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Fractal compression is a lossy compression method for digital images, based on fractals. The method is best suited for textures and natural images, relying on the fact that parts of an image often resemble other parts of the same image.[citation needed] Fractal algorithms convert these parts into mathematical data called "fractal codes" which are used to recreate the encoded image.
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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
fractal image compression
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.
fractal image compression
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
fractal image compression
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
fractal 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
fractal image compression
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.
fractal image compression
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
fractal image compression
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.
fractal image compression
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.
fractal image compression
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.
fractal image compression
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
fractal image compression
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
fractal image compression
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
fractal image compression
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.
fractal image compression
Zoom Using Fractal Decoding Original image zooming
2x 2x
4x 4x
fractal image compression
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
fractal image compression
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.
fractal image compression
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)
fractal image compression
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
fractal image compression
QUESTIONS ?
fractal image compression
THANK YOU!