34

CHAPTER 3

Tetrolet Based Adaptive Edge-preserving Image

Denoising

3.1 Introduction

In the context of denoising, the wavelet transforms have been successfully proven as efficient

means due to their ability of decorrelation (separation of noise and useful signal). The basic

concept related to the noise reduction based on the wavelet transform is to compute the

multiscale wavelet decomposition of the corrupted image into the wavelet coefficients and to

modify the obtained wavelet coefficients. The modified coefficients are obtained by applying a

predetermined threshold on them according to a shrinkage rule. Reconstruction from these

modified coefficients then produces the desired denoised image. A flow diagram to illustrate the

main stages in waveletbased denoising process has been depicted in Fig. 1.2 and detail review of

these stages has been presented in Section 2.4.

The basic issue with most of the wavelet transforms is that the multiscale decomposition of

image into the wavelet coefficients is not adaptive i.e. local structures of image are not taken into

account during decomposition. Although this issue is resolved by an adaptive Haar wavelet

transform (also called tetrolet transform) [Krommweh (2010)] but this tetrolet system is suitable

only for sparse image representation due to its non-redundant nature, while for image denoising

redundant information is helpful. Thus, it is very much needed to exploit redundancy by a

denoising method based on tetrolet transform.Apart from this, in conventionalthresholding

35

schemes, a universal (global) threshold is generally used to shrink small wavelet coefficients.

However, such procedure may also suppress high frequency details, such as edges. Also the

noise variance which is used in the computation of threshold (s), is usually kept fixed through all

resolution scales. However, the noise strength decreases with the raise in resolution scale.

To deal with the aforementioned issues,a new edge-preserving image denoising method using

tetrolet transforms is proposedin this chapter. Although the method is motivated by non-

redundant tetrolet system [Krommweh (2010)] but the underlying approach has a higher degree

of redundancy. This redundancy helps in achieving better denoising. The proposed method also

improves the conventional wavelet denoising methods as instead of using a global threshold, an

adaptive threshold is calculated in a subband-dependent manner to characterize local features of

the image. In addition, instead of using fixed noise variancewhich is used in the computation of

thresholds, it is estimatedlocally for each decomposition level (resolution scale). An adaptive

epsilon-median(e-median) filtering, which uses above computed threshold, is employed to

perform noise suppression in tetrolet coefficients.

There are some reasons that motivate us to use tetrolet transform [Krommweh (2010)] to

obtain a multiscale decomposition of an input image into wavelet (tetrolet) coefficients. First,

local geometrical structures of image are taken into account during the decomposition process,

i.e. the decomposition of an image into coefficients using tetrolet transform is adaptive to the

image contents. Second, the underlying idea of adaptive tetrolet decomposition algorithm is

simple but is fast and effective.

Apart from the above discussed issues related to the wavelet transform and the threshold being

applied, there may also an issue with the thresholding rule being used. Most simple non-linear

thresholding rules assume that the wavelet coefficients are independent. However, it is observed

36

that the wavelet coefficients of natural images have significant statistical dependencies. Sendur

and Selenick (2002a) have suggested four joint shrinkage functions that utilize the dependencies

between wavelet coefficients through two adjacent resolution scales to enhance thedenoising

performance. In [Sendur and Selenick (2002b)], the authors further enhanced their earlier

proposed bivariate shrinkage function in [Sendur and Selenick (2002a)] and proposed a locally

adaptive thresholding method in which the thresholding parameters are computed in a local

neighborhood.

Inspired by their approach, another edge-preserving image denoising technique in tetrolet

domain is proposed in this chapter. Indeed, this method is an extension of our earlier mentioned

method, i.e. it inherits the merits of previous method and incorporates some more aspects, such

as the proposed approach employs a locally adaptive (that is, coefficient dependent) thresholding

which exploits the interscale statistical dependencies between tetrolet coefficients and computes

the thresholding parameters in a local neighborhood.

3.2 Related Work

The tetrolet transform has also been in other denoising approaches as main stream. In Singh

(2010), a new approach was proposed to the denoising problem based on the tetrolet transform.

This approach uses the ideas of adaptiveness and 4 × 4 block mechanism described in

Krommweh (2010) and works on each 4 × 4 block independently and adapts to image

characteristics automatically.

Li et al. (2010) introduced a new class of denoising function that has continuous derivatives for

image denoising. The authors have presented a new algorithm in which, firstly, the noisy image

is decomposed into tetrolet coefficients by using a discrete tetrolet transform given in

Krommweh (2010). Secondly, an adaptive method based on SURE risk is presented by using a

37

new denoising function. Instead of using a universal threshold for noise suppression, the

threshold is obtained by minimizing an estimate of the mean square error through an adaptive

genetic algorithm.

Further, Zhang et al. (2013) presented a new denoising techniquein tetrolet domain. In this

technique,the authors provided improvement over the Bivariate Model (BM) [Sendur and

Selenick (2002b)]. This improved model fits the joint distribution of parent-child tetrolet

coefficients with a Scale Variable Parameter Bivariate Model (SVPBM). Corresponding non-

linear bivariate shrinkage function is derived from SVBPM by using maximum-a-posteriori

(MAP) estimator.

Recently, Dai et al. (2013) improved the BM3D algorithm [Dabov et al. (2006)] by combining

BM3D with tetrolet prefiltering. Authors resolved the limitation with BM3D method of having a

sharp drop in denoising performance at higher values of noise standard deviation. In this

approach, firstly the tetrolet filtering of the strongly noisy image is done to remove part of the

noise and then BM3D filtering is carried out over this partially filtered image.

3.3 Tetrolet Transforms

Tetrolets are Haar-type wavelets whose supports are the shapes called tetrominoes. The

tetrominoes are some geometric shapes in the famous computer game ‘Tetris’ [Breukelaar et al.

(2004)]. All the tetrominoes are made by connecting four equal-sized squares. Disregarding

rotations and reflections there are five basic free tetrominoes as given in following figure:

Figure 3.1:Five free tetrominoes

When applying the tetrolet transform to an image 𝐠 = [g(𝑥, 𝑦)]𝑥,𝑦=1𝑁 with N = 2

K, K N, we

divide an image into 4 4 blocks. Then each block is covered with any four free tetrominoes,

38

while rotations and reflections of tetrominoes are taken into consideration, which depends on the

local structure in the block. These four tetrominoes, which form the adaptive basis, are denoted

by {I0,I1,I2, I3} and the four indices in each tetromino subset Iv are mapped to a unique order {0,

1, 2, 3} by applying a bijective mapping L. On the basis of these definitions, for each tetromino

subset Iv, Krommweh (2010) defined following discrete basis functions:

𝐼𝑣

[𝑥′, 𝑦′]: = { 1/2, (𝑥′, 𝑦′) ∈ 𝐼𝑣

0, otherwise

(3.1)

𝜓𝐼𝑣

𝑙 [𝑥′, 𝑦′]: = { [𝑙, 𝐿(𝑥’, 𝑦’)], (𝑥′, 𝑦′) ∈ 𝐼𝑣

0, otherwise

(3.2)

for l = 1, 2, 3. Due to the underlying tetromino support, 𝜓𝑣𝑙 are called tetrolets and

𝑣 the

corresponding scaling function. The function values [𝑙, 𝐿(𝑥’, 𝑦’)] in the tetrolet definition come

from the Haar wavelet transform matrix

W = ([𝑚, 𝑛])𝑚,𝑛=03 =

1

2(

1 1 1 1 1 1 − 1 − 1 1 − 1 1 − 1 1 − 1 − 1 1

) (3.3)

The basic idea of tetrolets is inherited from the conventional 2-D Haar wavelets. Infect tetrolets

are improved version of conventional Haar wavelets. The 2-D Haar wavelets allow the covering

of a 4 4 block with four fixed 2 2 squares. This is a very inefficient way of covering the

block because the local image structures are not taken into account during the covering. On the

other hand, for the tetrolets, it has been proved that there are 117 possible ways of covering a 4

4 block with any four tetrominoes [Krommweh (2010)]. This inefficiency of conventional Haar

wavelet transform motivates to use tetrolet transform method because it allows more partitions

than the conventional Haar wavelets and also finds most appropriate one based on local features

of the image. Fig. 3.2(a) shows the example of the fixed square partitioning of a 4 4 block by

39

the 2-D Haar wavelets. Fig. 3.2(b) shows one of the 117 possible tetromino partitions. If the local

structure of the block to be partitioned matches with Fig. 3.2(c), the solution in Fig. 3.2(b) is

obviously more appropriate than the fixed squares in Fig. 3.2(a).

(a) (b) (c)

Figure 3.2:Two examples of covering a 4 4 block and the local structure of the 4 4 block:

(a)The fixed squares of the 2-D Haar wavelets; (b) One of the 117 solutions for disjoint covering

of a 4 4 block with tetrominoes; (c) Example of the local structure

Following steps are followed to obtain the multiscale tetrolet decomposition of given input noisy

image 𝐠 = [g(𝑥, 𝑦)]𝑥,𝑦=1𝑁 with N = 2

K, K N, while given a tetromino covering c and number of

decomposition levels J. We start with the input image 𝐠0 = 𝐠 = [g(𝑥, 𝑦)]𝑥,𝑦=1𝑁 . In the j-th level,

𝑗 = 1, … … , 𝐽, we do the following computations.

1. Divide the low-pass image 𝐠𝑗−1 into 4 4 blocks, 𝑄𝑚,𝑛, 𝑚, 𝑛 = 1, … … ,𝑁

2𝑗+1.

2. In each block Qm,n, we compute the low-pass part for tetromino covering c by

𝐠𝑗,(𝑐) = (g𝑗,(𝑐)[𝑣])𝑣=0

3with g𝑗,(𝑐)[𝑣] = ∑

𝐼𝑣(𝑐)[𝑥′, 𝑦′]g𝑗−1[𝑥′, 𝑦′]

(𝑥′,𝑦′)∈𝐼𝑣(𝑐)

(3.4)

as well as the three high-pass parts for l = 1, 2, 3

𝐒𝑙𝑗,(𝑐)

= (S𝑙𝑗,(𝑐)[𝑣])

𝑣=0

3

withS𝑙𝑗,(𝑐)[𝑣] = ∑ 𝜓

𝐼𝑣(𝑐)

𝑙 [𝑥′, 𝑦′]g𝑗−1[𝑥′, 𝑦′]

(𝑥′,𝑦′)∈𝐼𝑣(𝑐)

(3.5)

where 𝐼𝑣

and 𝜓𝐼𝑣

𝑙 are defined in (3.1) and (3.2), respectively.

40

For each block 𝑄𝑚,𝑛 we save the covering c, which is used for partition, since this information is

required at the time of reconstruction.

3. In order to be able to apply further levels of the tetrolet decomposition algorithm, we

rearrange the entries of the vectors 𝐠𝑗,(𝑐) and 𝐒𝑙𝑗,(𝑐)

into 2 2 matrices using a reshape

function R,

𝐠|𝑄𝑚,𝑛

𝑗= 𝑅(𝐠𝑗,(𝑐)) = (

g𝑗,(𝑐)[0] g𝑗,(𝑐)[1]

g𝑗,(𝑐)[2]g𝑗,(𝑐)[3]) ,

and in the same way 𝐒𝑙|𝑄𝑚,𝑛

𝑗= 𝑅(𝐒𝑙

𝑗,(𝑐) ), 𝑙 = 1, 2, 3.

4. After finding a tetrolet decomposition in every block 𝑄𝑚,𝑛, 𝑚, 𝑛 = 1, … … ,𝑁

2𝑗+1, we store the

low-pass matrix 𝐠𝑗 = (𝐠|𝑄𝑚,𝑛

𝑗)

𝑚,𝑛=1

𝑁

2𝑗+1and the high-pass matrices 𝐒𝑙

𝑗= (𝐒𝑙|𝑄𝑚,𝑛

𝑗)

𝑚,𝑛=1

𝑁

2𝑗+1, 𝑙 =

1, 2, 3, replacing the low-pass image 𝐠𝑗−1by the matrix (𝐠𝑗𝐒2

𝑗

𝐒1𝑗𝐒3

𝑗). In this way we obtained one

low pass image (subband) 𝐠𝑗 and three highpass subbands 𝐒1𝑗,𝐒2

𝑗and 𝐒3

𝑗 in the decomposition

level j.

3.4 Adaptive Image Denoising Methods

In this section anedge-preserving image denoising methods for noise suppression in tetrolet

coefficients. The method is based on adaptive epsilon-median filtering which provides subband-

dependent thresholding andits variant is based on locally adaptive thresholding i.e. it provides a

coefficient dependent thresholding.

3.4.1 Image Denoising using Adaptive Epsilon-Median(E-median) Filtering

Inspired by an adaptive tetrolet decomposition algorithm [Krommweh (2010)] and the epsilon-

median (e-median) filtering method [Haseyama et al. (2000)], we proposed a novel adaptive

41

edge-preserving image denoising method. The main stages of the proposed method are shown in

Fig. 3.3.

Let 𝐟 = [f(𝑥, 𝑦)]𝑥,𝑦=1𝑁 , 𝑁 = 2𝐾 , 𝐾 Ndenote 𝑁 × 𝑁 original image to be recovered. During

transmission, the image f is corrupted by independent and identically distributed (i.i.d.) zero

mean white Gaussian noise according to the following model:

𝐠 = 𝐟 + 𝐧, (3.6)

where n represents the noise and g the observed image. The goal is to estimate f from the noisy

observation g.

Figure 3.3: Image denoising using adaptive epsilon-median filtering

Initially, the input image g corrupted by additive Gaussian noise, is decomposed into

tetroletcoefficients through a Discrete Tetrolet Transform (DTT) 𝐖T given in previous section,

expressed as

𝐆T = 𝐖T(𝐠) = {𝐠𝐽; 𝐒1𝑗; 𝐒2

𝑗; 𝐒3

𝑗, 𝑗 = 𝐽, 𝐽 − 1, … … ,1} (3.7)

where j indicates the decomposition level (or resolution scale) of tetrolet transformand J is the

largest scale in the decomposition.The low-frequency subband 𝐠𝐽 at largest resolution scale J

noisy image

g

denoised

image 𝐟

GT

e-median

filtering

subband

adaptive

threshold

estimation

�̂�

𝐖T−1

inverse

tetrolet

transform

discrete

tetrolet

transform

WT

local noise

variance

estimation

42

corresponds to a coarsest approximation of the image signal while the high-frequency subbands

𝐒1𝑗, 𝐒2

𝑗, and 𝐒3

𝑗, respectively correspond to horizontal, vertical and diagonal details of the image

signal at scale j.Figure 3.4 illustrates the subband regions of the 2-D critically sampled tetrolet

transform.

Let us consider a detail subband S at scale j, i.e.𝐒 = 𝐒𝑙𝑗, 𝑙 = 1, 2, 3, 𝑗 = 1, … … , 𝐽.

Note that the additive noise model for image g in spatial domain in Eq. (3.6) is also applicable

for the subband 𝐒𝑙𝑗 in wavelet domain. Thus, we have

𝐒𝑙𝑗

= 𝐰𝑙𝑗

+ 𝐧𝑙𝑗, (3.8)

where the noisy subband 𝐒𝑙𝑗 is obtained after introducing the noise 𝐧𝑙

𝑗 in its noiseless counterpart

𝐰𝑙𝑗. The aim of denoising method is to obtain �̂�𝑙

𝑗 (the estimate of noiseless counterpart 𝐰𝑙

𝑗) from

the given noisy subband 𝐒𝑙𝑗.

Figure 3.4: Subband regions of critically sampled tetrolet transform

Let 𝑦𝑘 ∈ 𝐒𝑙𝑗, 𝑤𝑘 ∈ 𝐰𝑙

𝑗 and 𝑛𝑘 ∈ 𝐧𝑙

𝑗. Then,

𝑦𝑘 = 𝑤𝑘 + 𝑛𝑘 , 𝑘 = 1 … no. of tetrolet coeffs. (3.9)

𝐠3 𝐒23

𝐒2

2 𝐒1

3 𝐒33

𝐒21

𝐒1

2 𝐒32

𝐒1

1 𝐒31

43

where 𝑦𝑘 is the noisy observation of wk and nk is the noise sample. The objective of the proposed

adaptive e-medianfiltering method (3 × 3 window size based) is to obtain �̂�𝑘, the estimate of 𝑤𝑘,

from its noisy observation 𝑦𝑘. The proposed adaptivee-median filtercan be defined as

�̂�𝑘 = 𝑦𝑘𝑚 + 𝑋(𝑦𝑘 − 𝑦𝑘

𝑚) (3.10)

where 𝑦𝑘 represents the degraded data, 𝑦𝑘𝑚 represents the median filtered data.

The function 𝑋 is defined as

𝑋(𝑥) = { 𝑥, |𝑥| > 𝜆𝑙

𝑗

0, otherwise

(3.11)

where 𝜆𝑙𝑗 represents the threshold value.

The e-median filter preserves edges while removing noise [Haseyama et al. (2000), Koç and

Ergelebi (2006)]. The threshold 𝜆𝑙𝑗 on a given subband 𝐒𝑙

𝑗 is computed by using the BayesShrink

[Chipman et al. (1997), Chang et al. (2000a)] as threshold estimation criterion, expressed as

𝜆𝑙𝑗

=�̂�𝑛𝑜𝑖𝑠𝑒,𝑗

2

�̂�𝑠𝑖𝑔𝑛𝑎𝑙,𝑙𝑗

, (3.12)

where �̂�𝑛𝑜𝑖𝑠𝑒,𝑗2 is the local noise variance which can be estimated from the diagonal detail

coefficients at the same scale j as the subband under consideration (that is, the coefficients in 𝐒3𝑗

subband).

�̂�𝑛𝑜𝑖𝑠𝑒,𝑗 = 𝑚𝑒𝑑𝑖𝑎𝑛(|𝑦𝑖|)

0.6745, 𝑦𝑖 ∈ 𝐒3

𝑗 (3.13)

The expression on right-hand side of Eq. (3.13) is a robust median estimator [Donoho and

Johnstone (1994)] often used to estimate noise variance from diagonal detail coefficients at the

finest scale (that is, the coefficients in subband 𝐒31). The term �̂�𝑠𝑖𝑔𝑛𝑎𝑙,𝑙

𝑗 in Eq. (3.12) is the local

estimated signal deviation on the subband under consideration, estimated as

44

�̂�𝑠𝑖𝑔𝑛𝑎𝑙,𝑙𝑗

= √max(�̂�𝑦2 − �̂�𝑛𝑜𝑖𝑠𝑒,𝑗

2 , 0) (3.14)

where �̂�𝑦2 =

1

𝑁s∑ 𝑦𝑘

2𝑁𝑠𝑘=1 and 𝑁s is the number of tetrolet coefficients 𝑦𝑘 on the subband under

consideration.

The threshold computed in (3.12) will be used obtain the elements �̂�𝑘 of �̂�𝑙𝑗 (the estimate of

noiseless counterpart 𝐰𝑙𝑗

of the given noisy subband 𝐒𝑙𝑗

) according to (3.10). The above

procedure will be used to obtain the estimates of noiseless counterparts for all the detail

subbands at each level and the final thresholded result �̂� of complete image in wavelet domain

can be obtained as

�̂� = {𝐠𝐽; �̂�1𝑗; �̂�2

𝑗; �̂�3

𝑗, 𝑗 = 𝐽, 𝐽 − 1, … ,1} (3.15)

Now an Inverse Discrete Tetrolet Transform (IDTT) is applied at last stage, expressed as

𝐟𝑟𝑒𝑐 = 𝐖T−1(�̂�) (3.16)

where 𝐖T−1 denotes the inverse discrete tetrolet transform and 𝐟𝑟𝑒𝑐 is the reconstructed image.

For the reconstruction of the image, we use the low-pass coefficients from the coarsest level and

the thresholded tetrolet coefficients from all levels obtained in above stage as usual.

Additionally, the information about the respective covering in each level and block is used. We

now apply the mechanism of tetrolet decomposition process elaborated earlier in reverse order.

Table 3.1 gives the algorithm for the proposed adaptive e-median filtering-based denoising

method. In this algorithm, the whole procedure iteratively executed for each admissible

tetromino covering. As there are 117 possible tetromino coverings that can be used for

partitioning image blocks, so many samples (instead of one) are obtained for a pixel value. This

situation leads to redundancy in samples for the image pixels. Such redundant information is

45

helpful in achieving better denoising. The average of all the collected samples is taken to obtain

the desired denoised version of a noisy pixel.

Table 3.1: Algorithm for the proposed adaptive e-median filtering-based denoising

method[Jain and Tyagi (2015a)]

Algorithm’s prerequisite: Extend the input noisy image (if needed) to make its size as N N

with N = 2K, K N. After denoising, the image is cropped to get the original size. The extended

image is used for further analysis.

Inputs: Noisy image 𝐠 = [g(𝑥, 𝑦)]𝑥,𝑦=1𝑁 with N = 2

K, K N, the number of decomposition levels

J (default value, log2N – 1), number of admissible tetromino coverings num_cov (default value,

117).

Output: The denoised image 𝐟.

Initialization: Initialize 𝐟 = 0. Set c = 1.

Iterate as follows:

Step (1) Considering image g, tetromino configuration c, and the number of decomposition

levels J as input arguments, obtain the multiscale tetrolet decomposition

𝐆T = {𝐠𝐽; 𝐒1𝑗; 𝐒2

𝑗; 𝐒3

𝑗, 𝑗 = 𝐽, 𝐽 − 1, … … ,1} using (3.7).

Step (2)For each decomposition level (j = 1J ):

(a) Calculate the local noise variance �̂�𝑛𝑜𝑖𝑠𝑒,𝑗2 using (3.13).

(b) For each subband (S = 𝐒𝑙𝑗, l = 1, 2, 3):

(i) Compute �̂�𝑠𝑖𝑔𝑛𝑎𝑙,𝑙𝑗 using (3.14).

(ii) Compute the threshold 𝜆𝑙𝑗 using (3.12).

(iii) For each tetrolet coefficient, yk S (k = 1no. of tetrolet coefficients in

subband S):

Estimate each coefficient using 𝜆𝑙𝑗 in (3.10).

Step (3) Get the thresholded wavelet output �̂� using (3.15).

Step (4) Obtain the reconstructed image 𝐟𝑟𝑒𝑐 using (3.16).

Step (5) 𝐟 = 𝐟 + 𝐟𝑟𝑒𝑐.

Step (6) Increase c (c = c + 1) and go to Step (1).

Termination criterion: If c > num_cov, stop iteration.

Output: 𝐟= 𝐟 / num_cov

3.4.2 Image Denoising using Locally Adaptive Thresholding

Inspired by an adaptive tetrolet decomposition algorithm [Krommweh (2010)] and the locally

adaptive denoising algorithm [Sendur and Selesnick (2002b)], we have proposed a novel locally

adaptive image denoising method to preserves edges. This method is a variation of the earlier

proposed method. That is, it inherits some relevant aspects of previous method while includes

46

some new aspects to achieve gain in denoising performance. All of these aspects can be

summarized as follows:

1. Redundancy is exploited: Although our method is motivated by non-redundant tetrolet

system [Krommweh (2010)] but proposed approach has a higher degree of redundancy. This

redundancy helps in achieving better denoising.

2. Interscale dependency is exploited: Sendur and Selenick (2002a) have suggested four joint

shrinkage functions that utilize the dependencies between wavelet coefficients through two

adjacent resolution scales in order to improve the denoising performance. Inspired by their

approach, the proposed thresholding scheme also exploits the interscale statistical

dependencies between coefficients obtained by applying discrete tetrolet transform.

3. Local noise variance is computed: It can be observed that noise strength decreases with the

raise in resolution scale. Therefore, instead of using fixed noise variance, which is used in the

computation of thresholds, it is estimated locally for each resolution scale.

4. The thresholding parameters are estimated in a local neighborhood: Sendur and Selenick

(2002b) further shown improvisation over their earlier proposed bivariate shrinkage function

in [Sendur and Selenick (2002a)] and proposed a locally adaptive thresholding method in

which the thresholding parameters are computed in a local neighborhood. Inspired by their

approach, our thresholding scheme also determines the thresholding parameters in a local

neighborhood around each pixel position.

The implementation outlines of the proposed method are as follows: First, we start with the

decomposition of input noisy image into the tetrolet coefficients by applying a discrete tetrolet

transform. Second, we estimate the noise variance locally for each resolution scale by using a

robust median estimator [Donoho and Johnstone (1994)]. Third, we have computed the threshold

47

for the subband under consideration by using the local noise variance (estimated in previous

step) and the coefficients which are present in the neighborhood of that coefficient. Fourth, a

bivariate shrinkage function which exploits the interscale dependency between the coefficients,

is employed to threshold each coefficient, which is analytically dependent on three parameters,

namely the coefficient itself, its parent coefficient (the coefficient at the same position, but at the

next coarser scale) and the threshold computed earlier. Lastly, the thresholded coefficients are

transformed back to the original domain by applying an inverse discrete tetrolet transform.

The main stages of this variant are illustrated in Fig. 3.5. The notations and definitions for the

noisy image 𝐠 , the subbands 𝐒𝑙𝑗, 𝐰𝑙

𝑗, 𝐧𝑙

𝑗, �̂�𝑙

𝑗 and the local noise variance �̂�𝑛𝑜𝑖𝑠𝑒,𝑗

2 that have

defined earlier for previous method, are also reused for this method.

Figure 3.5: Image denoising using locally adaptive thresholding

Let w2k represents the parent of w1k (w2k is the tetrolet coefficient at the same position as the k-th

tetrolet coefficient w1k, but at the next coarser scale). Then

𝑦1𝑘 = 𝑤1𝑘 + 𝑛1𝑘 𝑦2𝑘 = 𝑤2𝑘 + 𝑛2𝑘 (3.17)

noisy image

g

denoised

image 𝐟

GT

bivariate

shrinkage

function

T

coefficient

adaptive

threshold

estimation

�̂�

𝐖T−1

inverse

tetrolet

transform

discrete

tetrolet

transform

WT

local noise

variance

estimation

48

where 𝑦1𝑘 and 𝑦2𝑘 are noisy observations of 𝑤1𝑘 and 𝑤2𝑘 ; and 𝑛1𝑘 and 𝑛2𝑘 are noise samples.

We can write

𝐲𝑘 = 𝐰𝑘 + 𝐧𝑘, 𝑘 = 1 … no. of tetrolet coeffs. (3.18)

where 𝐰𝑘 = (𝑤1𝑘, 𝑤2𝑘), 𝐲𝑘 = (𝑦1𝑘, 𝑦2𝑘) and 𝐧𝑘 = (𝑛1𝑘, 𝑛2𝑘).

Let us define subband P(S), which is the subband of the parents of the coefficients of the

subband S. For example, if S is 𝐒31 , then P(S) is 𝐒3

2 , or if S is 𝐒22 , then P(S) is 𝐒2

3 . In the

observation model given in Eq. (3.17), 𝑦1𝑘, 𝑤1𝑘, 𝑛1𝑘 S and 𝑦2𝑘, 𝑤2𝑘, 𝑛2𝑘 P(S).

The estimate of 𝑤1𝑘can be obtained as

�̂�1𝑘 = 𝑻(𝑦1𝑘, 𝑦2𝑘, 𝜆𝑘) =(√𝑦1𝑘

2 +𝑦2𝑘2 −𝜆𝑘)

+

√𝑦1𝑘2 +𝑦2𝑘

2. 𝑦1𝑘 (3.19)

which can be interpreted as a bivariate shrinkage function.The function (a)+ is defined as

(a)+ = { 0, if a < 0 a, if a > 0

(3.20)

The term 𝜆𝑘 in Eq. (3.19) is the threshold for the k-th coefficient, computed as

𝜆𝑘 =√3�̂�𝑛𝑜𝑖𝑠𝑒,𝑗

2

𝜎𝑘, (3.21)

where the local noise variance �̂�𝑛𝑜𝑖𝑠𝑒,𝑗2 is estimated similarly as in Eq. (3.13).

The estimator in Eq. (3.19) uses the threshold 𝜆𝑘 which in turn requires the prior knowledge of

the local noise variance �̂�𝑛𝑜𝑖𝑠𝑒,𝑗2 and the marginal variance 𝜎𝑘

2 for each tetrolet coefficient. In this

procedure, the marginal variance for the k-th coefficient will be estimated using neighboring

coefficients in the region N(k). Here N(k) is defined as all coefficients within a square-shaped

window that is centered at the k-th coefficient as illustrated in Fig. 3.6.

49

Figure 3.6:Illustration of neighborhood N(k)

Let us assume that we are trying to estimate the marginal variances 𝜎𝑘2 for the k-th wavelet

coefficient. From the observation model in Eq. (3.17), we have

𝜎𝑦𝑘2 = 𝜎𝑘

2 + 𝜎𝑛𝑜𝑖𝑠𝑒,𝑗2 (3.22)

where 𝜎𝑦𝑘2 is the marginal variance of noisy observations y1k and y2k. Since y1k and y2k are modeled

as zero mean, 𝜎𝑦𝑘2 can be found empirically by

�̂�𝑦𝑘2 =

1

|𝑁(𝑘)|∑ 𝑦𝑖

2

𝑦𝑖∈𝑁(𝑘)

(3.23)

where |𝑁(𝑘)|is the size of the neighborhood N(k). Then using Eq. (3.22), 𝜎𝑘 can be estimated as

�̂�𝑘 = √(�̂�𝑦𝑘2 − �̂�𝑛𝑜𝑖𝑠𝑒,𝑗

2 )+

(3.24)

The threshold computed in Eq. (3.21) will be used to obtain the elements �̂�1𝑘 of �̂�𝑙𝑗 (the estimate

of 𝐰𝑙𝑗) according to the shrinkage function T in Eq. (3.19). The final thresholded result �̂� and the

reconstructed image 𝐟𝑟𝑒𝑐 are obtained similar to as in Eqs. (3.15) and (3.16), respectively. Table

3.2 gives the algorithm for the proposed locally adaptive thresholding-based denoising method.

y2k (noisy parent) scale j + 1

coarser

scale j

y1k (noisy child)

N(k): neighbor

coefficients

k-th pixel

subband P(S)

subband S

50

In this algorithm, whole above procedure is iterated for every admissible tetromino covering, i.e.

we obtained the reconstructed data for every admissible tetromino covering. Such iterative

method leads to redundancy in the reconstructed results. The pixel values of the desired denoised

image are obtained by the averaging of these results over total number of tetromino coverings

considered in the experiment. Fig. 3.7 demonstrates the flow of execution of above three stages

for a given tetromino covering.

Table 3.2: Algorithm for the proposed locally adaptive thresholding-based denoisingmetho

Method [Jain and Tyagi (2015d)]

Algorithm’s prerequisite: Similar as in Table 3.1.

Inputs: The input noisy image 𝐠 = [g(𝑥, 𝑦)]𝑥,𝑦=1𝑁 with N = 2

K, K N, the number of

decomposition levels J (default value, log2N – 1), number of admissible tetromino coverings

num_cov (default value, 117), and size |𝑁(𝑘)| of the local neighborhood N(k) (default size, 7

7).

Output: The denoised image 𝐟.

Initialization: Initialize 𝐟 = 0. Set c =1.

Iterate as follows:

Step (1) Considering image g, tetromino configuration c, and the number of decomposition

levels J as input arguments, obtain the multiscale tetrolet decomposition

𝐆T = {𝐠𝐽; 𝐒1𝑗; 𝐒2

𝑗; 𝐒3

𝑗, 𝑗 = 𝐽, 𝐽 − 1, … … ,1} using (3.7).

Step (2)For each decomposition level (j = 1J ):

(c) Calculate the local noise variance �̂�𝑛𝑜𝑖𝑠𝑒,𝑗2 using (3.13).

(d) For each subband (S = 𝐒𝑙𝑗, l =1, 2, 3):

For each tetrolet coefficient, y1k S (k = 1no. of tetrolet coefficients in

subband S):

(i) Compute �̂�𝑦𝑘2 using (3.23).

(ii) Compute�̂�𝑘 using (3.24).

(iii)Compute the threshold 𝜆𝑘 using (3.21).

(iv) Estimate each coefficient using 𝜆𝑘 in (3.19).

Step (3) Get the thresholded wavelet output �̂� using (3.15).

Step (4) Obtain the reconstructed image 𝐟𝑟𝑒𝑐 using (3.16).

Step (5) 𝐟 = 𝐟 + 𝐟𝑟𝑒𝑐.

Step (6) Increase c (c = c + 1) and go to Step (1).

Termination criterion: If c > num_cov, stop iteration.

Output: 𝐟= 𝐟 / num_cov

51

Figure3.7: Flow of sequential execution for a given tetromino configuration

3.5 Experimental Results

The proposed image denoising method is applied to several test images corrupted by simulated

additive Gaussian white noise at six different power level 𝜎 [10, 15, 20, 25, 30, 35]. The

I/P noisy image

Divide into 4 4 blocks

Obtain the four low-pass and

twelve high-pass coefficients in

each block

Rearrange low/high

frequency coefficients into a

2 2 block

Store

tetromino

covering

Store

subbands

Store

subbands

Store

subbands

Reconstructed

image

Tetrolet

reconstruction

Further

decomposition

N

Y

Subbands of

high-pass

tetrolet

coefficients

Finish the

decomposition

?

Locally adaptive

thresholding

Subbands of

thresholded

high-pass

tetrolet coefficients

Subbands of

low-pass

tetrolet

coefficients

Modified

tetrolet

image

52

denoising process has been iterated for ten different noise realizations for each standard

deviation and the results are averaged over these ten runs. The test set comprises total 52 test

images in which 49 images have been taken from the standard grayscale image dataset

[http://decsai.ugr.es/cvg/CG/base.htm] and restthree are well-known images Lena, cameraman,

and peppers, respectively. A subset of test images, shown in Fig. 3.8, is considered in the

subsequent discussions. The graphs in Fig. 3.9, 3.11 and 3.12 are obtained using 128128

images.

(a) Lena (b) Cameraman (c) Barbara (d) Man (e) Boat (f) Peppers

Figure 3.8: Images used in experiments

3.5.1 Experimental setup

We estimate a set of parameters used by the algorithmspresented in Tables 3.1 and 3.2: number

of decomposition levels J, number of admissible tetromino coverings num_cov, and size

|𝑁(𝑘)|of the local neighborhood named N(k). These parameters are estimated using a set of

images different from those shown in the comparisons. Once the parameters are set, they are kept

fixed throughout the comparisons to other methods.

In the proposed method, wavelet coefficients are obtained using Haar-type wavelets (tetrolets)

and the results are obtained using four decomposition levels (i.e. J = 4). Therefore, for all other

wavelet-based methods (SURELET, Bayes, Bivariate), the Haar (Daubechies-1) wavelet with

four decomposition levels are used for fair comparison.

The denoising results are obtained by averaging the collected samples over a number of

tetromino coverings used. Different values of num_cov are considered in the experiments. In

53

Fig.3.9, the PSNR responses of denoised images that are obtained with our locally adaptive

thresholding-based denoising method,are plotted against number of tetromino coverings being

averaged.

It can be observed that redundancy improves the denoising performance by a huge factor. The

denoising performance improves as more and more tetromino coverings are averaged. Though,

the performance improves rapidly at the beginning, but gets saturated after a certain point. The

primary reason for this is the duplication in the generated tetrolet coefficients. Figure 3.10 shows

the duplication in the coefficients generated by selecting different tetromino coverings. It shows

that after a certain point, further raise in number of tetromino coverings do not improve the

image quality by much. Dropping them from consideration improves the speed with very little or

no cost to the image quality. Considering this fact and based on experimental observations, the

appropriate choice for num_cov is set to be 16.

Another important parameter is the size |𝑁(𝑘)| of the neighborhood windowwhich is a crucial

parameter used in the experiment. We have used |𝑁(𝑘)| as 7 7 in our experiments. However,

we have also tested different options for |𝑁(𝑘)|, but 7 7 resulted in best performance.

3.5.2 Comparisons

The proposed methods are compared to state-of-the-art methods to assess the denoising

effectiveness. For convenience in deciphering, we are using the conventions Proposed 1 and

Proposed 2, respectively for proposed method and its variant. The Proposed 1 method is

compared with the wavelet-based methods, namely, SURELET [Blu and Luisier (2007)],

Bivariate [Sendur and Selesnick (2002b)] and Bayes [Chang et al. (2000a)] while the variant,

Proposed 2 method, along with these wavelet-based methods, is also compared with the non

wavelet-based methods, namely, LLSURE [Qiu et al. (2013)], guided image filter (GIF) [He et

54

al. (2010)], fast bilateral filter (FBF) [Yang et al. (2009)] and total variation (TV) [Rudin et al.

(1992)]. All these state-of-the-art denoising approaches have been briefly discussed in Section

2.3. The PSNR (in dB), SSIM [Wang et al. (2004)] and FOM [Pratt (2012)] values of the

denoised images relative to their original images using such methods are reported in Tables 3.3,

3.4 and 3.5, respectively. Mathematical background for the performance metrics PSNR, SSIM

and FOM has been discussed in Section 1.5. Values marked in bold indicate the best results

among all compared methods; values with underline indicate the best results among the wavelet-

based methods compared with Proposed 1 method and values with italic indicate the best results

among the wavelet-based methods compared with Proposed 2 method.

The right segment in each table shows the results of wavelet-based denoising methods. The

results obtained with both the proposed methods reveal significant gain when compared with

such methods, specially considering Bivariate and Bayes methods. Both the proposed methods

achieve better results with PSNR and FOM, while close to the SURELET method regarding the

SSIM measure.When we focus on Proposed 2 method, the results are comparable to those

obtained with LLSURE and superior to GIF, FBF and TV.

In Fig. 3.11 which shows the comparison of the PSNR performances of the Proposed 1 method

with other state-of-the-art denoising methods, starting from (a) to (f) it can be observed that the

this method performs better in comparison to others in most of the cases. Though, at times the

SURELET method marginally beats it. In Fig. 3.12 which shows the comparison of the PSNR

performances of the Proposed 2 method with other state-of-the-art denoising methods, starting

from (a) to (f) it can be observed that the proposed method performs better in comparison to

others in most of the cases. Though, at times the LLSURE method marginally beats it. Figures

3.13, 3.14, 3.15 and 3.16 demonstrate the visual comparison among all the methods, including

55

the proposed method with respect to the sample test images: Lena, Barbara, Boat and Peppers,

respectively.

Bayes wavelet-based methods tend to produce smoothed results in homogeneous regions.

Nevertheless, certain features such as edges are affected. As the proposed denoising methods that

respectively employ an epsilon-median filtering which provides a subband-dependent

thresholding and a bivariate shrinkage function which provides a locally adaptive (coefficient-

dependent) thresholding, it is observed that such adaptive thresholding schemes, in conjunction

with the tetrolet concept and the computation of local noise variance, effectively reduces noise

while preserving features of the image.

The SURELET method produces a similar result on edges. However, as can be perceived

through Fig. 3.13, 3.14, 3.15 and 3.16, the proposed methods outperform SURELET in

homogeneous regions, producing smoother results. That can be clearly observed in the various

homogeneous regions in Fig. 3.16.

The GIF, FBF, Bivariate and Bayes methods fail to smooth images when noise increases to

higher levels. These produce good results at lower 𝜎 values, but give poor denoised images at

higher noise levels. The TV method tends to oversmooth the image. Due to this reason some fine

structures of the original image are not being preserved in the filtered output image. All such

issues are greatly resolved by proposed method and its variant.

56

Table 3.3: Performance of various methods as measured by PSNR

Non wavelet-based methods

Wavelet-based methods

LLSURE GIF FBF TV

SURELET Bivariate Bayes Proposed 1 Proposed 2

method method method method

Method method method method method

Lena

= 10 31.33 30.56 30.53 30.47

30.94 30.02 29.97 30.45 30.72

= 15 29.09 28.23 28.21 28.40

28.50 27.12 27.30 28.78 29.45

= 20 27.76 26.56 26.12 26.92

26.99 25.14 25.96 27.25 28.25

= 25 26.38 25.52 24.14 25.84

26.07 23.84 23.75 26.31 26.38

= 30 25.37 24.35 23.12 24.94

25.04 23.10 23.09 25.54 25.70

= 35 24.45 23.89 22.78 24.34

24.42 22.20 22.51 24.73 24.81

Cameraman

= 10 32.93 31.86 32.10 31.21

32.35 30.96 30.91 32.01 32.02

= 15 30.65 30.10 29.60 29.02

29.79 27.93 28.12 30.57 30.97

= 20 29.20 27.72 27.71 27.68

28.27 25.88 26.56 28.62 28.74

= 25 27.75 25.93 26.07 26.55

26.85 24.35 25.25 27.24 27.55

= 30 26.55 24.65 24.72 25.69

25.66 23.29 24.16 26.18 26.34

= 35 25.54 23.62 23.70 25.13

24.63 21.75 23.38 25.03 25.23

Barbara

= 10 30.57 30.03 29.81 29.89

30.30 29.24 29.56 29.42 29.92

= 15 28.20 27.32 27.10 27.68

26.84 26.43 27.12 28.15 28.68

= 20 26.65 25.78 25.86 26.26

26.19 24.40 24.93 26.61 26.81

= 25 25.64 24.13 24.88 25.25

25.25 23.26 23.96 25.78 25.78

= 30 24.83 23.09 23.98 24.36

24.24 22.02 22.52 24.45 24.49

= 35 23.39 22.36 23.14 23.58

23.58 21.42 22.14 23.72 23.72

Man

= 10 30.02 29.32 29.34 29.25

29.74 29.17 28.90 30.81 31.45

= 15 27.76 26.90 26.89 27.25

27.29 26.37 26.63 28.15 28.49

= 20 26.26 25.19 25.47 25.86

25.83 24.57 24.98 26.70 26.93

= 25 25.64 24.16 24.08 24.80

24.98 23.19 23.72 25.24 25.34

= 30 24.71 23.52 23.26 24.04

23.97 22.50 22.55 24.35 24.42

= 35 23.80 22.68 22.44 23.26

22.80 21.81 21.72 23.26 23.31

Boat

= 10 30.88 30.03 29.95 29.64

30.68 29.96 29.76 30.87 30.55

= 15 28.60 27.62 27.69 27.43

28.07 27.12 27.16 28.48 28.60

= 20 26.87 25.42 25.48 26.02

26.51 25.00 24.99 27.18 27.29

= 25 25.76 24.32 24.49 24.97

25.34 23.90 24.07 25.52 25.42

= 30 24.48 23.29 23.51 24.19

24.61 22.53 23.07 25.01 25.03

= 35 23.81 22.59 22.87 23.55

23.95 21.85 21.95 24.40 24.42

Peppers

= 10 30.91 30.36 30.37 29.94

30.53 29.62 29.58 30.06 30.06

= 15 28.60 27.71 27.72 27.61

28.01 26.63 27.04 28.29 28.31

= 20 27.35 26.07 26.15 26.08

26.58 24.70 24.96 26.83 26.96

= 25 25.86 24.48 24.73 24.87

25.24 23.27 23.99 25.58 25.77

= 30 24.74 23.65 23.86 23.86

24.50 22.41 22.57 24.89 25.03

= 35 23.97 22.76 22.89 23.07 23.86 21.13 21.59 24.28 24.39

57

Table 3.4: Performance of various methods as measured by SSIM

Non wavelet-based methods

Wavelet-based methods

LLSURE GIF FBF TV

SURELET Bivariate Bayes Proposed 1 Proposed 2

Method method method method

Method method method method method

Lena

= 10 0.92 0.91 0.91 0.91

0.90 0.85 0.88 0.91 0.92

= 15 0.89 0.89 0.87 0.87

0.85 0.76 0.82 0.87 0.88

= 20 0.85 0.83 0.84 0.83

0.80 0.67 0.77 0.83 0.84

= 25 0.81 0.80 0.79 0.80

0.76 0.60 0.65 0.78 0.78

= 30 0.78 0.76 0.74 0.77

0.72 0.54 0.62 0.74 0.72

= 35 0.75 0.74 0.71 0.74

0.68 0.48 0.56 0.70 0.65

Cameraman

= 10 0.93 0.91 0.92 0.91

0.90 0.80 0.82 0.90 0.91

= 15 0.89 0.89 0.90 0.88

0.85 0.67 0.75 0.85 0.87

= 20 0.86 0.85 0.83 0.86

0.80 0.57 0.69 0.81 0.82

= 25 0.82 0.80 0.81 0.83

0.76 0.50 0.65 0.76 0.76

= 30 0.79 0.77 0.75 0.81

0.72 0.44 0.62 0.72 0.73

= 35 0.76 0.73 0.72 0.80

0.68 0.40 0.59 0.70 0.70

Barbara

= 10 0.92 0.90 0.91 0.91

0.91 0.87 0.89 0.91 0.92

= 15 0.88 0.87 0.89 0.86

0.85 0.77 0.83 0.87 0.89

= 20 0.83 0.81 0.81 0.82

0.80 0.69 0.75 0.83 0.84

= 25 0.80 0.80 0.79 0.78

0.76 0.62 0.71 0.78 0.79

= 30 0.77 0.78 0.77 0.74

0.73 0.55 0.62 0.74 0.73

= 35 0.74 0.71 0.72 0.71

0.69 0.49 0.60 0.70 0.69

Man

= 10 0.90 0.91 0.92 0.89

0.89 0.87 0.88 0.89 0.89

= 15 0.85 0.84 0.86 0.83

0.83 0.78 0.81 0.84 0.85

= 20 0.80 0.79 0.78 0.78

0.78 0.73 0.75 0.80 0.81

= 25 0.75 0.72 0.70 0.74

0.74 0.62 0.68 0.75 0.76

= 30 0.71 0.69 0.68 0.70

0.69 0.55 0.61 0.71 0.70

= 35 0.69 0.68 0.66 0.66

0.65 0.50 0.56 0.67 0.67

Boat

= 10 0.92 0.91 0.92 0.90

0.90 0.85 0.87 0.91 0.90

= 15 0.87 0.88 0.89 0.85

0.85 0.75 0.80 0.86 0.88

= 20 0.83 0.82 0.81 0.81

0.79 0.66 0.68 0.81 0.83

= 25 0.79 0.81 0.79 0.77

0.74 0.58 0.66 0.76 0.77

= 30 0.74 0.72 0.70 0.73

0.70 0.52 0.62 0.72 0.71

= 35 0.71 0.69 0.69 0.70

0.66 0.47 0.52 0.68 0.67

Peppers

= 10 0.93 0.91 0.90 0.93

0.91 0.86 0.88 0.91 0.93

= 15 0.90 0.88 0.89 0.89

0.86 0.77 0.82 0.88 0.90

= 20 0.87 0.86 0.83 0.86

0.82 0.68 0.75 0.83 0.86

= 25 0.84 0.82 0.81 0.83

0.78 0.62 0.72 0.78 0.81

= 30 0.81 0.79 0.78 0.80

0.74 0.56 0.63 0.74 0.76

= 35 0.79 0.76 0.74 0.77 0.71 0.51 0.58 0.72 0.73

58

Table 3.5: Performance of various methods as measured by FOM

Non wavelet-based methods

Wavelet-based methods

LLSURE GIF FBF TV

SURELET Bivariate Bayes Proposed 1 Proposed 2

Method method method method Method method method method method

Lena

= 10 0.92 0.92 0.91 0.90

0.94 0.96 0.94 0.90 0.91

= 15 0.90 0.90 0.89 0.88

0.92 0.95 0.93 0.89 0.90

= 20 0.86 0.85 0.86 0.83

0.91 0.93 0.93 0.90 0.91

= 25 0.84 0.84 0.83 0.81

0.89 0.90 0.90 0.91 0.93

= 30 0.84 0.82 0.83 0.80

0.88 0.88 0.88 0.88 0.89

= 35 0.82 0.82 0.81 0.76

0.88 0.83 0.88 0.87 0.88

Cameraman

= 10 0.88 0.88 0.87 0.88

0.90 0.95 0.94 0.91 0.92

= 15 0.86 0.85 0.85 0.86

0.89 0.92 0.93 0.89 0.89

= 20 0.83 0.83 0.85 0.81

0.89 0.89 0.88 0.88 0.89

= 25 0.81 0.82 0.83 0.77

0.87 0.78 0.78 0.86 0.87

= 30 0.78 0.80 0.81 0.75

0.84 0.65 0.72 0.83 0.84

= 35 0.79 0.80 0.78 0.72

0.80 0.64 0.69 0.80 0.82

Barbara

= 10 0.92 0.93 0.92 0.89

0.93 0.96 0.95 0.91 0.92

= 15 0.89 0.90 0.90 0.84

0.92 0.94 0.94 0.91 0.92

= 20 0.82 0.86 0.85 0.81

0.90 0.92 0.92 0.90 0.91

= 25 0.82 0.85 0.83 0.78

0.89 0.90 0.90 0.90 0.91

= 30 0.80 0.81 0.81 0.75

0.87 0.88 0.88 0.89 0.91

= 35 0.76 0.77 0.80 0.72

0.86 0.86 0.87 0.88 0.90

Man

= 10 0.92 0.90 0.91 0.89

0.93 0.96 0.95 0.92 0.93

= 15 0.87 0.86 0.86 0.84

0.91 0.95 0.94 0.91 0.92

= 20 0.82 0.84 0.83 0.82

0.90 0.92 0.92 0.91 0.92

= 25 0.77 0.78 0.81 0.76

0.88 0.89 0.90 0.89 0.90

= 30 0.75 0.74 0.78 0.72

0.87 0.87 0.88 0.88 0.89

= 35 0.73 0.72 0.76 0.69

0.86 0.84 0.85 0.87 0.88

Boat

= 10 0.95 0.93 0.92 0.92

0.95 0.94 0.96 0.94 0.94

= 15 0.92 0.91 0.91 0.90

0.94 0.92 0.93 0.93 0.95

= 20 0.90 0.90 0.91 0.84

0.92 0.88 0.90 0.92 0.93

= 25 0.85 0.86 0.87 0.83

0.87 0.85 0.85 0.90 0.93

= 30 0.83 0.84 0.82 0.75

0.85 0.81 0.82 0.86 0.88

= 35 0.80 0.79 0.80 0.75

0.82 0.75 0.79 0.82 0.84

Peppers

= 10 0.94 0.94 0.92 0.93

0.96 0.96 0.96 0.93 0.94

= 15 0.90 0.89 0.89 0.88

0.93 0.95 0.94 0.92 0.92

= 20 0.85 0.84 0.86 0.84

0.92 0.92 0.93 0.90 0.90

= 25 0.84 0.83 0.84 0.81

0.89 0.91 0.91 0.89 0.92

= 30 0.81 0.80 0.80 0.78

0.87 0.89 0.90 0.89 0.90

= 35 0.80 0.78 0.79 0.76 0.89 0.87 0.88 0.89 0.90

59

(a) sigma =10 (b) sigma = 15

(c) sigma = 20 (d) sigma = 25

(e) sigma = 30 (f) sigma = 3

Figure3.9: PSNR vs number of tetromino coverings being averaged

Figure3.10: Duplicate tetrolet coefficients in two different tetromino coverings

These pixels will generate same tetrolet

coefficients

60

(a) (b)

(c) (d)

(e) (f)

Figure 3.11: PSNR performance graphs of the Proposed 1 method for test images

61

(a) (b)

(c) (d)

(e) (f)

Figure 3.12: PSNR performance graphs of the Proposed 2 method for test images

62

(a) Original (b) Noisy (=30) (c) LLSURE

(d) GIF (e) FBF (f) TV

(g) SURELET (h) Bivariate (i) Bayes

(j) Proposed 1 (k) Proposed 2

Figure 3.13: Denoising results for image ‘Lena’

63

(a) Original (b) Noisy (=30) (c) LLSURE

(d) GIF (e) FBF (f) TV

(g) SURELET (h) Bivariate (i) Bayes

(j) Proposed 1 (k) Proposed 2

Figure 3.14: Denoising results for image ‘Barbara’

64

(a) Original (b) Noisy (=30) (c) LLSURE

(d) GIF (e) FBF (f) TV

(g) SURELET (h) Bivariate (i) Bayes

(j) Proposed 1 (k) Proposed 2

Figure 3.15: Denoising results for image ‘Boat’

65

(a) Original (b) Noisy (=30) (c) LLSURE

(d) GIF (e) FBF (f) TV

(g) SURELET (h) Bivariate (i) Bayes

(j) Proposed 1 (k) Proposed 2

Figure 3.16: Denoising results for image ‘Peppers’

66

3.6 Conclusions

In this chapter, we have presented an adaptive method and its variant in tetrolet domain to

achieve edgepreserving image denoising. In the proposed method, we have employed a subband

adaptive epsilon-median filtering scheme which is based on estimation of local noise variance, to

threshold the tetrolet coefficients. This method has several desirable features. First, redundancy

of tetrolet coefficients is exploited in the method to achieve significant gain in denoising

performance. Second, estimating the term noise variance, used in the computation of thresholds,

locally at each resolution scale makes it more beneficial as it takes the noise strength at that scale

into consideration. Third, thresholding is done in subband-dependent manner which shows

improvisation over conventional thresholding schemes that make use of a universal threshold.

In the variant of this method, a bivariate shrinkage function which provides a coefficient-

dependent thresholding, is employed. In fact, this method is an extension of the first method as it

inherits the aspects of exploiting the redundancy and computing the noise level from the first

method and incorporates some other aspects.Firstly, the interscale statistical

dependenciesbetween tetrolet coefficients are exploited, as this property of coefficients also leads

to achieve efficient denoising. Secondly, thresholding is done in coefficient-dependent manner

rather than the subband-dependent approach suggested in the first method, which can

characterize the local features of the image more efficiently.

The proposed method denoises square natural gray scale images with dimensions in the

exponential order of two. If the image is not a square then it has to be extended to make it a

suitable input image. The quantitative and qualitative analysis of the experimental results

indicates that the proposed methods produce superior results compared to the methods based on

the wavelet transform and results comparable to other well-known denoising methods.