chapter 4 optimization of wavelet and bilateral filter based...

Chapter 4

Optimization of Wavelet and

Bilateral Filter based Denoising

Models using Genetic Algori thm

4.1 Introduction

Algorithms for function optimization are generally limited to convex regular func

tions. However, many functions are multimodal, discontinuous and nondifferen-

tiable. Stochastic sampling methods have been used to optimize these functions.

Whereas traditional search techniques use characteristics of the problem to deter

mine the next sampling point (e.g., gradients, hessians, linearity and continuity),

stochastic search techniques make no such assumptions. Instead, the next sampled

points are determined based on stochastic sampling decision rules rather than a set

of deterministic decision rules. Genetic algorithms have been used to solve difficult

problems with objective functions that do not possess "nice" properties such as

continuity, differentiability, satisfaction of the Lipschitz Condition, etc. [97]-[100].

These algorithms maintain and manipulate a family, or population of solutions

and implement a "survival of the fittest" strategy in their search for better so

lutions. This provides an implicit as well as explicit parallelism that allows for

the exploitation of several promising areas of the solution space at the same time.

The implicit parallelism is due to the schema theory developed by fioiland, while

the explicit parallelism arises from the manipulation of a population of points -

96

4.2. Gene t ic Algor i thms 97

the evaluation of the fitness of these points is easy to accomplish in parallel.

This provides the genetic algorithm with a local improvement operator which

is shown in [101], can greatly enhance the performance of the genetic algorithm.

Many researchers have shown that GAs perform well for a global search but per

form very poorly in a localized search [97], [100]-[102]. GAs are capable of quickly

finding promising regions of the search space but may take a relatively long time

to reach the optimal solution.

Genetic algorithms are different from normal search methods employed in

engineering optimization in the following ways:

(i) GAs work with a coding of the parameter set not the parameters themselves,

(ii) GAs search from a population of points, not a single point,

(iii) GAs use probabilistic transition rules, not deterministic transition rules.

A typical GA search starts with a random population of individuals. Each

individual (a string) is a coded set of parameters that we want to optimize. An

objective (fitness) function is used to determine how fit these individuals are. A

suitable selection strategy is employed to select the individuals that will reproduce.

Reproduction (or a new set of individuals) is achieved by using crossover and

mutation operators. The GA then manipulates the most promising strings in its

search for improved solutions.

Section 4.2 presents the basic genetic algorithm and section 4.3 briefly de

scribes the proposed approach and its Matlab implementation and discussion on

the results after the use of the GA is made in section 4.4.

4.2 Genetic Algorithms

Genetic algorithms search the solution space of a function through the use of

simulated evolution, based on the survival of the fittest strategy. In general, the

fittest individuals of any population tend to reproduce and survive to the next

generation, thus improving successive generations. However, inferior individuals

can, by chance, survive and also reproduce. Genetic algorithms have been shown

to solve linear and nonlinear problems by exploring all regions of the state space

4.2. Genetic Algorithms 98

and exponentially exploiting promising areas through mutation, crossover and

selection operations applied to individuals in the population [100].

Though details of implementation of the algorithm are varied but a generic

view of a GA would include:

1. [Start] Generate random population of n chromosomes (suitable solutions

for the problem). Initialization of the initial population (either randomly or

from a best guess or previous partial solution).

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.

4. [Selection] Select two parent chromosomes from a population according to

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

5. [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.

6. [Mutation] With a mutation probability mutate new offspring at each locus

(position in chromosome).

7. [Accepting] Place new offspring in the new population.

8. [Replace] Use newly generated population for a further run of the algorithm.

9. [Test for termination] If the condition for termination is satisfied, stop, and

return the best solution in current population, otherwise go to next step.

10. [Loop] Go to step 2.

The use of a genetic algorithm requires the determination of six fundamen

tal issues: chromosome representation, selection function, the genetic operators

making up the reproduction function, the creation of the initial population, ter

mination criteria, and the evaluation function. These are discussed in subsequent

sections.


4.2.1 Solution Representation

For any GA, a chromosome representation is needed to describe each individual

in the population of interest. The representation scheme determines how the

problem is structured in the GA and also determines the genetic operators that

are used. Each individual or chromosome is made up of a sequence of genes from

a certain alphabet. An alphabet could consist of binary digits (0 and 1) and

floating point numbers, integers, symbols (i.e.. A, B, C, D), matrices, etc. In

Holland's original design, the alphabet was limited to binary digits. Since then,

problem representation has been the subject of much investigation. It has been

shown that more natural representations are more efficient and produce better

solutions [100]. One useful representation of an individual or chromosome for

function optimization involves genes or variables from an alphabet of floating point

numbers with values within the upper and lower bounds of variables. Michalewicz

[100] has done extensive experimentation comparing real-valued and binary GAs

and shown that the real-valued GA is an order of magnitude more efficient in

terms of CPU time. He also shows that a real-valued representation moves the

problem closer to the problem representation which offers higher precision with

more consistent results across replications [100].

4.2.2 Selection Function

The selection of individuals to produce in successive generations plays an extremely

important role in genetic algorithm. A probabilistic selection is performed based

upon the individual's fitness such that the better individuals have an increased

chance of being selected. An individual in the population can be selected more

than once with all individuals in the population having a chance of being se

lected to reproduce into the next generation. There are several schemes for the

selection process: roulette wheel selection and its extensions, scaling techniques,

tournament, elitist models and ranking methods [98], [100]. A common selection

approach assigns a probability of selection, Pj, to each individual, j based on its

fitness value. A series of N random numbers are generated and compared against

the cumulative probability Cj = Yl]=i P] ' of * hc population. The appropriate in

dividual, ?:, is selected and copied into the new population if Q - i < [/(0,1) < Q .

4.2. Genet ic Algor i thms 100

Various methods exist to assign probabihties to individuals: roulette wheel, linear

ranking and geometric ranking. Roulette wheel, developed by Holland [99] was

the first selection method. The probability, Pi, for each individual is defined by:

P[Individual i is choosen] = —PovLze— i^-^) E,=i F,

where Fj equals the fitness of individ\ial i. The use of roulette wheel selec

tion limits the genetic algorithm to maximization since the evaluation function

must map the solutions to a fully ordered set of values on 3?"*". Extensions, such as

windowing and scaling, have been proposed to allow for minimization and nega

tivity. Ranking methods only require the evaluation function to map the solutions

to a partially ordered set, thus allowing for minimization and negativity. Ranking

methods assign Pi based on the rank of solution i when all solutions are sorted.

Normalized geometric ranking [103], defines P, for each individual by:

P{Selecting the z"* individual) = q'{l - qY~^ (4.2)

where

q ~ the probability of selecting the best individual.

r = the rank of the individual, where is the best.

P = the population size.

Tournament selection, like ranking methods, only requires the evaluation

function to map solutions to a partially ordered set, however, it does not assign

probabilities. Tournament selection works by selecting j individuals randomly,

with replacement, from the population, and inserts the best of the j into the new

population. This procedure is repeated until TV individuals have been selected.

4.2.3 Genetic Operators

Genetic Operators provide the basic search mechanism of the GA. The operators

are used to create new solutions based on existing solutions in the population.

There are two basic types of operators: crossover and mutation. Crossover takes

two individuals and produces two new individuals while mutation alters one indi

vidual to produce a single new solution. The application of these two basic types

4.2. Genet ic Algorithms 101

of operators and their derivatives depends on the chromosome representation used.

Let X and Y be two m-dimensional row vectors denoting individuals (par

ents) from the population. For X and Y binary, the following operators are defined

binary mutation and simple crossover.

Binary mutation flips each bit in every individual in the population with

probability pm according to equation (4.3).

.X = < (4.3) 1-Xi, i{U{0,l)<Pm

Xi, otherwise

Simple crossover generates a random number r from a uniform distribution

from 1 to m and creates two new individuals X and Y according to equations

(4.4) and (4.5).

.X = <

Xi, if i < r

y,, otherwise

Vi if z < Vi

(4.4)

(4.5) Xi, otherwise

Operators for real-valued representations, i.e., an alphabet of floats, wore

developed by Michalcwicz [100]. For real X and Y the following operators are

defined: uniforiri mutation, non-uniform mutation, multi-non-uniform mutation,

boundary mutation, simple crossover, arithmetic crossover and heuristic crossover.

Let a, and bi be the lower and upper bound, respectively, for each variable i.

Uniform mutation randomly selects one variable, j and sets it equal to an uniform

random number U{ai,bi):

X = < (4.6) U{ai,bi), iii = j

Xi, otherwise

Boundary mutation randomly selects one variable, j , and sets it equal to

either its lower or upper bound, where r = f/(0,1):


.X- = <

flj, \i i = j , r < 0.5

bi, if z = j , r > 0.5

rr,. otherwise

(4.7)

Non-uniform mutation randomly selects one variable, j , and sets it equal to

an non-uniform random number:

X = <

Xi + {bi-x^)f{G), i f r i < 0 . 5

Xi-{xi + ai)f{G), i f r i > 0 . 5

Xi, otherwise

(4.8)

where

fiG)= [Ml-G

G„ (4.9)

r l , r 2 =: a uniform random number between (0,1),

G = the current generation,

Gmax = the maximum number of generations,

6 = a shape parameter.

The multi-non-uniform mutation operator applies the non-uniform operator

to all of the variables in the parent X. Real-valued simple crossover is identical

to the binary version presented above in equations (4.4) and (4.5). Arithmetic

crossover produces two complimentary linear combinations of the parents, where

r = U{0,1)

X' = rX + {I - r)Y (4.10)

Y' = {l-r)X + rY (4.11)

Heuristic crossover produces an linear extrapolation of the two individuals.

This is the only operator that utilizes fitness information. A new individual, X', is

created using equation (4.12) where r = f/(0,1) and X is better than Y in terms

of fitness. If X' is infcasible i.e., feasibility equals 0 as given by equation (4.14)

then generate a new random number r and create a new solution using equtxtions


(4.12) and (4.13) otherwise stop. To ensure halting, after t failures, let the children

equal the parents and stop.

X' = X + riX -Y) (4.12)

Y' = X (4.13)

1, if x[ > a,, x[ < bi, \/i feasibility = ^ (4.14)

0, otherwise

4.2.4 Initialization, Termination, and Evaluation Functions

The GA must be provided with an initial population as indicated in step 1 of the

algorithm. The most common method is to randomly generate solutions for the

entire population. However, since GAs can iteratively improve existing solutions

(i.e., solutions from other heuristics and/or current practices) the beginning pop

ulation can be seeded with potentially good solutions, with the remainder of the

population being randomly generated solutions. The GA moves from generation

to generation selecting and reproducing parents until a termination criterion is

met. The most frequently used stopping criterion is a specified maximum num

ber of generations. Another termination strategy involves population convergence

criteria. In general, GAs will force much of the entire population to converge to

a single solution. When the sum of the deviations among individuals becomes

smaller than some specified threshold, the algorithm can be terminated. The al

gorithm can also be terminated due to a lack of improvement in the best solution

over a specified number of generations. Alternatively, a target value for the evalua

tion measure can be established based on some arbitrarily acceptable "threshold".

Several strategies can be used in conjunction with each other. Evaluation func

tions of many forms can be used in a GA, subject to the minimal requirement

that the function can map the population into a partially ordered set. As stated,

the evaluation function is independent of the GA (i.e., stochastic decision rules).

For the optimization of the test function two different representations are

used. A real valued alphabet is employed in conjunction with the selection, mu-

4.3. Proposed Approach 104

tation and crossover operators with their respective options. Also a binary repre

sentation may be used in conjunction with the selection, mutation and crossover

operators with their respective options.

The floating point genetic algorithm (FPGA) is used as the optimization

tool throughout this work.

To optimize the performance, the fitness function is designed in such a man

ner that both the values of the PSNR and IQI should be maximum. The fitness

function is also formed using PSNR and IQI with equal weightagc. The PSNR

is calculated by equation (2.49) and IQI by equation (2.50) for all the considered

models with the considered parameters as discussed above.

4.3 Proposed Approach

Out of 48 models formed through hybridization of wavelet baaed filtering and

bilateral filters in different configurations as reported and discussed in chapter 3,

the best three models are considered for optimization. In addition, three models,

one with only wavelet based filters, the other with only bilateral filter and the third

one as proposed by Zhang and Gunturk [8] are considered for drawing comperative

performance. The parameters of six (three plus three) filter models are optimized

using floating point GA (FPGA).

Structural variations of the models are shown in figs. 4.1 through 4.6. The

models are identified as model nos. 1 to 6 as shown in the figs. 4.1 through 4.6

respectively.

(i) Model with Wavelet thresholding based method only (model no. 1 in fig. 4.1).

(ii) Model with Bilateral filter only (model no. 2 in fig. 4.2).

(iii) Model proposed by Ming Zhang and Bahadur Gunturk (Zhang-Gunturk

method) (model no. 3 in fig. 4.3).

(iv) Hybridized model with bilateral filter after the reconstruction of the image

after the completion of the wavelet decomposition and thresholding (model

no. 4 in fig. 4.4).


(v) Hybridized model with bilateral filter before and after the wavelet decom

position, thresholding and reconstruction (model no. 5 in fig. 4.5).

(vi) Hybridized model with bilateral filter before the wavelet decomposition (model

no. 6 in fig. 4.6).

Input . image

r-^ LL - < , , ) - ^ W - < >-^

^ LH -^ly^ w -<T)-

-^ HL -<i>-> w -^y^

U HH -^i)-^ W -<f)-^

7> Output image

I S I BUateral filter

I W I Wavelet thresholding

Figure 4.1: Model No. 1

Input image B Output

image

m Bilateral filter

"y^ I Wavelet thresholding


While working with the models, the parameters w, ad and ar of bilateral

filters are varied over a wide range of values as there is no explicit rules that can

guide the tuning of these parameters. At the same time the threshold value for

the wavelet based filter is also varied.

The following issues are considered to carry out the work:

(i) Different standard availal)le images along witli satellite and tc;I(;scoj)ic images

are considered throughout this work. All the models are applied on each and


Input image

Input inuge

> B

KHGHZM)-I H } - > ( D - ® - < [ H

lO-^Q-^wHIH HJTMD- rwHD-J

B

Bilateral filter

I W Wavelet thresholding

Bilateral filter H] I 'W? I Wavelet thresholding


^ B —> LH

^ LL HIHwHIH - (I)-fwH!>~>

H T M I H W H T ) ^

HH h®--[wMH

^ B

Bilateral filter

I W^ I Wavelet thresholding

^ Output image

Input image


p LL ^Q-^[wHlH - LH - < i ) - w -Kiy-^

-^ HL -<i)-^ w Aiy^ U HH -^1)-^ W ^Cf)-

-^ B ^ r>iitpirt

image

Output image


(ii) (a) After wavelet decomposition, soft thresholding or bilateral technique is

applied on each and every subband of the image,

(b) For a particular image and a model, a wide range of values of w, 04

4.4. Results and Discussion 107

Input image B

L L ha>-s-®- L H

H L

H H

h-0--[wHD-h^H^HD-|-aHw]-©-J

Output image

I Q I Bilateral filter

I W I Wavelet thresholding


and Or including thresholding parameter are considered for evolutionary op

timization using FPGA.

(iii) The threshold value for wavelet based filters is also optimized with FPGA.

(iv) The values of iw, a^ and Or of bilateral filter are aJso optimized using FPGA.

4.4 Results and Discussion

The first model is developed with wavelet based thresholding algorithm [104]. In

this model, the images are decomposed into four subbands. The soft thresholding

method is then employed on each of the four subbands wherein the thresholding

value is optimized using FPGA. As most of the researchers have used ^68 filters for

image denoising, the same has been considered in this work too. The second model

is with bilateral filter [88] only. The parameters, w, aj. and ar are optimized using

FPGA for finding the optimal performance. The third one is the model proposed

by Ming Zhang and Bahadur Gunturk [8]. As proposed by the authors, the images

arc decomposed into four subbands and the both wavelet based and bilateral filters

are used as given in model no. 3 in fig. 4.3.

The models from 4 to 6 are hybridized ones that are reported in the previous

chapter.

In model 4, the images are first decomposed into four subbands using dbS

filters in Matlab. In this level l,h(> wavc lot ba-scnl soft throsholdiug is api:)li(Kl on all


the subbands. The results obtained after thresholding are then used to reconstruct

the image. In the next level, bilateral filter is applied to get the final denoised

image.

In model 5, first the image is denoised with bilateral filter followed by decom

position into four subbands using dbS filters. And wavelet thresholding is applied

on all the subbands. The results obtained after thresholding are then used to

reconstruct the image. In the last level, again bilateral filter is applied to get the

final denoised image.

Model 6 is similar to model 5 except there is no bilateral filter at the output

side of model 6. The wavelet thresholding is applied on all the subbands. The

results obtained after thresholding are then used to reconstruct the denoised image.

All the models are experimented with standard pictures like Lena, Barbara,

Einstein, satellite picture like pyramid and two astronomical telescopic images

Astrol and Astro2.

Model

Model 1

Model2

Models

Image

Lena

Barbara

Einstein

Pyramid

Astrol

Astro2

Lena

Barbara

Einstein

Pyramid

Astrol

Astro2

Lena

Barbara

GA optimized

PSNR

39.0803

39.0103

39.0146

36.7269

40.5669

38.4527

43.3514

44.5522

43.3897

46.8700

41.6157

41.8386

36.1404

25.3825

IQI

0.9692

0.9788

0.9748

0.9967

0.9633

0.9503

0.9800

0.9880

0.9846

0.9991

0.8743

0.9513

0.9412

0.9550

Not GA optimized

P S N R

35.1217

34.9942

35.0108

34.2453

35.1464

35.1210

38.3243

39.2712

38.3424

38.7753

39.3953

39.3918

30.4803

17.5528

IQI

0.9222

0.9464

0.9358

0.9925

0.8666

0.8903

0.9260

0.9540

0.9677

0.9953

0.7698

0.9058

0.9465

0.9080

Continued on next page

4.4. Results and E >iscussion 109

Table 4.1 - continued from previous page

Model

Model4

Models

Modeie

Image

Einstein

Pyramid

Astrol

Astro2

Lena

Barbara

Einstein

Pyramid

Astrol

Astro2

Lena

Barbara

Einstein

Pyramid

Astrol

Astro2

Lena

Barbara

Einstein

Pyramid

Astrol

Astro2

GA optimized

P S N R

28.9590

13.7369

12.7607

19.5767

45.0978

50.4463

38.4985

45.1144

45.0999

45.5710

39.4876

41.3640

40.7318

45.4617

40.2209

39.7477

49.2344

44.8573

47.4786

51.4413

48.1736

41.6635

IQI

0.9564

0.8383

0.4413

0.6030

0.9899

0.9974

0.9880

0.9994

0.9550

0.9853

0.9583

0.9755

0.9697

0.9992

0.9998

0.9333

0.9954

0.9907

0.9951

0.9998

0.9909

0.9554

Not GA optimized

P S N R

24.9062

12.3090

4.9742

5.7261

40.6265

40.3110

32.9937

36.9293

34.9593

35.1238

38.6548

39.7834

39.4389

39.3131

36.8370

35.4243

39.5561

39.6969

39.5870

39.7432

39.8931

39.6700

IQI

0.9386

0.7857

0.3355

0.5833

0.9683

0.9777

0.9800

0.9969

0.8227

0.9276

0.9541

0.9677

0.9696

0.9979

0.9998

0.8905

0.9664

0.9777

0.9740

0.9976

0.8539

0.9359

Table 4.1: Performance Comparison of the Different

Models with Different Images

The value of soft-thresholding and w, a,i and ar for models 3 to 6 are opti

mized for optimal performance using FPGA.

4.4. Resul t s and Discussion 110

Floating point GA (FPGA) features used:

(i) Heuristic crossover

(ii) Non-uniform mutation

(iii) Norm.alized geometric select function

Tuned parameters for the FPGA algorithm:

Population Size = 20

Maximum Iterations = 200

Penalty multiplier = 10.

The GA toolbox, GAOT, in Matlab as proposed by C.R. Houck, J. Joines,

and M. Kay [105] is used after modification for the optimization of the problem.

The PSNR and IQI values obtained for all the models with filter parameters

optimized using FPGA for all the images are given in table 4.1.

The performance in terms of PSNR and IQI is illustrated in figs. 4.13 and 4.14

respectively.

The six images, Lena, Barbara, Einstein, Pyramid, Astrol and Astro2, that

are denoised by the six models without FPGA optimization and with FPGA opti

mization, are depicted in the figs. 4.7, 4.8, 4.9, 4.10, 4.11 and 4.12 respectively.

Figure 4.7: Denoised images by Model 1: Upper row is without FPGA and Lower

row is with FPGA

It is evident in figs. 4.13 and 4.14 that there is significant improvement in the

performance of most of the models because of optimization of the filter parameters

using FPGA in lieu of manual trial and error method as reported in the previous

chapter.


E , , • " . - . ? - " , j ;


row is with FPGA


row is with FPGA


row is with FPGA

All the models have been found to be capable in denoising all the images

to some level. However, there is some difference in their performance in terms

of PSNR and IQL The comparative performances of FPGA optimized different

models on diffbrcnt tyi)es of images are depicted bellow.

For standard image Lena, model 6 achieved the best PSNR and IQI values

followed by model 4. Models 5, 2, 1 and 3 follow in descending order in terms

of PSNR as is evident in figs. 4.13 and 4.14 and in table 4.1. In the case of

Barbara, the performance of the model 4 is the best in terms of PSNR and IQI,


c


row is with FPGA


row is with FPGA

followed by model 6 and 2 in terms of PSNR but in terms of IQI model 6 is better

than model 2. For image Einstein, model 6 generates the best PSNR as well as

best IQI.

When the satellite image, Pyramid, is considered, the best model in terms

of PSNR and IQI is again 6 followed by model 2. Models 4 and 5 follow them.

However, all the models except model 3 have performances in terms of IQI only.

For the astronomical telescopic image, Astrol, model 6 gives the best results

in terms of PSNR while model 5 performs best in terms of IQI. And for another

image Astro2, model 4 achieves the best results in terms of PSNR and IQI.

Generally it is observed that the overall performance of model 6 is more

consistent especially in terms of IQI on all types of images foUowed by model 4.

And the performance of model 3 is worst on all the images.

Another general observation of the overall comparative performance among

the six models investigated in this work is that the performance of the filter with

the wavelet based thresholding filters only on all the decomposed subbands of any

image is far better than the model with bilateral filters applied in different ways

4.4. R e s u l t s and Discussion 113

60

50

40

Performance Comparison( PSNR)

GA NGA GA NGA GA NGA GA NGA GA NGA GA NGA

M o d e l l Model2 Models Mode(4 Models Modet6 I

Models—> (GA: GA opt imized, NGA: Not GA opt imized)

• Lena

K Barbara i I

: Bnstein I

K Pyranrid j

s A s t r o l 1

A5tro2 I

Figure 4.13: Performance comparison of the models with and GA optimization in

terms of PSNR

Performance Comparison (IQI)

I.

GA NGA GA NGA GA NGA GA NGA GA NGA GA NGA

M o d e l l Model2 Mode ls Model4 Models Models

Models—> {GA: GA opt imized, NGA: Not GA opt imized)

I Lena

I Barbara

Einstein

i Pyramid

I A s t r o l

Astro2

Figure 4.14: Performance comparison of the models with and GA optimization in

terniB of IQI

along with wavelet based thresholding filters on the decomposed subbands. But

when the bilateral filter is used before or after or both sides of the decomposition

(not on decomposed subbands) followed by reconstruction of an image as used

in models 4, 5 and 6, the performance of the filters improves as also reported in

the previo\is chapter. After FPGA optimization model 4 has transformed to be a

strong competitor for model 6 for most types of the images.

4.5. Conclusion 114

4.5 Conclusion

The filter parameters of all the filters including hybrid denoising models developed

through hj'bridization in three apparently best configurations are optimized using

FPGA and the performances of the thus GA based optimally designed filters are

tested on different types of noisy images. The performance of the models is eval

uated in terms of PSNR and IQI. Comparison is drawn in the performance of all

the six models with filter parameters optimized using trial and error method with

those with filter parameters optimized using FPGA. It has been observed that

the performance of all the models improves with FPGA optimized filter parame

ters even though the amount of improvement in different models is different. The

overall observation of the results reveals that the use of bilateral filters in diflFerent

configurations along with wavelet based thresholding filters on all the decomposed

subbands worsens the performance of the filter, whereas the bilateral filter be

fore or after or both sides of the decomposition (not on decomposed subbands)

followed by reconstruction of an image as used in models 4, 5 and 6, improves

the performance of the filter in denoising noisy images which is in line with the

inference claimed in previous chapter.

However, amongst the three models 4, 5 and 6, the performance of model 4 is

quite competitive with reference to model 6, which is better and more consistent on

all the images except on Astro2 type images. So, the model 6, wherein the bilateral

filter is applied before decomposition, is recommended as a well competent model

for denoising any type of images.

chapter 4 optimization of wavelet and bilateral filter based...

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