Kernel-Based Adaptive Image Retrieval Methods - Multimedia Database Retrieval: Technology and Applications - page 22

2.2.5 Nonlinear Model-Based Adaptive Method

The methods outlined above are referred to as linear-based learning and this restricts
the mapping function to quadratic form, which cannot cope with a complex decision
boundary. For example, the one-dimensional distance mapping function h

( d i ) in

Eq. ( 2.13 ) may take the following form:

w i d i h ( d i )= (2.17)

where d i = | x i − x qi | . This function has a small degree of nonlinear behaviour, i.e.,

∂ f q ( x )

d i =

2 w i d i (2.18)

∂

where w i is fixed to a numerical constant for the respective feature dimension.
To simulate human perception, a radial basis function (RBF) network [ 31 , 45 ]
is employed in this chapter. The input-output mapping function, f

, is employed
on the basis of a method called regularization [ 32 ]. In the context of a mapping
problem, the idea of regularization is based on the apriori assumption about the
form of the solution (i.e., the input-output mapping function f

( x )

). In its most
common solution, the input-output mapping function is smooth , in the sense that
similar inputs correspond to similar outputs. In particular, the solution function
that satisfies this regularization problem is given by the expansion of the radial basis
function [ 33 ]. In this case, a new inner product is expressed as a nonlinear kernel
function K

( x )

( x , z ) :

, = ( , )

x z K x z (2.19)

The Gaussian-shaped redial basis function is utilized:

exp

2

−

x − z

K ( x , z )= (2.20)

2

2 ˃

where z denotes the center of the function and ˃ denotes its width. The activity of

function K ,

which describes the degree of similarity between the input x and center of the
function. Under Gaussian distribution, this function reflects the likelihood that a
vector x may be mistaken to be another vector z .
To estimate the input-output mapping function

( x , z ) is to perform a Gaussian transformation of the distance x − z

, the Gaussian RBF is
expanded through both its center and width, yielding different RBFs which are then
formed as an RBF network. Its expansion is implemented via a learning process,
where the expanded RBFs can modify weighting, to capture user perception.

f ( x )