Optimizing the parameters of wavelets for pattern matching using ga

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976

– 6480(Print), ISSN 0976 – 6499(Online) Volume 3, Number 1, January - June (2012), © IAEME

77

OPTIMIZING THE PARAMETERS OF WAVELETS FOR PATTERN

MATCHING USING GA

1Manju B R ,

2Dr A R Rajan ,

3Dr V Sugumaran

1Research Scholar, Dept of Mathematics, University of Kerala, Thiruvananthapuram,

[email protected], 2Professor, Dept of Mathematics, University of Kerala, Thiruvananthapuram

3SMBS,VIT University, Chennai Campus, Tamil Nadu, India, [email protected]

ABSTRACT

Pattern matching has numerous applications in engineering. Wavelets have

been used as a tool for pattern matching of signals to identify pattern, specifically in

pattern recognition problems. To design wavelets for a given pattern, there are two

popular approaches namely, parametric approach and non-parametric approach. In

parametric approach, the wavelet is defined with a few parameters and designing a

wavelet for a given pattern is performed by choosing the right parameters which gives

minimum error. The selection of such parameters is done usually on trial –and –error

method. It is time consuming and laborious. In this paper a Genetic Algorithm based

approach is proposed to design parameters of the wavelet by minimizing the error

between the pattern and the designed wavelet. The method is illustrated with a

simulated sine wave for filter lengths of 4, 6, 8, and 10. The results are encouraging.

Keywords: Parametric wavelet design, GA, Pattern matching

1. INTRODUCTION

Compactly supported orthogonal systems of wavelets were introduced by

Daubechies, and she proved the existence of a multidimensional family of such wavelet

systems. She also proved specific wavelet systems with maximum vanishing moments

and with smoothness properties of a specific type. The wavelets are classified in a

rough manner by the nonvanishing coefficients in the fundamental difference

equation which defines them. This number is an integer N, N ≥ 2 and the support of the

scaling function has length N-1. As the number of coefficients increases, the support

gets larger and larger, and the smoothness also increases. In case if the wavelet system

INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET)

ISSN 0976 - 6480 (Print)

ISSN 0976 - 6499 (Online)

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is in C∞ then there are infinite number of coefficients and the support in the entire real

axis. In general, for an arbitrary even N, D = (N-2)/2 tells that there is a D – parameter

family of wavelet systems.

The parameterization has the advantage of global coordinates in R , the explicit

smoothness dependencies on the parameters and a simple geometry should facilitate

the optimization problems. The general goal would be to optimize the wavelet systems

adapted to a specific problem. There are two ways this can be made possible.

• Choose a wavelet system for an application, optimizing over Rn or

• Dynamically optimizing over R by modifying the choice of wavelet at successive

stages as the application progresses with respect to some parameter.

To develop a parameterized wavelet we will consider the properties of a wavelet. We

have to design an impulse response, i.e., filter coefficients and these filter coefficients

convolve with input signal to get the desired signal information. Effective filter design is

important to capture the desired information from a non-stationary signal.

2. LITERATURE SURVEY

Daubechies [1] Mayer[2] gives an insight of compactly supported orthonormal

wavelets. Daubechies [9] introduced a general method to construct compactly

supported wavelets based on scaling function which satisfy the dilation equation.

Parameterzing all possible filter coefficients that correspond to compactly supported

orthonormal wavelets has been studied by several authors [10, 11,12,14,15,16] A

discussion of scaling function with six filter coefficients depending on two parameters

were carried out in [3]and [4]. Paper [5] gives a discussion of parameterization of filter

coefficients with scaling functions and compactly support orthonormal wavelets with

several vanishing moments.[6] demonstrates a technique that determines the best

wavelet for each image from the class of orthogonal wavelets with fixed no of

coefficients.[7] presents a formal description of the algorithm for the construction of N

parametric equations.[8] gives a method for the construction of wavelet coefficients in

an algebraic extension field Q. Applications of parameterized wavelets to compression

are discussed in [17] and [13].

3. PROBLEM DEFINITION

The parameters of the parametric wavelets have to be found such that the error

between the pattern and the designed wavelet is minimum. Here, a sine wave is taken

for illustration and parametric wavelets are designed for lengths of 4, 6, 8 and 10.

The parameters of the wavelets are optimized for minimal error for pattern

matching application.

4. DESIGN OF PARAMETRIC WAVELETS

In this section we normalize the wavelet coefficients by setting hk =ak/2. We let

h = (h0 , h1 ,….hN-1) be a point of the moduli space MN, under this renormalization which

makes the analysis simpler and will eliminate many factors of powers of 2. Let h ∈MN,

then associated with h = (h0 , h1 ,….hN-1),is trigonometric polynomial H(ζ) =



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ζ. One method for constructing compactly supported scaling

k

x

(x) h (2x k)φ = φ −∑

with a finite number of coefficients. In order for a function of this type to be orthogonal

to its integer shifts, the coefficients in the dilation equation must satisfy H(z) = 1 and

|H(z)|2 +|H(-z)|

2 = 1, z = e

iw. Where H(z) = k

k

x

h z−∑ is the associated polynomial. The

necessary conditions for orthogonality yield a set of linear and nonlinear equations

which necessarily imply that various sums of dilation coefficients form perfect squares.

This leads to the introduction of parameters. Then we give the parameterization of

the length 4, 6, 8, 10 filters for completeness.

Length four Solution : Let H4(z) = a0 + b0z + a1z2

+ b1z3.

Lemma 1: H4(z) satisfies H4(1) =1and|H4(z)|2 +|H4(-z)|

2 =1 for all z = e

iw, w∈R if and

only if for all α∈R ; a0= 1 1cos

4 2 2+ α , a1= 1 1

cos4 2 2

− α , b0= 1 1sin

4 2 2+ α ,

b1= 1 1sin

4 2 2− α

Length six solution; H6(z) = a0 + b0z + a1z2 + b0z

3 + a2z

4 + b2z

5 be the trigonometric

polynomial of z = eiw

, w ∈ R

Lemma 2: H6(z) satisfies H6(1) = 1 and |H6(z)|2 + | H6(-z)|

2 = 1for all z = e

iw , w ∈ R

if and only if a0 = 1 1 pcos cos

8 24 2+ α − β , a1

= 1 1cos

4 2 2− α ,

a2 = 1 1 pcos cos

8 24 2+ α − β

b0 = 1 1 pcos sin

8 24 2+ α + β

b1 = 1 1sin

4 2 2− α ,b2 = 1 1 p

sin sin8 24 2

+ α − β

where p = for any α ,β € R

Solution for length 8

Lemma 3 :Suppose a, b , c , d ∈ R and a2+ b

2+ c

2+ d

2 = 1 if and only if

a = cos βcosγ, b = cosβcosγ , c = sin βcos θ, d = sin βsin θ for β, γ, θ ∈ R .

Let H8(z) =a0 + b0z + a1z

2 + b1z

3 + a2z

4 + b2z

5 + a3z

6 + z3z

7 be a trigonometric polynomial

of z = eiw

w∈R.

Lemma 4: H8(z) satisfies H8(1) = 1 and | H8(z)|2 +| H8(-z)|

2 = 1 for all z = e

iw, w∈R

if and only if a0 = 1 1 1cos cos cos

8 4 2 2 2+ α + β γ ,a1 = cos α + 1

2 2

cos β sin γ



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a2 = 1 1

8 4 2+ cos α - 1

2 2

cos β cos γ , a3 = cos α - 1

2 2

cos β sin γ

b0 = 1 1

8 4 2+ sin α + 1

2 2

sin β cosθ ,b1 = sin α + 1

2 2 sin β sin θ

b2 = 1 1

8 4 2+ sin α - 1

2 2

sin β cos θ ,b3 = sin α - 1

2 2

sin β sin θ where

γθβα ,,, ∈ R satisfy 2 cos θ sin β - 2 cos θ sin α sin β + 2 cos β (cos γ - sin γ) - 4

cos2β cos γ sinγ - 2 cos α cos β (cos γ + sin γ)- 2 sin β sin θ - 2 sin α sin β sin θ - 4 cos θ

sin2 β sin θ = 0.

Length 10 Solution

H10(z) satisfies H8(1) = 1and H8(z)2 + H8(-z)2

= 1 ∀ z = eiw

, w∈R

if and only if a0 = 1 1

16 8 2+ cos α + 1

4 2

cos β cos γ + r

2 cos δ

a1 = 1 1

8 4 2− cos α + 1

2 2 cos β cos , a2 = 1 1

8 4 2+ cos α - 1

2 2

cos β cos γ

a3 = 1 1

8 4 2− cos α - 1

2 2cosβcos γ,a4 = 1 1

16 8 2+ cos α + 1

4 2cos βcos γ - r

2 cos δ

b0 =1 1

16 8 2+ sinα + 1

4 2sinβcos θ +

r

2sin δ, b1 =

1 1

8 4 2− sin α + 1

2 2 sin β sin θ

b2 = 1 1

8 4 2+ sin α - 1

2 2 sin β cos θ, b3 = 1 1

8 4 2+ sin α - 1

2 2 sin β sin θ

b4 = 1 1

16 8 2+ sin α + 1

4 2 sin β cos θ - r

2 sin δ

where r = 2 2

i i

1a b

2− Σ − Σ and α, β, γ, δ, θ ∈ R satisfy

cos β [cos γ ( 2 -2 cos α) - 8 2 r cos δ sin γ)] +sin β [cos θ ( 2 -2 sin α) - 8 2 r sin δ sin

θ)] = 0

5. GENETIC ALGORITHM

Genetic algorithm or adaptive heuristic search algorithm is based on

evolutionary ideas of natural selection and genetics. It is a part of evolutionary

computing inspired by Darwin’s theory about evolution- “Survival of the fittest”. It uses

random search method to solve optimization problem. Although randomized it exploits

historical information to direct the search in to the region of better performance.

Genetic algorithms are good at taking large, potentially huge search pages and

navigating them, looking for optimal combinations of things, the solution one might not

find otherwise find in a lifetime [Salvatore Mangano, Computer design, May 1995].



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The optimization is a process that finds optimal solutions for a problem and is

centered around three factors:

1. An objective function which is to be minimized or maximized

2. A set of unknowns or variables that affect the objective function,

3. A set of constraints that allow the unknowns to take on certain values but

exclude others;

The GA optimizes a problem by mimicking the processes the nature uses, i.e.,

selection, cross-over, mutation and accepting. The Pseudo code of GA is given below:

BEGIN

INITIALISE population with random candidate solution.

EVALUATE each candidate;

REPEAT UNTIL (termination condition ) is satisfied DO

1. SELECT parents;

2. RECOMBINE pairs of parents;

3. MUTATE the resulting offspring;

4. SELECT individuals or the next generation;

END.

5.1 Outline of the Basic Genetic Algorithm

1. [Start] Generate random population of n chromosomes (i.e. suitable solutions for the

problem).

2. [Fitness] Evaluate the fitness f(x) of each chromosome x in the population.

3. [New population] Create a new population by repeating following steps until the

new population is complete.

(a) [Selection] Select two parent chromosomes from a population according to

their fitness (better the fitness, bigger the chance to be selected)

(b) [Crossover] With a crossover probability, cross over the parents to form new

offspring (children). If no crossover was performed, offspring is the exact copy

of parents.

(c) [Mutation] With a mutation probability, mutate new offspring at each locus

(position in chromosome).

(d) [Accepting] Place new offspring in the new population

4. [Replace] Use new generated population for a further run of the algorithm

5. [Test] If the end condition is satisfied, stop, and return the best solution in current

population

6. [Loop] Go to step 2.

6. RESULTS AND DISCUSSION

A sine wave is taken as a pattern for which a matching wavelet has been

designed using genetic algorithm. The parameters of the parametric wavelets are

derived and the final results alone are presented in section 4. The wavelets of filter co-

efficients of length 4, 6, 8 and 10 are considered in the present study. Section 4 gives

the general filter co-efficients that can be tuned to any pattern of interest. In the



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present study, the designed wavelets are tuned for the sine wave - a representative

pattern for illustration. Here, the tuning of wavelet parameter is modeled as an

optimization problem. The objective function is defined as the error between the

original pattern (sine wave) and designed wavelet. The number of variable parameters

will depend on the length of the filter co-efficients. For filter co-efficient length of 4,

the number of variable parameter is one and that happens to be ‘α’ (see section 4).

Similarly, for other lengths it may go up to 5 for the filter lengths considered here.

Tuning the wavelet for a pattern means finding the right values/ combination of values

such that the error is minimum. Hence, it is modeled as minimization problem. Matlab

optimization toolbox was used for optimization. The objective function here is an m-file

which calculates the error between the pattern (sine wave) and the designed wavelet.

The variable parameters are changed in random order and new error is found. The

process continues till we get a predefined small error or consecutively getting same

error for many iterations.

The Table 1 shows the simulation results of GA with filter co-efficient length of

four. The initial population is filled with random number. However, the optimization

converges in just 51 iterations giving α value of -4.05 with an error of 139.71. Here the

error is not expressed as percentage. It is root mean squared error (RMS error). The

error, however is high in this case.

Table 1. GA parameters for sine wave with filter length of 4

α Error No. of Iterations

-4.05 139.7156 51

-4.056 139.7165 51

-4.056 139.7166 51

-4.056 139.7166 51

-4.038 139.7165 51

-4.045 139.7164 51

The Table 2 shows the simulation results of GA with filter co-efficient length of

six. The initial population is filled with random number. However, the optimization

converges in just 58 iterations giving α value of -1.361 and β value of 1.044 with an

error of 5.7998 (RMS). The test results of various trials are shown in Table 2.

Table 2. GA parameters sine wave with filter length of 6

α β Error No. of

Iterations

-1.361 1.044 5.7998 58

-1.386 1.047 6.3047 67

-1.37 1.085 10.7789 59

-1.369 1.023 6.4553 51

-1.394 1.023 5.997 51



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The Table 3 and Table 4 show the simulation results of GA with filter co-efficient

length of eight and ten respectively. The initial population is filled with random number

and results of many trials were recorded. The optimization converges faster in case

length 10 compared to that of length 8; however, the error is very high in case of length

10. On careful observation, from Table 3, one can find GA has converged once with an

RMS error of 314.36. This is because GA has found the local minimum while the

solution that is in search is global minimum. This is a general problem with GA and this

can be solved by using multiple trials of GA. In the present study, the multiple use of

GA gives better result with an RMS error of 0.002. Looking for the similar pattern in

Table 4, it is clear that GA finds it difficult to optimize and give closed values. For the

chosen pattern, filter length of eight gives best results. The optimal values of each filter

co-efficient lengths is plotted against the pattern and shown in Fig. 1. One can confirm

pictorially from looking at the waveform matching for pattern matching application.

Table 3. GA parameters sine wave with filter length of 8

α β Γ Θ Error No. of

Iterations

-7.274 -4.79 -0.263 0.355 0.0028 5

-3.995 0.45 -0.119 1.062 0.1538 5

-6.481 0.377 -0.023 1.409 0.006 5

0.597 -0.724 2.904 1.137 314.36 5

-6.297 0.383 0.968 1.316 0.0073 5

Table 4. GA parameters with filter length of 10

α β Γ δ θ Error No. of

Iterations

0.292 0.539 -0.001 0.003 0.426 345.2606 2

0.785 1.775 0.922 0.218 0.65 180.4064 2

1.708 1.224 0.132 0.023 2.309 162.5255 2

0.786 -0.125 0.144 1.232 0.881 216.7954 2

0.1785 20.442 -26.782 -21.935 6.656 187.8904 4

1.796 -1.824 -1.475 -1.485 -0.253 68.9987 4

1.178 4.123 0.002 -0.001 -2.212 197.044 3



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Fig. 1 Pattern and designed wavelets

7. CONCLUSION

In the present study a parametric based wavelet design is taken up. The

parameters of the parametric wavelets were derived for filter lengths of 4, 6, 8 and 10.

These parameters are optimized for a chosen illustrative pattern using GA. From the

above results and discussion, one can easily conclude that GA can be used effectively

for design of parametric wavelets. Further, in this specific case, length eight gives best

wave let for pattern matching applications.

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Optimizing the parameters of wavelets for pattern matching using ga