Fractal Image Compression Using Wavelet

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IJRIT International Journal of Research in Information Technology, Volume 3, Issue 5, May 2015, Pg. 504 -510

Suman chaudhary, IJRIT-504 International Journal of Research in Information Technology

(IJRIT)

www.ijrit.com ISSN 2001-5569

Fractal Image Compression Using Wavelet

Suman chaudhary* kulwinder singh**

*M. Tech. Scholar, E.C.E, Bhai Maha Singh College Of Engineering, Muktsar (Punjab) [email protected]

**Associate Professor, E.C.E, Bhai Maha Singh College of Engineering, Muktsar (Punjab).

ABSTRACT

Fractal image compression is a comparatively recent technique based on the representation of an image by a contractive transform, on the space of images, for which the fixed point is close to the original image. This broad principle encompasses a very wide variety of coding schemes, many of which have been explored in the rapidly growing body of published research. While certain theoretical aspects of this representation are well established, relatively little attention has been given to the construction of a coherent underlying image model that would justify its use. Most purely fractal-based schemes are not competitive with the current state of the art, but hybrid schemes incorporating fractal compression and alternative techniques have achieved considerably greater success. This review represents a survey of the most significant advances, both practical and theoretical in original fractal coding scheme. In this paper, we review the basic principles of the construction of fractal objects with iterated function systems (IFS).

Keywords: Image compression, DWT, PSNR, MSE, Compression ratio.

1. INTRODUCTION

For recent years, the application of fractal image coding has become more and more popular.

Fractal coding is based on fractal geometry, it has a character of big compression ratio and a fast decoding speed, but it cannot be used for real time processing[2]. It is its blocks searching and matching that makes its long time. As wavelet can get good space frequency multi resolution, the energy mainly concentrated in low frequency sub images, and the images with same directions

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Suman chaudhary, IJRIT-505 but different resolutions have self similarity, which is consistent with fractal’s nature properties.

Recently, much reaching work has focused on fractal coding by using wavelet[6]. It is just at the beginning, but some research results improved this method is particle. The combination of wavelet and fractal is firstly proposed by Pentland and Horowitz. They wanted to find the redundancy of subimages decomposed after wavelet. Later, Rinaldo and Calvagno proposed a new method. First,decompose a image by wavelet, and then code the sub image with minimum resolution, and predict the other subimages. Finally, we’ll finish the compression. Jin Li introduced a new method [7].They firstly computed the bytes of fractal predicting, and only predicted when economization. But the methods above are all time consumption, and the reconstructed images are not always good.This paper proposed a new blocks searching method based on fractal. Firstly, we transform the image by wavelet, then divide it into blocks. Before matching, we first reduce the amount of domain blocks and the range blocks to lessen the block pools, then following the contractive mapping transformation [8].

2. FRACTAL IMAGE CODING a) Collage theorem

Collage theorem is the technique core of fractal coding.For a certain image X, we can choose a certain number of contractive mapping, such as N, and we can get number N sets by transformed for N times, in which every set is a small image. If the reconstructed image collaged by these N small images is very similar to X, we get the right IFS.

Supossed { R : w , i 1, 2 , , P} i

T = L is a contractive transform set, IFS, and R is a real set. To any V Ì R T , e > 0 , if the largest contractive gene s Î ( 0, 1 ) , and h (V , W (V )) < e is satisfied, we will get h (V , A ) < e /(1 - s ) A is the attractor of IFS, and h(A,B) is the Hausdorff distance. Collage theorem supplies a up bound value between V and IFS attractor, which represents the degree of approximation, the upbound value of collage error.Collage theorem provides the theoretical basis for image compression with IFS[2]. A binary image can be considered as a R2 dimensional tight subset.

And a gray image can be considered to be carried out by sampling and quantization from an original gray curve. Even we cannot make the original image be the attractor of a IFS,{W (V ) R : w i , i 1, 2, , P}

T = L , we can regard V as a good approach, if W(V) is much close to V, and Wi (i=1,2,…，P)

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Suman chaudhary, IJRIT-506 is a contractive mapping.

b) Partition

X should be devided into some range blocks (Ri) and some domain blocks(Di), and a Di should contain more pixels than a Ri to ensure the mapping, Wi：Di→Ri, is contractive. Generally, if a Ri is b×b, a Di should be 2b×2b.

c) Computation of IFS

Three dimensional affine transformations can be expressed as:

The transformation above is a synthetic of two. It is the matching process of Ri and Di including geometric transformation and gray transformation. Where ei, fi is the starting position of scope blocks, ai, bi, ci and di are the coefficients of geometric position transformation matrix.

MSE is defined as :

MSE = 1

N(R_,− s. D_,− 0)

,

S is the ratio; o is the offset, s and o are defined as :

S = N ∑ D_, _,R_,− ∑ D_, _,∑ R_, _, N∑ D_, _,− (∑ D_, _,)

O = 1

N( R_,− s. D_,

,

)

,

The matching is succeed if MSE is less than the defined error, otherwise the matching continues.

We should record the Jacquin transformation operator code, ratio,offset,Ri and Di.To each Ri, we should save the corresponding address of Di, contrast ratio, brightness offset and style of matching spiral approach.

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 =

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i i i

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t x

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3. Fractal image decoding approach

The fractal decoding process is relatively simple, that is, for all R blocks, act the corresponding affine transformation on any one initial image,

according to fixed point theorem, this iteration process would converge to a fixed attractor, and this attractor is the decoding image. Adaptive threshold quad tree fractal compression approach is the same as the basic fractal image encoding method, the main difference is, when doing image match, we don’t find the global best

adaptive threshold to judge whether the current R block matches D block, if the two’s matc error International Journal of Graphics, MSE is smaller than threshold, we can consider that current D block matches R block, otherwise, the two are not matching.

We can see that threshold is proportional to the image block’s variance. When block’s varia is larger, i.e., block has a high complexity, and has much high

greater distortion in recovery image block. When block’s variance is smaller, i.e., block has a low complexity, and has much low

level.

4. RESULTS: Fig 1.1 a shows an image Leena.Tiff of size 276 KB and dimensions 512x512.

This image is compressed and decompressed using Fractal Image Compression Technique in Wavelet Domain.Fig1.2 shows the Reconstru

Fig 1.1

The encoding time is 7.8940 seconds, decoding time is 10.2340 and Compression ratio achieved is of 85.1563.

Suman chaudhary 3. Fractal image decoding approach

The fractal decoding process is relatively simple, that is, for all R blocks, act the corresponding affine transformation on any one initial image, iterate for several times(commonly ten times) , according to fixed point theorem, this iteration process would converge to a fixed attractor, and this attractor is the decoding image. Adaptive threshold quad tree fractal compression approach the basic fractal image encoding method, the main difference is, when doing image match, we don’t find the global best-match block in D pool, and instead, we use an adaptive threshold to judge whether the current R block matches D block, if the two’s matc error International Journal of Graphics, MSE is smaller than threshold, we can consider that current D block matches R block, otherwise, the two are not matching.

can see that threshold is proportional to the image block’s variance. When block’s varia is larger, i.e., block has a high complexity, and has much high-frequency signal, we can allow a greater distortion in recovery image block. When block’s variance is smaller, i.e., block has a low complexity, and has much low-frequency signal, we must limit the distortion to a certain

Fig 1.1 a shows an image Leena.Tiff of size 276 KB and dimensions 512x512.

This image is compressed and decompressed using Fractal Image Compression Technique in Wavelet Domain.Fig1.2 shows the Reconstructed Image of Leena.

Fig 1.2

The encoding time is 7.8940 seconds, decoding time is 10.2340 and Compression ratio achieved

Suman chaudhary, IJRIT-507 The fractal decoding process is relatively simple, that is, for all R blocks, act the corresponding iterate for several times(commonly ten times) , according to fixed point theorem, this iteration process would converge to a fixed attractor, and this attractor is the decoding image. Adaptive threshold quad tree fractal compression approach the basic fractal image encoding method, the main difference is, when doing match block in D pool, and instead, we use an adaptive threshold to judge whether the current R block matches D block, if the two’s match error International Journal of Graphics, MSE is smaller than threshold, we can consider that

can see that threshold is proportional to the image block’s variance. When block’s variance frequency signal, we can allow a greater distortion in recovery image block. When block’s variance is smaller, i.e., block has a limit the distortion to a certain

Fig 1.1 a shows an image Leena.Tiff of size 276 KB and dimensions 512x512.

This image is compressed and decompressed using Fractal Image Compression Technique in

The encoding time is 7.8940 seconds, decoding time is 10.2340 and Compression ratio achieved

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Suman chaudhary, IJRIT-508 The following Table 1.1 shows the results of fractal Image Compression using Wavelet and its comparison with Quad tree Decomposition and Huffman coding.

Image Compression Methods

Test Images

Encoding Time

Compression Ratio

Decoding Time

PSNR (DB)

Using Quadtree Decomposition and huffman coding, Reference [5]

LEENA 15.86 2.02 20.20 29.92

Using Fractal Image Compression Using Wavelet

LEENA 7.8940 85.1563 10.2340 23.06666

Table 1.1

4. CONCLUSION

The field of fractal compression is relatively new, as is the study of fractals, and as such there is no standardized approach to this technique. The main concept in this compression scheme is to use Iterated Function Systems (IFS) to reproduce images. An important property of fractals is that they exhibit self-similarity. By partitioning an image into blocks, typically 8x8 or 16x16 pixels, it becomes possible to map small portions of an image to larger portions. In addition, the smaller portions are reproduced by use of affine transformations. These transformations effectively map squares to parallelograms through translation, scaling, skewing, rotation, etc. In this way an image can be stored as a collection of affine transformations that can be used to reproduce a near copy of the original image. The process is iterative in that detail is added after each pass through the function set. The process is computationally intensive but can yield much improved compression ratios. Fractal compression area is great. It should be possible to take advantage of the large compression ratios achieved from fractal compression and produce a trade-off of compression ratios but we have o little compromise with the image quality.

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