eigenmodels
The problem of combining W Null
opt
and W Range
opt
is similar to the problem of merging
two different eigenmodels obtained from two different sets of feature vectors XNull
and XRange. Among two methods for combining them, first method uses XNull and
XRange to obtain a set of combined discriminative directions W Dual , while the other
one forms a orthogonal basis W Dual for a matrix [W Null
opt , W Range
opt
] using QR decom-
position. Henceforth, we will refer to the first technique as Covariance Sum method
[96] and the second technique as Gramm-Schmidt Orthonormalization, adopted from
modified Gramm-Schmidt Orthonormalization technique [42] used for computing QR
decomposition of a matrix. A brief description for Covariance Sum method and
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