Image Processing Reference
In-Depth Information
15
Reducing the Dimension of Features
For many applications, a dimensionality reduction results in improved signal sepa-
ration in the presence of noise. We will discuss the underlying concept in the prin-
cipal components analysis section below. However, dimension reduction is a general
problem that has been the subject of intensive study in different disciplines. While
principal components analysis is the earliest and simplest technique, and is widely
used for its efficiency, there are other approaches, such as independent component
analysis, neural networks, and self-organizing maps [29, 117, 142], that have proven
to be more powerful in many applications. For reason of limited scope, we must,
however, restrict our discussion to principal components analysis. An example of its
use in face recognition is discussed in Sect. 15.2, whereas another example for which
reduced dimension is a prerequisite, texture analysis, will be illustrated in Sect. 16.7
when discussing clustering and boundary estimation in textures.
15.1 Principal Component Analysis (PCA)
There are many names to principal component analysis (PCA), as it has been used in
numerous disciplines and applications. Examples include color representation, tex-
ture segmentation, multispectral image classification, face recognition, source sepa-
ration, visualization, and image database queries [100, 172, 179, 202, 218, 220]. Syn-
onyms of PCA include
Karhunen-Loeve (KL) transform
, [133], Hotelling transform,
eigenvalue analysis, eigenvector decomposition, and spectral decomposition. In im-
age analysis it is used to reduce dimensions, and to find subspaces in which recog-
nition works better than taking the full space. Not only does PCA reduce the size of
the data for the sake of efficient storage, transmission, and processing advantages.
Assume that we have observed a set of
K
vectors
O
=
{
f
k
}
(15.1)
i.e., the
observation set
,inan
M
-dimensional vector space.
The coordinates of the observation set can be represented by
M
scalars in some
basis
