chapter 15 7

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CHAPTER IV

RESTORATION IN TRANSFORM DOMAIN

4.1 INTRODUCTION

Image transform plays an important role in digital image processing system.

Various image transforms such as Discrete Cosine Transform (DCT) [191][192], Discrete

Sine Transform (DST) [193], Singular value Decomposition (SVD) transform [194].

Curvelet transform (CT), and Discrete Wavelet Transform (DWT) [66][108] are

employed for various scientific and engineering applications. The DCT (2D) is a very

efficient transformation technique for achieving sparse representation of digital image

blocks for natural images. Its performance is very close to the optimum karhunen-loeve

transform (KLT) [02][195]. Thus, the DCT has been successfully implemented as the key

element in the various image-processing applications [196]. However, in the presence of

singularities or edges in an image, such near-optimality fails due to lack of sparsity, edges

cannot be restored effectively, and ringing artifacts arising in the restored image.

Meanwhile, within the category of DCT based restoration, it is observed that

sophisticated strategies such as shape adaptation [104][196] and weighted estimation

[197] have their own merits but the gain is often modest. So, the transform domain image

restoration techniques can be further classified based on linear, nonlinear operations and

their coefficient modeling.

4.2 CLASSIFICATION OF TRANSFORM DOMAIN RESTORATION

There are two basic approaches to image restoration, spatial domain restoration

and transform domain techniques. In transform domain, DWT is one of the powerful

mathematical CAD tool in image processing system. Image restoration techniques using

wavelet transform are effective because of its ability to capture most of the energy of

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image signal in a significant transform coefficient. A particular technique that has been

introduced for considerable attention in the last decades based on wavelet thresholding or

shrinkage: technique to killing coefficients of less significant which having less

magnitude relative to some threshold. From the literature of the transform domain

restoration techniques, we can classify various techniques in the different categories

according to their basis functions. Classification of restoration techniques in the

transform domain and outline of this chapter is as shown in figure 4-1.

Transform Doamin

Image Restoration Techniques

Non-Adaptive

Wiener Filtering

Technique

Perspective, Analysis and Conclusion

Nonlinear Restoration

Techniques

(Threshold Filtering)

Wavelet Coefficient

Model

Non-Orthogonal

Wavelet Transform

Linear Restoration

Techniques

Adaptive

FoE/Gaussian FoE

Restoration Technique

Visu Shrink

BayesShrink

SUREShrink

Hard/Soft/Global

Threshold

Bilateral/ NL Mean

LMMSE Filtering

Technique

Random Marquo

FIeld (MRF)UnDecimated

Wavelet

Transform

Tetrolet Transform

Contourlet Transform

Figure 1: Classification of Image Restoration Techniques in spatial domain and

outline of this chapter.

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4.2.1 LINEAR RESTORATION TECHNIQUES

Linear restoration process in the transform domain such as wiener filter in the

wavelet domain yield optimal results when the image signal degradation can be modeled

as a Gaussian process and accuracy criterion is the MSE [198][199]. This particular

filtering operation successfully minimizes the MSE [200]. The field of expert (FoE)

technique is one of the linear restoration processes using wavelet transform. It has very

heavy-tailed derivative histograms, and response from random linear filters have very

heavy-tailed responses [201].

4.2.2 NON-LINEAR RESTORATION TECHNIQUES

Non-linear thresholding filtering techniques can be divided into two categories

based on number of data points: one is non-data adaptive thresholding and second is

adaptive thresholding filtering technique. Most of the investigated image restoration

techniques in transform (Wavelet) domain are the non-linear coefficient thresholding

based techniques. The procedure in which small transform coefficients are removed while

others are left untouched is called hard thresholding [111]. To overcome the demerits of

hard thresholding, wavelet transform using soft thresholding was also introduced [111].

The various thresholding and shrinkage techniques proposed in the literature are

VisuShrink [111] [112], SureShrink [113][115], Bayes Shrink [114], hard and soft

thresholding [202], and Global thresholding [203] etc. The windowing techniques such as

LAWML are also available in the literature where statistical relationship of transform

coefficients in a neighborhood is considered while restoring an image. The wavelet

domain methods are suitable in retaining the detailed structure, some time they

introduced Mat-like structures in the smooth region of the restored image [203][204]. The

predominant nonlinear techniques in transform domain are explained in the next

subsection of this chapter.

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4.2.2.1 HARD THRESHOLDING

These methods used to determine the clean wavelet coefficients based on

thresholding. If the absolute value of coefficient is less than a threshold, then it is

assumed zero, otherwise it is unchanged. Mathematically it is represented as below

[111][205]:

ˆ ( )( .*( ( ) )..........(4.1)x sign y y abs y

4.2.2.2 SOFT THERESHOLDING

Hard thresholding causes Gibbs effect in the restored image. To overcome the

same, Donoho [111][112] introduced the soft thresholding method. If the absolute value

of a wavelet coefficient is less than a threshold ' ' then it is assumed to be zero

otherwise its value is shrunk by ' ' . It can be represented as given below:

ˆ ( ).*(( ( ) )*( ( ))).........(4.2)x sign y abs y abs y

This removes the discontinuities, but degrades all the other coefficients which

tends to blur image.

4.2.2.3 VISUSHRINK

This is also called as universal thresholding technique. A threshold given by as

given below:

2log( ).........(4.3)universalT M

Where, ‘M’ is the number of samples, and it is asymptotically yields a mean square error

estimate as ‘M’ tends to infinity [114][205].

4.2.2.4 SURESHRINK

SureShrink is an adaptive thresholding technique where the transform coefficient

are treated in level-by-level fashion [205]. In each particular level, when there is an

information that the wavelet representation of that level is not sparse [205], a threshold

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that minimizes Stein’s Unbiased Risk Estimate (SURE) is applicable. It is used to

suppress the degradation in transform domain where the threshold is employed for

restoration. In this case, threshold parameter SURET can be expressed as below:

argmin ( ( ; ))..........(4.4)Sure THT Sure TH Y

2 2 2

1

1( ; ) 2 .* : ( , ) ..........(4.5)

L

n n i i

i

Sure TH Y i Y TH Min Y THL

This implies that the reconstruction is smooth wherever the function is smooth and it

jumps wherever there is discontinuity in the function. This method can generate very

sparse wavelet coefficient resulting in an inadequate threshold [205].

4.2.2.5 BAYESSHRINK

This restoration technique is based on the Bayesian mathematical framework. A

generalized Gaussian distribution (GGD) models the wavelet coefficients of natural

images. This is used to calculate the threshold using a Bayesian framework [114]. S.

Grace Chang et. al. [114] has been presented an approximation and simple formula for

the threshold as given below: 2( )

.........(4.6)nh

s

T

If s is non-zero otherwise it is set to some predetermined maximum value.

2 2(( ) ( ) , )..........(4.7)s y nMax o

21( ..........(6.8)y nW

N

The noise variance n is estimated from HH band [114][205] as median ( ) 0.6745,nW

where nW represents the wavelet coefficient after subtracting the mean [206].

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4.2.2.6 SHRINKAGE TECNHIQUES

These types of techniques based on shrink the wavelet coefficient given:

ˆ( .* ), : 0 1x y y where y . It is the shrinkage factor. Further, some important methods

belong to this class has been explained.

4.2.2.7 MMSE TECNHIQUES

Michak et. al. have been proposed the linear minimum mean square estimation

(MMSE) method using a locally estimated variance [200]. An optimal predictor for the

clean wavelet transform coefficient at location ' 'k is given below:

2 2 2

, .ˆ *( ) / ( )..........(4.9)k k x k x k nx y

Where, .x k is the image signal variance estimated at location ' 'k and n is the

degradation variance, ky represents the noisy coefficients and ˆkx represents the

estimated wavelet transform coefficients. There are two approaches are presented to

estimate the local variance. The first approach is used an approximate maximum

likelihood estimator (MLE) as shown below: 2 2

( , ) ( . )arg max ( )..........(4.10)x k i x kP y

The second is used the maximum a posteriori estimator (MAPE) as given below: 2 2 2

( , ) ( , ) ( , )arg max( ( ( ))). ( )..........(4.11)x y i x k x kP y P

Where 22 ( )

( , )( ) .x kP e is empirically chosen [207][208].

4.2.2.8 HAAR WAVELET TRANSFORM BASED TECNHIQUES

The Haar transform is one of the most simple wavelet transform. The scaling and

wavelet function for Haar transform are defined as follows:

1 0 1

0( ) ..........(4.12)for t

otherwiset

1 0 0.5

0.5 0.5 1( ) ..........(4.13)for t

for tt

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The above equations shows the scaling and wavelet function at different scales and

translation indices [31][207].

4.2.2.9 TETROLATE TRANSFORM BASED TECNHIQUES

Jens Krommweh et. al. [208] has been proposed a new method for image

restoration using an adaptive Haar transform. It is also called as tetrolate transform. In

this restoration process, images are subdivided into 4x4 blocks. The features of this

image restoration technique are as follows: i) simplicity ii) less storage iii) redundant

coefficient iv) scalability. A finer image restoration technique is tetromino partitions are

picked and the averages of such restored images are taken [207].

4.2.3 WAVELET COEFFICIENT MODEL

Under this particular approach focused on exploiting the multiresolution

properties of wavelet transformation technique. Its identification is to close correlation of

image signal at various resolutions by obtaining the same across the multiple resolutions

[201]. These types of image restoration process can be classified into two categories: one

is deterministic and second is statistical modeling of wavelet coefficient [35].

4.2.3.1 DETERMINISTIC COEFFICIENT MODEL

In this particular modeling, process the tree structure of wavelet coefficients is

used. Tree structure is representing the scale of transformation and nodes are representing

the wavelet coefficients [4]. The optimal tree approximation displays a hierarchical

interpretation of wavelet decomposition [4][201].

4.2.3.2 STATISTICAL COEFFICIENT MODEL

Statistical approach is to restore a digital image focused on the some more

interesting and appealing properties of the wavelet transform such as multiscale

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correlation between the wavelet coefficients, local correlation between neighborhood

coefficients [209]. This particular technique can be subdivided into two categories: one is

marginal probabilistic model and other is joint probabilistic model to restore the digital

image in the transform domain [4][209].

Marginal probabilistic model based restoration technique will works on the

distribution of coefficients. And that distribution process is highly kurtotic, and usually

have a marked peak at zero and heavy tails [209][210]. The Gaussian mixture model

(GMM) and generalized Gaussian model (GGD) are commonly used to model the

wavelet coefficients distribution. Chang et. al. [114] have been proposed the use of

adaptive wavelet thresholding for image restoration, by modeling the coefficients as a

generalized Gaussian random variable, whose parameters are estimated locally within

given neighborhood [114][200].

Joint probabilistic models in transform domain (wavelet) are efficient in capturing

inter-scale dependencies [35]. Marko Random Field (MRF) models are more efficient to

capture intrascale correlation in transform domain [211]. Marko Random Field (MRF)

based model can be used to capture higher order statistics in image data. We have

implemented Markov Random Field (MRF) based image restoration method and its

explained in details in the subsection 4.3 of this chapter.

4.2.4 NON-ORTHOGONAL WAVELET BASED TECNHIQUES

Undecimated wavelet transform (UDWT) based restoration also been used for

decomposing the image signal to provide visually better artifacts such as Pseudo-Gibbs

phenomenon. Though the improvement in result is much higher, use of UDWT adds a

large overhead of computations thus making it less feasible [203]. Shift invariant wavelet

packet decomposition (SIWPD) is exploited to obtain of basis function of the

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transformation process. Then using minimum description length principle the better basis

function was found out which yielded smallest code length required for description of the

given image data [212].

4.3 MRF BASED RESTORATION METHOD

Many problems in digital image processing specially in restoration or edge

detection acceptable into a general image labeling framework, where a given label is

assigned to each image pixel. In image restoration, the label that is assigned to a image

element is the exact gray value of its intensity level. The value of wavelet coefficient in

this particular framework can also be interpreted as labels assigned to the corresponding

elements of image in the whole data of digital image [213]. Markow random field (MRF)

based restoration process will provide a convenient way of modeling within local

interaction and it is described by statistical dependencies of an image element on the

labels in its local surrounding [214]. Further, it is introduced the notation and definitions

of Markow Random Field (MRF).

Let {1,2,3,... }N n is a finite index set of on a regular rectangular lattice. The

elements of N correspond to points at which an image is sampled, i.e. to the location of

image elements. A family of random variables 1 2 3{ , , ,... }nX X X X X defined on the set

N is called the image field. The notation X x will be used to abbreviate the joint event

1 1 2 3( , , ,... )nX x x x x . The vector 1 2{ , ,......, }nx x x x is a configuration of ' 'X ,

corresponding to a given realization of the image data field. The space of all possible

configurations of ' 'X will be denoted by ; a subscript in the notation of a vector will

be used to designate that only some variables are exists in the vector i.e.

{ : { }}.kN

l

X k N l

A random field is a family of random variables

1{ ,.......... }nX X X such that all of its possible configurations have firmly positive

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probability. A particular class of random field model called Markov random fields (MRF)

furthermore requires that the label of every image element be influenced only by pixels

that are its neighbors within the image [215][216]. It is not necessary, but regularly these

neighborhood are the elements that surrounding the current one. Formally, the

neighboring relationship of the image pixel is defined as follows:

A collection { : }l l N of subset of N is called a neighborhood system, if the

neighborhoods l associated with the sites l satisfy: )i l l and ) :ii l k if and only if

.k l , the sites l k are called neighbors of k .

The order coding of the neighborhood up to the order five is shown in figure 4.2

(a). There are two examples also shown in figure 4.2., it shows in figure 4-2(b), the

neighborhood structure known as the four point or we can say that the first order

neighborhood of the center element. Figure 4.2(c) shows eight point neighborhood or

second order neighborhood of that pixel. Irregular grids are also useful in specific

engineering applications in an image restoration and segmentation.

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3

1 4

1

3

6

1 3 6136

2

S

4 5

24

45

24 2 4

5 4 4 5

(a)

(b)

(c)

Figure 4-1: Graph of Marquov Model in vision, (a) Order coding of neighborhood

structure. (b) Simple 4-connected grid of image pixels. (first order) (c)

Grids with greater connectivity can be useful-for example to achieve more

geometrical detail-as here with 8-connected pixel grid (second order).

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In practical purpose, these two neighborhood structures are the most frequently

used in image restoration process. Formally, the definition of the MRF has been

explained as: The random field is a Markov field with respect to the neighborhood

system , if for all x X :

\( ) ( )..........(4.14)l l N l l l l lP X x P X x x

According to the above definition, the probability of a image element label lx ,

given all other labels within an image, reduces to a function of neighboring labels only.

Through choosing an arbitrary large neighborhood, the MRF model can be applied to

every image for any various sizes. MRF models that are used in image processing are

often homogeneous [217][218] i.e. strictly stationary, meaning that the distribution

( )l l l lP X x x is the same for all pixels l .

4.4 SIMULATION METHODOLOGY

The state of art transform domain image restoration techniques: linear restoration

techniques, nonlinear restoration techniques (non-data adaptive and adaptive thresholding

based), wavelet coefficient model based techniques, and non-orthogonal wavelet

transform based techniques are simulated on MATLAB 7.8.0.347 platform over a

WINDOW-XP operating system. Moreover, stochastic model based technique i.e.

Markov Random Field (MRF) is also simulated on the same platform. All transform

domain restoration techniques are simulated with the combinations of different

degradations (Gaussian, Speckle, Poisson, and Salt & Pepper noise) and various types of

images from diversified fields (Medical, Natural, Aerial, and Underwater). Standard test

images from various fields of sizes 256x256, 512x512 are used for simulation. The

universal quality parameter (PSNR, MSE) and other difference distortion, correlation

distortion metrics has been adapted to evaluate the performance of all restoration

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techniques. All degradations with same standard deviation (SD) are used for simulation

purpose. These synthetic degradations are added to all images in controlled fashion.

Moreover, noisy images were obtained for processing. The best performing technique

was decided according to the value of PSNR and MSE to the combinations of specific

noise and image from particular field.

4.5 RESULTS AND DISCUSSION

In this chapter, standard transform domain image restoration techniques are

simulated and compared its performance. We have analyzed one thousand five hundred

and ninety six (1596) different combinations of various fields (03), types of image (07),

different synthetic degradations (04), and various transform domain restoration

techniques (19). The performances obtained from these combinations are tabulated in

table 4-1. The analysis is based on universal quality parameters i.e. peak signal to noise

ratio (PSNR) and mean square error (MSE). Then we tried to find out the optimal

restoration technique in transform domain to particular combination of noise and image

from specific field. Further, we are also trying to specify the category to optimal

restoration technique. Out of these combinations, we have concluded some optimum

selection of techniques according to their performance and it is tabulated in table 4-2.

From table 4-1 and table 4-2, the following conclusions are drawn according to the

performance of various transform domain image restoration techniques.

There is no single restoration technique, which is highly suitable to all

combination of different types of noise and various images from diversified fields.

However, we have found few image restoration techniques, which perform finest for a

particular type of image irrespective of the type of noise. Similarly, we have also found

few image restoration techniques that perform best for specific noise irrespective of the

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type of the images from various fields. It is observed that adaptive local filtering

technique is providing better performance to major combinations of images from these

three fields and four types of noise and its shown in table 4-2.

Va

rio

us

Fie

lds

of

Ima

ges

Types of Noise

Suitable Image Restoration Techniques

Gaussian Noise Poisson Noise Speckle Noise Salt & Pepper

Noise

Medical

Field

Undecimated DWT

Filtering (10)

NL-Mean Filtering

(09)

Non-Parametric Bayesian

Dictionary Learning (01)

Adaptive Local

Filter (DWT) (14)

Adaptive Local

Filter (DWT) (14)

Natural

Field Adaptive Local

Filter (DWT) (14)



Adaptive Local

Filter (DWT) (14)

Adaptive Local

Filter (DWT) (14)

Aerial

Field Adaptive Local Filter

(DWT) (14)

Adaptive Local Filter (14)



Adaptive Local

Filter (DWT) (14)

Adaptive Local

Filter (DWT) (14)

Table 4-2: Combination of different field and specific noise according to performance

of various transform domain image restoration techniques for optimum

selection (broad analysis).

The performance of adaptive local filter (discrete wavelet transform) is the better

as compared to all other transform domain restoration techniques in presence of any type

of degradations. There is not a single other technique in transform domain that performs

better than adaptive local filtering technique to any combination of aerial images

degraded by various type of noise. Adaptive local filtering technique is comes under the

non-linear filtering category of restoration technique in transform domain. Performance

of adaptive local filter is shown in figure 4-2. We have taken total ten images from aerial

fields to observe the performance of the same restoration technique. After considering

performance of this technique to all aerial images then we have concluded that the

adaptive local filtering technique is highly suitable to aerial field images.

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Figure 4-2: Analysis of various restoration techniques towards Aerial field images.

The performance of adaptive local filtering technique is better for the combination

of image from natural field and all types of synthetic degradations except Poisson noise.

For the combination of Poisson noise occurs in natural fields images, non-parametric

Bayesian dictionary learning filtering technique is highly suitable. This analysis is done

for all restoration techniques in transform domain. This inference is only for images from

natural field oriented. This technique is also in nonlinear class in transform domain.

Performance of this nonlinear technique is shown in figure 4-3.

Figure 4-3: Analysis of various restoration techniques towards Natural field images.

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Medical Natural Arial

PS

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Images & Fields

Analysis of Adaptive Local Filter Nonparametric Bayesian Dictionary

Learning filtering TechniqueNL-Mean Filtering Technique

Undecimated DWT Filtering

TechniqueAdaptive Local (DWT)

Filtering Technique

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PS

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Image & Fields

Analysis of ALF & NLBDLF NonparametricBayesian DictionaryLearning for Analysis

Adaptive Local(DWT) Filter

15

For images from medical field, no specific restoration technique perform superior

for any type of degradations. However, if we consider individual degradation (noise) then

we get restoration technique that suppress the specific type of noise. Undecimated DWT

filtering technique and NL-Mean filtering technique performs moderately for medical

field images degraded by AWGN. Similarly, non-parametric Bayesian dictionary

learning filtering technique performs better for medical images corrupted by Speckle, Salt

& Pepper noise as compared to other restoration techniques in transform domain [219]. It

is shown in figure 4-4.

Figure 4-4: Analysis of various restoration techniques towards Medical field images.

From table 4-1 and table 4-2, some inferences to noise oriented are as follows:

It is observed that adaptive local filtering technique is highly suitable to suppress

the Salt & Pepper noise from any type of image of various fields. Performance ALF

technique in the form of PSNR is shown in figure 4-5.

Non-parametric Bayesian dictionary learning filtering technique performs better

to all type of images from various fields contaminated by Poisson noise except aerial

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House (Trees

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PS

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Image and Fields

Analysis for Medical Field Nonparametric BayesianDictionary Learning for Analysis

NL-Mean

Undecimated DWT

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field images. ALF technique performs best in the presence of Poisson noise. However,

next to this particular technique non-parametric dictionary learning filtering technique

performs better as compared to other restoration techniques in transform domain. It is

shown in figure 4-6.

Figure 4-5: Analysis of various restoration techniques towards SPN and SN.

Adaptive local filtering technique is also suitable to suppress the Speckle noise

from all type of images from diversified field [219]. It is shown in figure 4-5.

Figure 4-6: Analysis of various restoration techniques towards AWGN and PN.

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Image & Fields

Analysis of ALF Technqiue for SN & SPN Adaptive Local

(DWT) Filter

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Image and Fields

Analysis to the various noise NonparametricBayesian DictionaryLearning forAnalysis

NL-Mean

Undecimated DWT

AdaptiveLocal (DWT) Filter

17

Adaptive local filtering technique performs best for any image corrupted with

AWGN except medical field images. For medical field images contaminated with

AWGN, Undecimated DWT (UDWT) filtering technique and NL-mean filtering

techniques performs finest as compared to all other restoration technique in transform

domain. It is shown in figure 4-6.

From table 4-1 and table 4-2, it is seen then the performance of ALF technique is

almost same for particular image irrespective of the any type of degradation. Therefore, if

we take any particular image and add various type of noise to it and after restored by

ALF technique we get the image that is having same PSNR for all type of noise [219].

So, for a specific image the performance in the form PSNR of ALF technique is constant

irrespective of the type of degradation. It is reflect in figure 4-5.

The performance of non-parametric dictionary learning filtering technique is

superior for any type of image in presence of Poisson noise but ironically its performance

comparatively poor for Salt & Pepper noise. It is shown in figure 4-7.

Figure 4-7: Analysis of non-parametric Bayesian Dictionary Learning filtering

technique.

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The performance of VisuShrink filtering technique is poor for aerial field images

in presence of any type of noise as compared to its performance to medical field and

natural field images degraded by any type of nose.

Figure 4-8: Analysis of VisuShrink: Nonlinear Thresholding filtering (Non-Adaptive)

technique.

The performance of hard thresholding and soft thresholding to the medical field

images is not up to the mark in presence of AWGN and Salt & Pepper noise as compared

to its performance in presence of Poisson and Speckle noise for same field images and its

shown in next figure 4-9.

Figure 4-8: Analysis of Hard and Soft Thresholding Filtering Techniques.

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Analysis of Hard and Soft Thresholding Restoration Techniques Hard Thresholding

Soft Thresholding

19

4.6 ANALYSIS OF MRF BASED RESTORATION TECHNIQUE

In previous subsection of this chapter, we have analyzed state-of-art transform

domain image restoration techniques to find out optimal technique to particular

combination of specific noise and certain image from diversified field. In this subsection,

joint probabilistic wavelet coefficient model using Marquov Random Field (MRF) based

restoration technique has been analyzed. The MRF theory gives a convenient and

consistent way of modeling context dependent entities such as image elements and

correlated features. While simulation the particular technique we have considered the

four type of synthetic degradations (Gaussian, Speckle, Poisson, and Salt & Pepper noise)

and various fields (Medical, Natural, Aerial, and Underwater). The performance in the

form of PSNR values is tabulated in table 4-3. The performance of MRF based

restoration technique to the different combinations is as shown from figure 4-9 to figure

4-13.

Figure 4-9: Analysis of MRF Based Image Restoration Technique Towards Medical

Field Images. (Objective analysis in the form of PSNR according to the

combinations of noise and images).

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Figure 4-10: Analysis of MRF Based Image Restoration Technique Towards Natural

Field Images. (Objective analysis in the form of PSNR according to the


Figure 4-11: Analysis of MRF Based Image Restoration Technique Towards Aerial Field

Images. (Objective analysis in the form of PSNR according to the


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Figure 4-12: Analysis of MRF Based Image Restoration Technique Towards Under

Water Field Images. (Objective analysis in the form of PSNR according to

the combinations of noise and images).

Figure 4-13: Analysis of MRF Based Image Restoration Technique Towards All

Combinations of all types of Noise and Field. (Objective analysis in the

form of PSNR according to the combinations of noise and images).

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Analysis of MRF Based Technqiue to All types of Images Medical Images

Natural Image

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From figure 4-9 to figure 4-13, out of these combinations of different images (40),

various noise (04), diversified fields (04), and one restoration technique, we have

concluded some optimum performance of MRF based technique to the particular

combination of noise and image, according to objective analysis and noise level

classification. These optimum selections are depicted in table 4-4.

Sr.No. Field Noise

Suitability of MRF based restoration technique to the

combination of noise and field according to the PSNR

value in dB. In addition, Noise Level classification

according to the table 1-1.

01 Any Field Gaussian

Noise

Highly Suitable

( Noise level in restored image very low)

02 Any Field Poisson

Noise

Moderately Suitable

(Noise level in restored image low)

03 Any Field Speckle

Noise

Suitable

(Noise level in restored image medium)

04 Any Field Salt &

Pepper Noise

Not Suitable

(Noise level in restored image high)

Table 4-4: Performance of joint probabilistic wavelet coefficient model using Marquov

Random Field (MRF) based restoration technique to the combination of

various noise and images from medical, aerial, natural, and under water

fields.

It is observed that from table 4-4, if any image from diversified fields degraded by

Gaussian noise then MRF based image restoration technique is highly suitable. From any

image to suppress the Poisson noise, MRF based technique can be utilize effectively. If

Speckle noise occurs in any image then this method can be utilize to reduce up to the

mark. According to the noise level occurs in the restored image that degraded by Salt &

Pepper noise, this technique is not showing a superlative performance [219] [220].

23

For proper judgment of the performance of MRF based image restoration

technique in transform domain, the subjective evaluation can be taken into consideration.

Performance of this technique on standard images (field wise) with various tones is

shown in figure 4-14 and figure 4-15.

(a) (b)

(c) (d)

(e) (f)

Figure 4-14: Subjective analysis of MRF based restoration technique using wavelet

transform; (a) & (b): Aerial images, (c) & (d): Natural images, (e) Medical

image (f) Under water image with Gaussian noise.

24

(a) (b)

(c) (d)

(e) (f)

Figure 4-14: Subjective analysis of MRF based restoration technique using wavelet

transform; (a) Poisson noise: Medical image, (b) Poisson noise: Natural

image, (c) Poisson noise: Underwater image (d) Salt & Pepper noise: Aerial

image (e) Speckle noise: Aerial image (f) Speckle noise: Medical Image.

Further, while analyzing both objective and subjective, it is observed that

performance of this restoration technique yields good visual quality of all restored images

from various fields degraded by Gaussian noise [220].

Inferences are drawn in the next subsection of this chapter.

25

4.7 CONCLUSION

The performance of various restoration techniques in transform domain viz. linear

and nonlinear thresholding filtering process based restoration techniques (Visu Shrink,

Sure Shrink, Bayes Shrink, hard and soft thresholding, global thresholding, Gaussian

field of expert (GFoE), Active random field (ARF)), Wavelet coefficient model based

deterministic and statistical (tree approximation, GMM, GGD, HMM and proposed MRF

based technique), non-orthogonal wavelet transform based undecimated wavelet

transform (UDWT) are studied. Above all, transform domain image restoration

techniques are analyzed with the combinations of synthetic noise and various fields.

From table 4-1 and table 4-2; we have found some suitable restoration techniques

to noise specific and image specific in transform domain. We have also shown from

figure 4-2 to figure 4-9 performances of specific techniques to the particular

combinations of synthetic degradations and image from various fields. We have specified

the category of restoration technique according to their processes to the combinations in

transform domain.

Table 4-3, shows the performance of joint probabilistic statistical wavelet

coefficient MRF model based image restoration technique in wavelet domain. A broad

inference has shown in table 4-3 and table 4-4.

From table 4-3 and figure 4-9 to figure 4-13, further major results of this chapter

lies in the estimation of MRF based restoration technique to suppress the synthetic

degradations from noisy image in transform domain. Simulation experiments indicate

that the technique and yielding results that are superior to specific combination and worst

for particular combination to those obtained by state-of-art restoration technique in

transform domain. It is observed that, significant and some insignificant gains are

achieved by using MRF at the expense of an increase in the computational complexity.

chapter 15 7

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