56 978-1—4244-1625-7/07/$25.00 © 2007 IEEE

Improving Quality Of Fractal Compressed Images

E.Nadernejad H.hassanpour M.Salarian Department of Computer and Electrical Engineering

Noushirvani Institute of Technology, University of Mazandaran P.O.Box 47144, Babol, Iran (Email: h_hassanpour@yahoo.com)

ABSTRACT - This paper proposes a new quality

improvement technique for fractal-based image compression techniques using diffusion equations. Fractal coding uses a contractive mapping scheme to represent an image. This process of contractive mapping causes artifacts and blocking effects in encoded images. This problem is severed when compression ratio is increased or there are high frequency regions in the image. Hence, to amulet the deficiency of fractal coding approaches in image compression, we propose using diffusion equations as a post processor. Diffusion equations are powerful tools for image enhancement. This technique has been examined on a variety of standard images. The obtained results indicate that the proposed method improves performance of fractal-based image compression techniques. Index Terms: fractal coding, quality image, diffusion equations.

I. INTRODUCTION Fractal image compression method is widely

used in image processing application such in image compression [1], feature extraction [2]. The main advantages of using fractal image coding are multi resolution property and fast image reconstruction [17]. A high compression ratio is another advantage of this method. These properties of fractal image coding make it very suitable for multimedia application. However, this technique like any image coding methods, more or less, suffers from coding artifacts. Therefore some researchers tried to enhance the coded image by adding a post processing step. This step usually uses denoising application, it might be very important that the denoising process has no blurring effect on the image and has no changes on image edges.

There are currently a number of techniques to reduce the artifact effects of image compression techniques. Using simple filtering, such as median filter and average filter are some of these techniques [5]. In another research a denoising process is applied in the wavelet domain by theresholding the wavelet coefficients[6]. These

methods, however, introduce smoothing effects on the image. In addition, in some of the existing methods it is necessary to know a prior distortion model, i.e., projection onto convex sets (pocs)[8] or constrained least squares(cls)[9].

In [11] partial differential equations are used as an alternate method for image denoising. These methods assume intensity of illumination on edges varies like geometric heat flow in which heat transforms from a warm environment to a cooler one until temperature of the two environments reach to a balance point. It was shown that these changes are in the form of Gaussian function [7]. As a result, sudden changes in edges might be due to the existing of noise. In fact, an image includes a series of regions in which different regions might have different standard deviation. This issue is considered as a diffusion equation [11]. The diffusion equations offer a strong tool for image denoising. The exiting diffusion-based methods suggest the use of side neighbors [10].

In section 2 of this paper we have a brief review of fractal image coding. In section 3 diffusion equation method is presented. Implementation and experimental results are addressed in Section 5 and 6.

II. FRACTAL IMAGE ENCODING

Jacquin [3] and Jacobs [4] described the first practical fractal image compression algorithm based on a partitioned iterated function system (PIFS). The PIFS procedure is as follows: Suppose the original image is of size NN × . at the first step the original image is partitioned into a set of non overlapping KK × block, named range block and a set of overlapping KK 22 × block named domain block. Therefore we have

2)/( KN range block. Before searching the best domain block, the size of range and domain blocks must be the same. Therefore we get averaging four pixels to one pixel for each domain block. Some

57

Fig 1.Quadtree partitioning. Fig 2. An image encoded using the quadtree algorithm.

transforms is applied to the domain blocks and then is compared with range block. If the error between two blocks is acceptable the transform parameter are saved as fractal code. The distance between a range block, R, and domain block, D is defined as follows:

The optimum affine parameters S and O can be obtained by the least squares method as follow:

2||1.||1.,1.

DDDDRRs

−>−−<=

(2)

DsRo −= (3)

That ><, R and D are: inner product, mean of R and mean of D respectively. But by using this method the quality of image is poor. In [17] fisher proposed a quadtree fractal encoding algorithm. In this algorithm if the error between the ranges block and the best matched domain block is larger than a predefined value, the range block partitioned into four quadrants. And we search the best domain block for each of them. This procedure can be continued recursively until the best matched domain block can be determined. Figure 1 show this partitioning scheme. To increase the compression ratio, we may select large value for error in each level. The figure 2 shows a region of

Lena image that encoded in high compression ratio by quadtree algorithm. As have shown in this image, the blocking artifact is seen easily. In this paper we use diffusion equation to improve image encoded by this method. In the next section we will introduce diffusion equation Algorithm.

III.DIFFUSION EQUATION FOR NOISE REDUCTION

The diffusion equation is based on repetition. The main idea of using diffusion equation in image processing is the use of a two-dimensional

Gaussian filter in which the image ( , )I x y is

convolved with a window ( , )K x yσ :

)2

||||exp(2

1),( 2

22

2 σπσσyxyxK +−=

(4)

where ( , )K x yσ is a Gaussian filter, and σ represents the standard deviation of the filter’s coefficients. By convolution operation, there would be the problem of edge smoothing. It has been previously shown that this problem can be resolved by treating intensity variations in an image as diffusion of heat flow [10]. The diffusion equation for an image ( , )I x y would be as follow:

(5)

2

2 2

2 2

( , , ) ( , , )

( , , ) ( , , ) ,

I x y t I x y tt

I x y t I x y tx y

∂ = ∇∂

∂ ∂= +∂ ∂

∑=

−+=n

iii rosdDRE

1

2)(),( (1)

58

Where 0( , ,0) ( , ) I x y I x y= as initial image ( 0)t = ,

and ( , , )I x y t is the image at 20.5 .t σ=

The above equation can be rewritten as below:

0

( , , ) ( ( , , ) ( , , ))

( , , 0) ( , ),

I x y t c x y t I x y tt

I x y I x y

∂ = ∇ ⋅ ∇∂

=

(6)

where∇ is the gradient operator, ),,( tyxc is the diffusion factor, and ⋅∇ is the divergence operator. If c has a constant value (independent to tyx ,, ), the obtained equation is called diffusion equation with isotropic diffusion factor. In this case, all points, even edges would be smoothed as there is no difference between a pixel on edge and other pixels. It is obvious that this is not an ideal condition. For resolving this deficiency, the diffusion factor could be considered as a function of x and y . Hence, the above equation is changed to a linear and anisotropic equation. If c is dependent to the image, the linear equation would be transformed to a nonlinear equation. This is the idea that was suggested in [11],[10]. In these researches two different equations for the diffusion factor were suggested as below:

(7) 2

2

2

2

1( , , )| |(1 )

| |( , , ) e x p ( )2

c x y tI

kIc x y tk

=∇+

∇= −

(8)

In this equation the diffusion factor c changes at different point in the image. In those points that the gradient of the image is large, this factor has a small value. Consequently, the diffusion factor would be small around the edges. In (7) and (8) k is used to control the diffusion factor.

IV. PROPOSED METHOD AND IMPLANTATION

We used some fractal coded images that have been encoded using quadtree method. These images are encoded in high compression ratio therefore the blocking effect is seen easily. For reducing this effect we applied diffusion equations.

There are a number of methods to solve diffusion equations. In [7], the first and second derivatives and also Laplacian of a current pixel are computed using the pixels in neighbors. Here, the most general numerical approach for a solution of these equations is presented [12]. To state the solution, equation (6) is rewritten as:

.

2 2( , , ) ( , , ) ( , , )( , , )( )2 2

( , , ) ( , , ) ( , , ) ( , , )

I x y t I x y t I x y tC x y tt x y

I x y t C x y t I x y t C x y tx x y y

∂ ∂ ∂= + +∂ ∂ ∂

∂ ∂ ∂ ∂⋅ + ⋅∂ ∂ ∂ ∂

(9)

The approximate solution of (9) is [15]:

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1

( , , ) ( , , ) ((

) (2

))

n n s s

e e w w ne ne se se nw nw sw sw

n n s s e e w w ne ne

se se nw nw sw sw

I x y t t I x y t t d c d cd c d c d c d c d c d c

d c d c d c d c d c

d c d c d c

αα

+∆ = +∆ + ++ + + + +

+ + + + + +

+ +

(10)

Here parameters d and c of equations (10) is written as below:

2

1( , 1, ) ( , , )1 ( )

n nn

d I x y t I x y t c dk

= − − =+

(11)

These parameters are the numerical equivalent for the derivation and diffusion factors for the four cardinal directions. In (10), the goal is to find a solution for the initial times before heat flow. It is obvious that the temperature at different point changes toward balancing. Therefore, in the case of a gray level image, the value of pixels converge which leads to enhancement. It needs to be noted that for an area containing an edge, a larger diffusion factor is chosen to preserve the edge. Considering (11), one can claim that in (10) the rate of enhancement is low for pixels with a large differentiation (d is large). In other words, for a large gradient value, which belongs to the edge, smoothing is not taking place. In fact, this is the way used in this technique to preserve edges. In equation (10) α is a factor representing the significance (plentitude) of slant edges in the image. In this equation, the new parameters

( , ,...ne sed d ) are computed similar to the parameter in (11). In this paper we choose 25.0=∆t , as it was suggested in [11],[10]. To find an optimal value for

α,k , we have performed a variety of experiments on images available in [13] and [14]. It was found

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that α and k are changed in ranges [0, 1] and [5, 100], respectively. In this method, to find the best possible solution, the algorithm needs to be repeated 10 to 100 times.

VI. EXPERIMENTAL RESULTS

To evaluate the performance of this algorithm, we use the two metrics:

A) Peak Signal to Noise Ratio

PSNR show the quality of decoded image and represents the fidelity of the algorithm. The PSNR is defined as follows:

2

10* 2

1

25510log ( )1 ( )

N

i ii

PSNRf f

N =

=−∑

(12)

Where N is the total number of pixels in image,

if and if ∗ are thi pixels gray level of the original and decoded images respectively.

B) Structural SIMilarity

The Structural Similarity (SSIM) factor can be used to measure the similarity between two images [16]. This factor consists of three different metrics. Let { }1,2,3,...,ix x i N= = , { }1,2,3,...iy y i N= = be the original and the test images, respectively. The SSIM is defined as:

2 2 2 2

4( )[( ) ( ) ]

XY

X Y

XYSSIM QX Y

σσ σ

= =+ +

(13)

Where:

2 2

1 1 1

2 2

1 1

1 1 1, , ( )

11 1

( ) , ( )( )1 1

N N N

i i x ii i i

N N

y i xy i ii i

X x Y Y x XN N N

y Y x X y YN N

σ

σ σ

= = =

= =

= = = −−

= − = − −− −

∑ ∑ ∑

∑ ∑ (14)

The dynamic range of SIMM is [-1,1]. The best value 1 is achieved if and only if i iy x= for all 1, 2,3,..., Ni = . The lowest value of -1 occurs when 2i iy X x= − for all 1,2,3,...,i N= . The

SIMM indicates any distortion as a combination of three different factors: loss of correlation, luminance distortion, and contrast distortion. On other words, Q in (13) can be rewritten as a product of three components: The first component is the correlation coefficient between x and y, which represents the degree of linear correlation between x and y, and its dynamic range is between -1 and 1. The best value 1 is obtained when i iy ax b= + for all

1,2,..., ,i N= where a and b are constants and a>0. Even if x and y are linearly related, there still might be relative distortions between them, which are evaluated in the second and third components. The second component, with a value range of [0, 1], measures how much the x and y are close in luminance. It equals 1 if and only if X Y= . Xσ and yσ can be considered as an

estimate of the contrast in x and y. Eventually, the third component indicates how similar the contrasts of the images are. Its values range between 0 and 1, where the best value 1 is

achieved if and only if X Yσ σ= . Parameters , ,α β γ are used to adjust the significance of each of the tree components. In practice to use the above measure, the image is windowed equally, then for each window the SSIM is computed to find the average SSIM as follows:

1

1( , ) ( , )M

j jj

SSIM X Y SSIM x yM =

= ∑

(16)

where X and Y are the original and the decoded images respectively, M is the number of local windows in the image, xj and yj are the image contents at the jth local window. Our algorithm applied on several standard image taken from reference [14] also we implemented this method in Matlab. In all cases our method increases the PSNR and SSIM significantly and reduced blocking effect and artifact without smoothing. Some experimental results are shown in table-1 for four familiar images. In each cases PSNR improvement are more than 2dB, also SSIM measure indicates that the proposed approach has often a better performance

1 2 3 2 2

2 2

2( ) ( )

2( , ) ( , ) ( , )

xy

x y

x y

x y

XYSSIM Q Q Q Q

X Y

s x y l x y c x yα β γ

σσ σ

σ σσ σ

= = = ×+

× = × ×+

(15)

60

a

b

c

d

Figure 3. (a) decode image in PSNR=31.2 (b) image improved by diffusion method with PSNR=33.4 (c),(d) zoom on a section of hat of (a),(b)

in preserving the structure of original images. That indicated our method is very effective. These results indicate that the proposed image enhancement techniques have a good performance. To judge the act of this algorithm from the human

eye’s view, one of the test images is shown in figure3.

VI. CONCLUSION

In this paper, a new technique has been proposed for improving performance of fractal-based image compression techniques. In this approach, diffusion equations are applied, as a post processor, on fractal coded images. Results in this paper have shown that the proposed post processing technique can considerably amulet deficiency of fractal-based approaches in images compression, by reducing artifacts and blocking effects.

Table 1 the results for some images

fractal diffusion PSNR SSM PSNR SSIM

lena 29.12 0.65 29.96 0.78

Boat 28.36 0.71 29.06 0.75

peppers 27.83 0.58 28.15 0.71

Babon 28.63 0.76 29.25 0.81

CameraMan 28.81 0.69 29.63 0.73

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