Kunchenkos Polynomials for Template Matching

The problem statement of matching data with template includes input data sets to analyse, specified template and sometimes search domain. It’s very important to represent available input data in appropriate form and define matching criteria.

In this paper, we consider 1D digital signal. Although this approach can be used with 2D images because several techniques to transform 2D images into one-dimensional representation have been proposed recently [2].

There are two very different approaches to find templates in available data set [3].

The first one is to use artificial neural networks after learning phase [4]. In terms of mathematics, neural network’s learning is a multi-parametric nonlinear optimization problem. A large number of right answers are required and that is not always possible.

The second approach consists in comparing either special features of template with data set (feature-based matching) or entire template with part of data set (template-based matching) [3]. As far as we consider one-variable function analysis, we don’t focus on feature-based template matching, for instance angles or edges at pictures [5,6]. Let’s take a look at technique of entire template search in input data set. This method is one of the first in history. In this sort of problems when comparing template with signal Euclidean metric, sum of squared differences, cross-correlation and several others are used [7][8][9]. However, we can try to solve this problem in another way -we may approximate input signal using polynomial template-based function [10]. In this case, areas of the most close signal and polynomial approximation that are over some threshold value can be considered as places of signal where template occurred.

It is well known that a Taylor polynomial is a polynomial to approximate a function the best way in a neighborhood of a given point. Though, the approximated function must have derivatives of appropriate order in this surrounding. Conditions that allow to approximate function with Fourier series over linearly independent orthogonal functions system that made up basis of considering space with inner product are much weaker. But from the template matching point of view Kunchenko’s polynomials using seems to be very prospective [11,12]. It is a result of the space where the polynomials are built. Such space is a subspace of Euclidean or Hilbert space that has particular element called generative element. Given template can be considered as generative element. After that template matching problem can be discussed as simpler problem of finding closest Kunchenko’s approximation polynomial using some metrics [10].

Let us assume function ( ) f x called generative function and defined on the interval [a, d]. After that we denote set of ordered generated functions as follows:

( ) φ [ ( )],

[ , ]

where φ ( ) v  -real functions fitted in some particular way.

In case of defining linear operations -addition and multiplication by number -set of all generated functions’ linear combinations will make linear space.

Since some of generated functions (1) can be linear dependent, let us define space-constructing set of functions. This set is constructed as union of linear independent generated functions from (1). Thus, we’ll call linear space over such set as linear space over independent generated functions (linear Kunchenko’s space) and denote LFKu. Now let us define inner product of two elements ( ) v u x and ( ) k u x in LFKu space as usual and call this product as correlant of these two elements:

According to (2) we can define distance ρ vk between two elements ( ) v u x and ( ) k u x from Kunchenko’s linear space as norm of difference between two functions:

LFKu space where set of generated functions consists only of linear independent and pairwise (totally or partly) nonorthogonal functions we’ll call Kunchenko’s space with generative element. Under partially nonorthogonal generated functions we consider such set of functions where some of them are pairwise nonorthogonal and the other part of functions is pairwise orthogonal.

Let us also take some generated function ( ) b u x and call it cardinal function. In this case we can construct Kunchenko’s polynomial from generated functions ( ), v u x v b  , such functions we’ll call additional functions:

Considering distance minimization between Kunchenko’s polynomial (4) and cardinal function (

As shown in [12, p. 75-76] coefficients α v , 0 v  can be found as solution of following linear equations system:

where centered correlants are 2 , , 0 ( )

Coefficient 0 α must be equal to next expression:

Using coefficients ( 6), ( 7) in ( 5) we’ll get:

where r J is so-called polynomial’s (4) inforkune that looks as follows:

At last, let us introd