Fractal Image Compression

FRACTAL IMAGE COMPRESSION

Presented by

Nithin SinkaranRoll No:57

Guided By

Mrs. Sheena S

fractal image compression

Overview

Basics of Fractal image compression

Why Fractal Image Compression

Mathematical Background

How does it work?

Examples

Possible Improvements


Some Important facts

Fractals have infinite details at every point

Self similarity between object part and whole object

Fractals are generated by iteratively applying transformation

function to a region of space(initiator).

Fractal image compression is NOT the compression of fractals

It uses the properties of fractals for compressing images.


BasicsProposed by M.Barnsley

Based on college theoremLet R²(Hausdorff space) be set of two real numbers, and L be an object

Let w1,w2,w3… be some affine transforms which maps the entire image to its subsets

W(L) =U wi(L)

If distance h(L, U wi(L) )= ɛ(a small value)

Then h(L,A)= ɛ/(1-c) where c is the contractility factor, A is the converging abstract set

Now L can be approximated to the abstractor A


Fern created using the fractal method(fig 1)

The highlighted portion of the fern is similar to the entire Image. Application of different affine transformation on That portion produces the entire fern(fig 2).

The fern is self similar

The fern creation requires only 28 numbers and can Achieve a large amount of compression.

The success of the compression depends on the amount ofSelf similarity found in that image.

Significance in image compression


Limitation of basic theory

All images are not self similar

The direct application of affine transforms to the whole set L(image) will not always maps to its subsets due to lack of self similarities.

Now the affine transforms are applied not to L, but to the part of L Or subset of L and tries to map them to self similar parts of the Same image

This method can be used for compressing any common image


Jacquin’s Compression AlgorithmSuppose that we are dealing with a 128 x 128 image in(called Range image).

1. We then reduce by averaging (down sampling and low pass-filtering) the original image to 64 x 64(Domain Image).

2. Partitioned both images into blocks 4x4 pixels.


3.performed the following affine transformation to each block

(Di,j)=α Di,j + t0

where α - contrast scaling t0-luminance shift ([−255,255 ]).

4.Compare each domain block with each range block

5.Find Min Σ (Ri,j ) m,n -T(Di,j ))m,n

6.The transformed domain blockwhich is found to be the best approximation for the current range block is assigned to that range block

7. The coordinates of the domain block along with value of α, t0 describing the transformations. This is what is called the Fractal Code Book


Decoding

Apply the transformations defined in fractal code book iteratively to

some initial image Winit, until the encoded image is retrieved back.

The transformation over the whole initial image can be described as follows

W1 = h(Winit)W2 = h(W1)W3 = h(W2)..... = ......Wn = h(Wn-1)

Wn will converge to a good approximation of original image after some iterations.

Greater the number of iterations greater will be the decoded similarity.


Quad-tree partition method

Take a starting image and divide it into small, non-overlapping, square blocks, typically called “parent blocks”.

Divide each parent block into 4 each blocks, or “child blocks.”

Compare each child block against a subset of all possible parent blocks.

(Need to reduce the size of the parent to allow the comparison to work.)

Determine which larger block has the lowest difference, according to some measure, between it and the child block.

Calculate a grayscale transform to match intensity levels between large block and child block precisely. Typically an affine transform is used (w*x = a*x + b) to match grayscale levels.


Encoding Upper left corner child block, very similar to upper right parent block.

Compute affine transform.

Store location of parent block and child block, affine transform components, etc .into a file(Fractal code book).

Repeat for each child block.

Lots of comparisons andcalculations.

For 256x256 original image and 16x16 sized parent blocks241*241 = 58,081 block comparisons.


Decoding Use any blank starting image of same size as original image

For each child block apply stored transforms against specified transform block

Overwrite child block pixel values with transform block pixel values

Repeat until acceptable image quality is reached.

Initial image First Iteration Second Iteration Fifth iteration Tenth Iteration

Original image


Compression of color images Color images are often built of several stacked color channels, each of them representing value levels of the given channel.

For example, RGB images are composed of three independent channels for red, green and blue primary color components.

Here RGB image is separated Into 3 color channels

Their gray scale equivalentsAre shown in the right side

The reverse transformationIs also possible

These gray scale parts can be compressed separately


SPEED-UP TECHNIQUES Boss, Fisher and Jacob's scheme

Each range and domain blocks are further divided in to 4 partsAverage intensities are calculated for each blockBased on the average intensities it falls into any one of the 3 major classes Comparison is done with blocks belonging to similar class only. Speed up factor:8

Nearest neighbour search scheme(D. Saupe and U. Freiburg)

fractal image compression is equivalent to the multidimensional nearest neighbour search.optimal domain-range pairs is equivalent to solving nearest neighbour problems in a suitable Euclidean spaceMulti-dimensional nearest neighbor searching operates in logarithmic timeSpeed up factor:1.3 up to 11.5


Properties

A fractal compressed image is actually stored as domain blocks and theirTransformations. Its now resolution independent.

Can be de compressed into any resolution needed without loosing much details.

Behaves almost like a fractal image

It can be zoomed at any magnitude without producing the jagged effect

‘ .FIF ’(Fractal Image File) is a commonly used format for fractal compressed images.


Zoom Using Fractal Decoding Original image zooming

2x 2x

4x 4x


EXAMPLES Image Details (JPEG) Original Size(KB) Compressed Size(KB)

Lena256X256(24bit) 84.3 17.5

66.2 9.91Brick256X256(24bit)

Leaf256X256(24bit)

45.9 5.91


Application

1. Image enlargements

2. Automated Semiconductor Defect Detection semiconductor devices always follows a definite patter. Fractal

coding can be used to identify any variation from the self similar part of the image(defects).

3. Image enhancement Images have a large amount of affine redundancy Any damaged part can be mapped into some other parts of the

same image

4.Texture compression Textures are self similar images. High compression can be achieved.


Some soft wares1. Fractal Imager

developed by iterated systems Inc

Used to create .FIF files from JPEG,PNG etc..

Fractal Imager showing

8:1

zooming of a Leaf.

Original image(left),

.FIF image (right)


2. Genuine Fractals

Developed by onOne software Inc.

Used as a plug-in to software like Adobe Photoshop, Adobe Light room

Images can be enlarged up to 1000 times its original size

Genuine fractals in Adobe Photoshop CS5


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Technology

Fractal Image Compression