Super Resolution Image Using Dictionary Technique

JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 17, ISSUE 2, FEBRUARY 2013

38

Super Resolution Image Using Dictionary Technique

Ghada M. Shaker 1, Alaa A. Hefnawy1, Fathi E. Abd El-Samie2 and Moawed I.Dessouky2

Abstract- This paper addresses the problem of generating a super resolution (SR) image from a degraded input image; using a

hybrid algorithm to form dictionary from low resolution (LR) image to produce high resolution (HR) one. A patch-based, locally

adaptive denoising method based on clustering the given degraded image into regions of a like geometric structure has been proposed. We will show the effectiveness of sparsity as a prior for regularizing where we utilize as features the local weight functions derived from steering kernel regression, to effectively perform such clustering. Next, we model each region (or cluster)

which may not be spatially contiguous these by “learning” a best basis describing the patches within that cluster. This learned basis (or “dictionary”) is then employed to optimally estimate the underlying pixel values using a kernel regression framework. We illustrate the overall algorithm’s capabilities with several examples. Sparse K-SVD algorithm is applied for optimization to speed up sparse coding. Comparison with sparse coding method shows sparse dictionary is more compact and effective.

Keywords- : Super-Resolution (SR), Sparse Coding, Sparse Representation, Learning-based, Sparse Dictionary.

1. INTRODUCTION

High-resolution images are important in

many fields, like in medical image diagnosis,

remote sensing, high-resolution video and other

fields; one of the main methods of obtaining HR

images is SR method. SR image reconstruction is

currently a very active area of research, as it offers

the promise of overcoming some of the inherent

resolution limitations of low-cost imaging sensors

(cell phone or surveillance cameras) allowing

better utilization of the emergent capability of

high-resolution displays (e.g., high-definition

LCDs) [1].

The SR task is direct as the inverse

problem of recovering the original HR image by

fusing multiple LR images of the same scene [2],

based on assumptions or prior knowledge about

the generation model from the HR image to the LR

images However, SR image reconstruction is

commonly a severely ill-posed problem because of

the insufficient number of LR images, ill-

conditioned registration and unknown blurring

operators and the solution from the reconstruction

limitation is not unique. Another method assumes

the LR images are wrapped, blurred and down-

sampled version of the corresponding high-

resolution image, and through modeling this

process, HR image can be inversed from the

sequence of LR mages. But some of the parameters

of the model are hard to define. Another category

of SR methods that can overcome this difficulty is

learning based approaches, which use a learned

contributed prior to predict the correspondence

between LR and HR image patches. Nevertheless,

the above methods typically need enormous

databases of millions of HR and LR patch pairs to

make the databases expressive enough.

Learning-based super-resolution method

builds association between both high-and low-

resolution image patches (or patch features), and

defines the association as prior for SR

reconstruction. There are many methods of

learning-based SR method; Freeman et al. [3]

assumes that an image consists of three

components: high, middle and low (h, m, l)

frequency component. Also, the method is based

on a hypothesis that the LR image is the result

which is discarded HF component from the

corresponding HR one. The aim of learning-based

method is to restore the HF component from low-

and middle-frequency component: maximizing the

probability ( / , )p h m l . A method that picks the

first order and second order gradients of Laplacian

and Gaussian pyramid of the image as its feature

space, this method may be called second kind

learning-based super resolution method.

Hertzmann et al. [4], Efros and Freeman [5]

presented a local feature transform method, the

idea of image analogies could also been used for

image super-resolution, which may be called the

third kind learning-based super-resolution 1 Electronics Research Institute

2 Faculty of Eng., Menofia Univ.


39

algorithm. In order to break through the limits,

Yang et al [6], presented a learning-based

algorithm based on sparse coding, which

effectively builds sparse association between high

and low-resolution image patches and gets

excellent results.

Most learning-based SR methods have two

steps [7]: matching K similar feature examples

from training set; optimization estimation based on

the K examples. This method is restricted by the

quality of the K candidates, which limited the

freedom of the model and wasted prior

information of other patches in the training set.

During sparse dictionary coding, the training

examples consist of feature patches rather than

original image patches. We select first-order and

second-order gradients of the LR patches as the

feature, which is same with the methods of Chang

et al. [8] and Yang et al. [9]. Rubinstein et al [10],

presented a parameter model called sparse

dictionary (SD) to balance efficiency and

adaptively, which decomposes each atom in the

dictionary by a basis dictionary. This model has a

simple and high adaptive structure. Follows works

in the papers [4, 10-14]. Learning-based super-

resolution algorithms have been intensively

studies. In this paper we suggest a framework for

denoising by learning an appropriate basis

function to describe image patches after applying

wavelet transform on noisy image. Using of such

basis functions to describe geometric structure has

been earlier explored.

2. SUPER-RESOLUTIONS FROM SPARSITY

In recent years, the beginning of such high

resolution imaging devices, the signal sensors are

becoming increasingly dense in terms of the

number of pixels per unit area of the sensor. This

means that relatively speaking; the overall sensor

is increasingly more prone to the effects of noise.

Hence, denoising remains an important research

problem in image processing. Before we deal with

the image denoising problem, we first define our

observation model as [14]

i iy = z ηi (1)

where iz is the original pixel intensity of the ith

pixel observed as iy after being corrupted by zero

mean independent identically ddiissttrriibbuutteedd aaddddiittiivvee

nnooiissee i . Many recently introduced denoising

methods are patch based in nature [15]-[16]. Hence,

it is useful to formulate the observation model in

terms of image patches as well. Decomposing the

image into overlapping patches, we can also write

the data model as [14]

i i iy=z+η (2)

where iz is the original image patch with the

ith pixel at its center written in a vectorized format

and iy is the observed vectorized patch corrupted

by a noise vector iη . Denoising the image is thus

solving the inverse problem to estimate the pixel

intensities iz . Many linear and nonlinear methods

have been proposed to solve this problem. Very

effective method that achieves excellent denoising

results is the K-SVD algorithm proposed by

Aharon [17]-[18]. In this method, an optimal over

complete dictionary of image patches adapted for

the observed noisy data is first determined.

Assuming that each image patch is sparse

representable, denoising is carried out by coding

each patch as a linear combination of only a few

patches in the dictionary.

Denoising the image is thus solving the

inverse problem to estimate the pixel intensities iz .

Proposed a simple patch-based algorithm that

takes advantage of the presence of repeating

structures in a given image and performs a

weighted averaging of pixels with similar

neighborhoods to suppress the noise we note that

“denoising” is a special case of the regression

problem where samples at all desired pixel

locations are given, but these samples are

degraded, and need to be restored [13]. While

kernel regression (KR) is a well studied method in

statistics and signal processing, KR is identified as

a nonparametric approach that requires minimal

assumptions, and hence the framework is one of

the appropriate approaches to the regression

problem. The steering kernel regression (SKR)

method is distinguished by the way it forms the

local regression weights which weights can be

considered to be a measure of similarity of a group

of pixels compared to a certain pixel or

neighborhood of pixels under consideration. The

SK in this particular case can be expressed as [13]:-


40

j

2 2

det( ) ( ) ( )exp{ }

2 2

i j j i j

ij

Tx x x x

h h

C Cw (3)

where ijw describes the similarity of the

jth pixel with respect to the ith pixel, 2, jix x R

denote the location of the ith and the jth pixels

respectively and h is a global smoothing

parameter which controls the support of the

steering kernel. The matrix jC denotes the

symmetric gradient covariance matrix formed from

the estimated vertical and horizontal gradients of

the jth pixel .It can be expressed as a matrix that

allows the Gaussian to align with the underlying

image structure by a combination of elongation

and rotation operators. Mathematically, it can be

expressed in the form [13]

j= j θj jT

θjC γU ΛU (4)

where θjU represents the rotation

operator that aligns the Gaussian to the

direction j of the underlying edge, jΛ denotes

the elongation operator, and jγ acts as a scaling

parameter as shown in Fig.1. It can be clearly seen

how the kernels are representative of the

underlying image structure.

Additionally, it can be seen that different

locations in the image having different intensities

but similar underlying structure still result in

similarly shaped kernels. For the SKR method the

data is modeled to be locally polynomial where the

image is assumed to be sufficiently smooth

(locally) to allow fitting of a polynomial of some

low degree (usually 0, 1 or 2). We can then rewrite

the data model of equation (2) [14]:-

i i iy=φβ+η (5)

where the dictionary φ is a matrix

whose columns are formed from polynomial basis

vectors and iβ is the vector of coefficients. This

paper presents a novel super resolution algorithm

based on sparse dictionary, which builds sparse

association between image feature patches to direct

super-resolution reconstruction.

3. PROPOSED METHOD

The weights go behind edge directions

and dictate the contribution of various pixels in a

local neighborhood of the pixel to be denoised.

However, this regression framework has two

inherent restrictions: the basis function remains the

same (polynomial) over the entire image, and the

order of regression is constant for the entire image.

These drawbacks force both smooth and textured

regions of any image to be reconstructed using the

same basis vectors (and, hence, the same order of

regression). Our proposed method here aims to

improve these problems by the use of regression

where both the type and number of basis vectors

are dictated by the given image data. They (basis

vectors) show that the second-order regression

generally leads to an improved denoising

performance, as compared to lower orders.

More specifically, once the β̂ parameters

are estimated, one can proceed to reconstruct a

vectorized version of each patch in the image as

ˆˆ iiz φβ .This method tends to find the best

global over complete dictionary and represents

each image patch as a linear combination of only a

few dictionary vectors (atoms) [19]. For each patch,

the dictionary consists of only the basis vectors (or

atoms) used for sparse coding. We note that

patches of similar geometric structure should

ideally have similar dictionaries. Our algorithm

aims to advance on the successful kernel regression

framework by eliminating the restrictions

mentioned before; namely, the static nature of the

dictionary, and the constancy of the approximation

order across the image.

A clustering-based algorithm which

consists of four stages shown in (Fig. 2): the

wavelet transform, the clustering step, the

i iγ

/i iγ

jγ

Scale

1/ i

θjR

Fig.1 A schematic representation illustrating the effects of steering matrix and

its components ( iC j θj jT

θjγ ΛR R ) on the size and shape of the

regression kernel footprint.

jΛ

i

Elongation

Rotation i


41

dictionary selection stage; and, finally, the

coefficient calculation stage.

A. Clustering

In the initial stage, our algorithm attempts

to achieve clustering to identify regions of similar

structure in the image. To achieve clustering we

need first to identify informative features from the

image. Commonly used low level features to

identify similar pixels (or patches) are pixel

intensities, gradient information, etc., or a

combination of these. We note that the steering

weights computed in a neighborhood are robust to

the presence of significant amounts of noise. At the

end of this stage we expect the image to be divided

into not necessarily contiguous regions ( )K , each

containing patches of similar structure. Hence, the

entire noisy image Y can be thought to be

composed of a union of such clusters [14]:-

K

i/j

K=1

Y = {y }KU (6)

Before we continue to perform clustering

on the weights, we need to identify a metric to

calculate the distance between two weight

functions (features). The easiest measure of

distance between the steering weights of two

patches would be to calculate the 2L or 1L norm

between them. In our work we use the 2L norm

as our distance metric. In our case, for now we

work with the simple 2L distance which proves to

be an effective distance metric to perform suitable

clustering for our algorithm.

While there are many clustering

algorithms available, we need a method that is

relatively fast and performs acceptable

segmentation of the image into geometrically

similar regions based on the steering weights. For

proposed method, we make use of a version of the

standard K-means algorithm. This clustering

algorithm is one of the simplest unsupervised

methods where the motivation of clustering is to

segment the image into some prefixed number

( )k of clusters such that for each class, the squared

distance of any feature (normalized weight) vector

to the center of the class is minimized.

K-means then aims to find an iterative

solution to compute the class centers (and, hence,

the classes K ) that minimizes the objective

function; we can see that the flat regions are

clustered in the same class, irrespective of the

difference in intensities in different regions. This

clearly shows how the clusters are formed by

considering geometric similarity between image

patches, even in the presence of considerable noise.

In general, the advantages of our clustering

scheme, especially under strong noise, easily

outweigh these shortcomings.

B. Dictionary Selection

Once the clusters are formed, we proceed

to form a dictionary best suitable to each cluster

independently. For each cluster we aim to find a

dictionary that best describes the structure of the

underlying data within that cluster. In other

words, for each image patch in a cluster K , we

want to find an estimate ˆiy which best

approximates the input vectorized patch iy .

Mathematically, we intend to find the optimal ( )k

φ and ιβ to minimize the cost function [14]:-

2 ( ) ( ) 2ˆ|| || || ( ) ||k k

k k

i i

i i

i ιy y y y φ β (7)

where ( )k

y is the mean vector of the

set /( ) { }i j KKY y ,

( )kφ denotes the dictionary

which best represents the patches in cluster K

and ιβ denotes a vector of coefficients for the

linear combination of dictionary atoms to

compute iy . Note that this allows us to learn a

Class 1

Class k

Clustering Stage

Dictionary Selection

Stage

Denoised Image

Coefficient Calculation

Stage

W Calculate Steering

Weight Noisy

Image

Y

^

Z

Fig.2. Block diagram of the proposed method.

.

.

.

.

.

.

.


42

dictionary whose form is dictated by the

underlying data and not restricted to be

polynomial, as is the case for SKR.

Choosing a dictionary ( )k

φ to minimize

this cost function is nontrivial since the

ιβ parameters are also unknown. One way to

estimate these two unknowns is by using a

numerical method of alternate minimization where

we first assume one of the two variables (say( )k

φ )

to be fixed and minimize the cost function of (7)

with respect to the other ( ιβ in our case). Once a

minimum is obtained for ιβ , it is kept fixed and we

minimize with respect to( )k

φ . This process is

carried on iteratively. However, such a method

requires good initial guesses for each of the

unknowns, where we first write down an

analytical expression for the estimate of ιβ

assuming ( )k

φ to be known. The estimate of ιβ

that minimizes the cost function in (7) is given by

[14]:-

1( ) (k)T (k) (k)T (k)

i iβ Φ Φ Φ y% (8)

This solution is then plugged into (7) and

we reformulate the problem as that of

minimization with respect to ( )k

φ alone. To

simplify the problem further, we enforce the

dictionary to be orthonormal, transforming the

optimization problem of (7) into [14]:-

( )

2ˆ arg min

k

ki

(k) (k) (k)i iΦ y -Φ β%

=( )

2

arg min )k

ki

(k) (k) (k)T (k) -1 (k)T (k)i iy -Φ (Φ Φ ) (Φ y% %

= ( )

2

arg min )k

ki

(k) (k) (k)T (k)i iy -Φ Φ y% % (9)

( ).suchthat (k) (k)TΦ (Φ = I

Moreover, constraining the dictionary to

be orthonormal satisfies the condition of ( )k

φ

having constant rank which is assumed by the

variable projection method.

C. Coefficient Calculation

Once the dictionary is formed for each

cluster, we proceed to estimate the ιβ parameters

under a regression framework. We pose this as an

optimization problem EQ (7). The dictionary now

is adapted to a specific class of image structure that

is captured by each cluster. Furthermore, the

number of principal components or dictionary

atoms that will be needed to fit a prespecified

percentage of data varies across the different

clusters. Here, for example, the smooth flat regions

of the first cluster need only one dictionary atom to

well describe each patch whereas regions with

sharp edges and texture may require more atoms.

These results in a varying order of regression

across locations depending on which cluster each

pixel belongs to. Once the ˆiβ parameters are

estimated, we can reconstruct the target patch as:-

( ) ( ) ( ) ( )

(1) (1) (2) (2) ( ) ( )ˆ ˆ ˆˆ .......k k k k

i i mk mk i iz y φ β φ β φ β

( ) ( ) ˆk k

i y φ β (10)

i K

The patches thus estimated are

overlapping, so we should perfectly optimally

combine the overlapping regions in some way to

form the final image. However, since the β̂

parameters are estimated taking into account the

local weights, the pixels in each of the estimated

patches having a high confidence in regions where

the local weights are high. As a result, the patch

reconstruction form of (10) is more accurate along

the edge directions and towards the center of the

patch under consideration (that is, wherever the

local weights are strong).

4. Simulation Results

To validate the proposed method, we

performed various experiments. We trained some

images that collected from [13], down sized them

to half of the original size then we artificially

added zero mean white Gaussian noise of different

standard deviations (variance) to produce noisy

images. The parameters that can be tuned for our

method are the number of clusters (K) for the

clustering stage and the smoothing parameter (H).

For the parrot and man image shown in Table 1,


43

the method was found to give the best K for each

standard deviation (from 5-25) when the image

was divided into different number of clusters

(from 1-13) according to each case. For each case

we calculate the MSE and PSNR value of image in

case our technique.

Figure 3 illustrates how the MSE varies

with K for additive white Gaussian noise of

standard deviation varies from (5-25) according to

table 1. Where we vary K from 1 to 13 and for each

case we calculate the MSE value of image in case

our technique. We iterate this process with

different values of standard deviation (σ) from 5 to

25 with step equal 5.which the top curve indicate

to parrot image case where K opt. (that occurs least

MSE ) are (3,7,8,13,2) for σ (5,10,15,20,25)

respectively and the down curve indicate to case of

man image where K opt. (that occurs least MSE )

are (1,1,2,2,2) for σ (5,10,15,20,25) respectively. We

put circle around optimal K on the figure 3.

Tables 2, shows the optimized values of H

(smoothing parameter) for parrot and man image

with different levels of noise (only) on image, and

compare the MSE values for noisy image with

denoised image resulted from proposed method

for different cases of K and standard deviation.

For the parrot image shown in Fig. 4, the

method was found to give the best results when

the image (with σ=25) was divided into 2 clusters

with H=3.2 and show the different level of quality.

For the man image shown in Fig. 5, the method

was found to give the best results when the image

(with σ=25) was divided into 2 clusters with H=2.8

and show the different level of quality.

Fig.3. Illustration of how the MSE varies with k for additive white

Gaussian noise of standard deviation varies from (5-25).

(c) (a) (b)

Fig.5 Comparison of denoising results on noisy parrot image

corrupted by additive white Gaussian noise of standard deviation 25.

(a) Original image, (b) noisy image (MSE=567.98) (c) Proposed

method (MSE =160.62).

(c) (b) (a)

Fig.4 Comparison of denoising results on noisy parrot image corrupted by

additive white Gaussian noise of standard deviation 25. (a) Original image,

(b) noisy image (MSE=571.29) (c) Proposed method (MSE =109.97).


44

Table 1: The MSE and PSNR values of different sigma and number of cluster for number of image.

image k 1 2 3 4 5 6 7 8 9 10 11 12 13 σ

P

A

R

R

O

T

MSE 15.24 13.69 13.48 13.56 13.53 14.81 14.88 14.02 14.05 13.65 13.51 13.52 14.39

5 PSNR 36.30 34.16 36.84 36.81 36.82 36.43 36.41 36.66 36.66 36.78 36.83 36.82 36.55

MSE 38.89 37.27 37.02 37.43 35.59 35.36 34.73 35.02 35.05 35.28 35.08 34.76 34.90

10 PSNR 32.23 32.41 32.44 32.39 32.62 32.65 32.72 32.69 32.69 32.66 32.68 32.72 32.70

MSE 61.32 62.41 62.57 61.36 61.01 60.59 59.96 59.25 60.28 59.59 59.05 60.36 59.86

15 PSNR 30.25 30.18 30.17 30.26 30.28 30.31 30.35 30.40 30.33 30.38 30.42 30.32 30.36

MSE 88.35 84.72 86.02 86.05 85.63 84.71 85.64 86.37 84.19 85.38 84.92 85.52 83.47

20 PSNR 28.67 28.85 28.78 28.78 28.80 28.85 28.80 28.767 28.878 28.82 28.84 28.81 28.92

MSE 112.3 109.84 113.21 112.95 113.34 114.73 112.20 112.46 111.72 111.49 111.2 111.17 112.16

25 PSNR 27.64 27.72 27.59 27.60 27.59 27.53 27.63 27.62 27.65 27.66 27.67 27.67 27.632

M

A

N

MSE 18.37 18.76 18.92 18.60 18.42 20.87 21.57 21.64 20.85 20.69 22.00 20.18 19.98

5 PSNR 35.49 35.39 35.36 35.44 35.48 34.94 34.79 34.78 34.94 34.97 34.71 35.08 35.13

MSE 49.73 50.23 51.07 54.05 53.68 53.93 52.70 52.46 52.29 52.04 52.51 51.92 51.27

10 PSNR 31.16 31.12 31.05 30.80 30.83 30.81 30.91 30.93 30.95 30.97 30.93 30.98 31.03

MSE 95.71 86.32 92.91 87.09 87.04 87.44 87.35 88.81 88.11 87.45 88.78 88.60 88.50

15 PSNR 28.32 28.77 28.45 28.73 28.73 28.71 28.72 28.65 28.68 28.71 28.65 28.66 28.66

MSE 145.8 138.04 142.40 142.41 141.06 140.61 140.91 140.48 140.10 139.97 139.02 138.50 138.73 20

PSNR 26.49 26.73 26.59 26.59 26.64 26.65 26.64 26.65 26.67 26.67 26.7 26.72 26.71

MSE 145.8 138.04 142.29 142.41 141.16 140.63 140.93 141.42 140.11 138.98 138.5 138.94 138.52

25 PSNR 26.49 26.73 26.59 26.59 26.64 26.65 26.64 26.63 26.667 26.70 26.70 26.70 26.72

Tables 3, we compare the MSE values for

noisy image with denoised image resulted from

wavelet denoised proposed method and proposed

method for different cases of K and standard

deviation. We note that wavelet technique can

perform well for high noise levels at (σ =25, 20, 15),

while it fail for the low one at (σ=10, 5), where the

MSE increases. In contrary the proposed technique

reduce amount of noise in all cases. In addition, we

observe that the number of clusters differ for both

images.

For the parrot image shown in Fig. 6, the

method was found to give the best results when

the image (with σ=25) was divided into 2 clusters

with H=3.2 and show the different level of quality

between wavelet denoised method and the

proposed method. For the man image shown in

Fig. 7, the method was found to give the best

results when the image (with σ=25) was divided

into 2 clusters with H=2.8 and show the different

level of quality between wavelet denoised method

and the proposed method.

However, for illustrative purposes, in this

paper we show results using a value of that

allowed us to achieve the least MSE. Moreover, to

eliminate dependence on the random initialization

of cluster centers for the K-Means algorithm, we

perform clustering using K-Means multiple times

and use the best clustering that produces the least

cost. A part from this, the bandwidth or the

smoothing parameter for the steering kernel also

needed to be tuned for optimality.

Further, we note that the patch-based

method of K-SVD performs better denoising

compared wavelet denoise where wavelet denoise


45

failed to decrease the noise in case of low noise

image and the proposed method appears to

perform better in the same cases .

Table 2: The H-opt and MSE values of different sigma and

number of cluster for number of image.

Image Σ H-

opt

K-

opt

MSE. Using

the noisy

image

MSE. Using

the proposed

method

parrot

5

10

15

20

25

3

3

2.8

2.8

3.2

3

7

8

10

2

24.98

97.82

214.98

373.65

571.29

13.97

34.85

59.66

85.34

109.97

man

5

10

15

20

25

3

3

2.8

3

2.8

1

1

11

4

2

24.21

94.79

210.11

368.49

567.98

18.26

49.77

88.49

123.71

160.62

Table 3: The MSE values of different sigma and number of

cluster for number of image for wavelet denoised method and

proposed method

Image σ H-

opt

K-

opt

MSE.

For

noisy

image

MSE.

Using the

Wavelet

denoise

MSE.

Using the

proposed

method

parrot

5

10

15

20

25

3

3

2.8

2.8

3.2

3

7

8

10

2

24.98

97.82

214.98

373.65

571.29

150.66

162.54

181.69

207.94

240.99

13.97

34.85

59.66

85.34

109.97

man

5

10

15

20

25

3

3

2.8

3

2.8

1

1

11

4

10

24.21

94.79

210.11

368.49

567.98

117.58

129.72

149.18

175.79

209.44

18.26

49.77

88.49

123.71

160.62

5. Conclusions

This paper presents a SR method based on

sparse dictionary, which builds sparse association

between image feature patches, and at the same

time carry on matching and optimization.

Compared with other sparse coding method,

sparse dictionary is more compressible and

efficient, and needs fewer examples for the same

quality. Comparison with other learning-based

super-resolution method shows that our method

superior in quality and computation. However,

there are some possibilities for future

improvement. Through the proposed method used

wavelet domain for clustering the image using an

important effect features that are able to capture

the underlying geometry in the presence of noise.

A dictionary is learned for each of the clusters and

a generalized kernel regression is performed to

produce a denoised estimate for each pixel, then

we expand our proposed method on image with

blurring and image degraded with both noise and

blurring, as a result the proposed method shows a

satisfied result. It presented the implementation of

the proposed method that consists of four stages,

namely wavelet transform, clustering, dictionary

learning, and coefficient calculation. However,

each of these blocks can be replaced by alternate

approaches that satisfy similar objectives. Our

outline is evaluated experimentally and compared

to some of the state of the art methods for learning

SR method. It can be seen that the performance of

the proposed method is viable, qualitatively as

well as quantitatively. For optimal performance, it

is necessary to tune a few parameters of our

framework like number of clusters and smoothing

parameter. Although our method may be useful to

use variants of K-Means that converge to the

optimal number of clusters automatically.

Fig.6 Comparison of denoising results on noisy parrot image

corrupted by additive white Gaussian noise of standard deviation 25.

(a) Noisy image (MSE=571.29), (b) wavelet denoised method

(MSE=240.99) and (c) Proposed method (MSE =109.97).

(c) (a) (b)


46

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Fig.7 Comparison of denoising results on noisy parrot image corrupted

by additive white Gaussian noise of standard deviation 25. (a) ) noisy

image (MSE=567.98), (b) wavelet denoised method (MSE=209.44) (c)

Proposed method (MSE =160.62).

(c) (a) (b)

Documents

Super Resolution Image Using Dictionary Technique