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SUBMITTED BY
ANKIT ASOKAN

motivation
Developments in information storing and transmitting.
Communication links have grown fast.
Compressing data saves space and faster communication.
Efforts for new compression algorithms.

introduction
Proposed by M Bransley.
Contemporary compression techniques use DCT algorithm eg: JPEG,MPEG
New technique used for image compression known as fractal compression.
Properties of fractals are used.
High compression rate.

Overview
Why Fractal Image Compression

Fractals

Mathematical Background

How does it work?

Examples

Possible Improvements

Why fractal compression
Different type of compression scheme worth exploring
Takes advantage of similarities within an image
Advanced detail interpolation
High theoretical compression rates
Fast decompression times

fractals
Term coined by Mandelbrot.
Natural objects - fractals include clouds, mountain ranges, cauliflower.

Mandelbrot set---

dimension
A point has dimension=0
A line has dimension =1
A plane has dimension =2
Objects live in spaces.
Dimension of a space is related with degrees of freedom of the space.

Dimension of natural objects
Dimension of a leaf.

Dimension of a fern.

Dimension of a cloud.

fractals
Fractals are geometrical objects with fractional dimension.

Mathematical background
Started with Michael Barnsley, and refined by A. Jacquin
Try and find a set of transforms that map an image onto itself.
The key is the Collage Theorem
States that if the error difference between the target image and the transformation of that image is less than a certain value the transforms are an equivalent representation of the image.

Iterated function systems
An IFS is the collection of contractive affine transforms

Partitioned iterated function system
Proposed by Arnaud Jacquin (1988).
Image is not the collage of copies of entire image but of smaller copies of itself.
For encoding of images.
It is divided into domains and ranges.

Binary image
Binary images can be represented using simple affine transforms because there are only two values.

Colour images
Color images are combination of grayscale images. In addition depth also comes.

Encoding - How does it work?
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 overlapping blocks of parent block size.

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

Compute affine transform.

Store location of parent block (or transform block), affine transform components, and related child block into a file.

Repeat for each child block.

Lots of comparisons can calculations.
256x256 original image
16x16 sized parent blocks
241*241 = 58,081 block comparisons

Decoding - How does it work?
Read in child block and tranform block position, transform, and size information.
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.

Fractal vs jpeg
JPEG is better at low compression ratios, and Fractal Compression is better at high.
Crossover point at 40:1
The fractal compressed images have a much more natural looking noise than JPEG.
Same decompression time as JPEG, sometimes faster.
Fractal compression much slower compression time than JPEG.
Can zoom on the fractal image and the image will stillhave a natural look -> higher effective compression ratios.

Possible Improvments
Greatest weakness is time for encoding
Possible speed ups
Order transform blocks into domains based off of average intensity and variance
Only search through blocks with similar structures
Do not search all possible blocks
Reduce number of child blocks
Quality and Compression Improvements through
Quadtrees or HV Trees
Rotations of Transform Blocks during comparison
Improved grayscale transforms

conclusion
So why isn't everyone using Fractal Image Compression ?
Fractal Image Compression is still under development.
new algorithms to reach shorter encoding time
Fractal Image Format is not standardized.
currently no public domain documentation available.

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