5.7. Fractal compression Overview

1 Prof. Dr. Paul Müller, University of Kaiserslautern

5.7. Fractal compression Overview

1. Introduction

2. Principles

3. Encoding

4. Decoding

5. Example

6. Evaluation

7. Comparison

8. Literature References


Introduction (1) - General

Use of self-similarities between

picture ranges

Geometrical and brightness

modification to describe a range

Perception and efficient coding

of self-similarities (↔ redundant

information) through fractal

transformations

No pixel values are stored, only

transformation parameters


Introduction (2) - General

Concept is based on the Fixpoint Theorem from Banach and IFS (Iterated Function Systems)

Reconstruction is independent of original resolution, (e.g. enabling a zoom function without negative side effects such as interpolation)

Lossy and extremely asymmetric compression method

Developed by Michael F. Barnsley in 80´s at the Georgia Institute of Technology

Barnsley"s company "Iterated Systems" owns the licenses for full-automatic fractal compression


Introduction (3) - MRCM

The Multi Reduction Copy Machine Optional number of lenses

Optional rate of diminution

Optional arrangement of copies

Example: Sierpinsky Triangle 3 Lenses

Diminution to 25%

Pyramidal arrangement

The geometric transformation of the MRCM can be expressed by six parameters:

The affine coefficients a,b,c,d cause a mirroring, rotation and diminution. The offset T (e,f) describes the geometric displacement.


Introduction (4) - MRCM Samples

after 0 iteration after 1 iteration after 2 iterations

after 3 iterations after 4 iterations after 5 iterations


Introduction (5) - The "inverse" problem

Iteration converges to Sierpinsky Triangle ("Attractor")

Optional size / resolution

Independent of input image

Compression – "inverse" problem

extension of MRCM necessary for efficient coding


Principles (1) - Partitions

2 Partitions Range blocks Ri (often uniform

size, non-overlapping)

Domain blocks Di (optional size, overlapping)

surjective function Di → Ri

Idea: Manipulation of (relatively) big domain blocks to describe range blocks

Task: Finding the best matching domain-block for each range block

Matching criteria essential for complexity


Principles (2a) - Transformations

Geometric transformation of a domain block i (see MRCM):

Transformation matrix (a,b,c,d) Usually operations are applied to rectangular areas (often quadratic areas) are used. Therefore the geometric transformations for a,b,c,d are limited to the following 8 operations:


Principles (2b) - Transformations

The luminance transformation tolerates a variance and change of the leading sign of the contrast, also the mean value of the brightness could be changed. To achieve a higher similarity between domain and range blocks, the brightness and variance are corrected after the geometric adaption. This can be done using the method of the smallest quadrates, by minimizing the following expression:

whereby d(x,y) is the domain block and r(x,y) is the range block.

From the geometric and luminance transformation follows the resulting transformation:


Principles (3) - Fixpoint Theorem

Idea based on the Fixpoint Theorem from Banach also

known as "Contraction Mapping Theorem" (CMT)

Quotes that:

any contractive auto-mapping function within a metrical space

contains a fix point

Iterative application of function converges to fix point

Original picture is defined a "Banchachscher fix point" and

transformation needs to fulfill the CMT accomplishments

to ensure reconstruction

Quality of result after each iteration is usually measured

by the "Collage Theorem"


Principles (4) - Metrics

Fractal Compression produces a certain error since

perfect reconstruction is (usually) not possible

Error can be kept on an appropriate level by any function

accomplishing metrical properties (e.g. accumulation of

single pixel differences)

Finite input (as pictures) is commonly based on the L2

metric

Based on the applied metric, the so called "Peak Signal to

Noise Ratio" (PSNR), describes the difference between

two picture (-areas)


Principles (5) - PSNR

Peak Signal to Noise Ratio

Measured in dB

The higher the value, the smaller the perceivable

difference

For a value ≥ 40 dB, no difference can be perceived

between copy and original anymore


Principles (6) - Collage Theorem

Fractal Compression is based on sub-optimal parameters → error occurrence

Collage Theorem defines an upper error limit

Basically a conclusion from the Contraction Mapping Theorem:

The distance |x-w(x)| is also called "Collage Error"

Task: To apply a contraction W : F → F that minimizes the distance |x-xf|

Minimization is achieved by minimizing the error of each partial transformation (also called Collage Coding)


Encoding (1) - Range block Partition

Basically 3 different methods are used for the range block

partition

Uniform Partition:

Same size for each block (e.g. 4x4, 8x8 pixel)

Simple but inefficient

Quadtree Partition:

Based on squares with adaptive sizes to information content

relatively simple and relatively efficient

HV- Partition:

Based on quadrangles with adaptive sizes to information content

Advanced but efficient


Encoding (2) - Range block partition example


Encoding (3) - Domain block partition

By defining a set of range blocks the choice of potential

domain blocks becomes reduced

Compromise between number and size has to be taken

Too many and/or too small domain blocks:

Very good results

Poor performance

Too few and/or too big domain blocks:

Poor results

Great performance

Optimal block matching strategy is not reasonable (see

example)


Encoding (4) - Codebook generation

Assignment of domain to range blocks is also known as

"Codebook Generation" (thus, the assignment table itself

is called "Codebook")

Pixels are vertically and horizontally scanned

Legal operations are applied on domain blocks and best

match is written in codebook


Decoding (1) - iterative buffer strategy

Decoding requires a small part of the coding complexity only

Decoding is based on two scalable buffers A and B

The following 5 steps are executed until the implemented loop

condition (specifying a quality parameter) is fulfilled:

Step 1: block boarders of domain- and range blocks are read and

written into buffer (starting with Buffer A)

Step 2: Resolution reduction of domain blocks to size of range

blocks, eventually geometric adaptation

Step 3: Multiplication of reduced domain blocks with contrast and

addition of brightness value on each pixel of the reduced range

blocks

Step 4: Saving of the resulting picture in the opposite buffer

Step 5: if differences between buffer A and B are beyond the

specified tolerance go back to Step 1 using the buffer with the

resulting picture.


Decoding (2) - Alternative Strategies

Alternative decoding modes are e.g.:

Thumbnails: provides smaller preview pictures

Pyramidal Arrangement: resolution is doubled each iteration

(reasonable and requires less space)

Best Time Algorithms: optimize the total decoding time for

reconstruction of complete picture

Remark:

Maximum quality of the reconstructed picture can only be

slightly influenced by the choice of the decoding method.

The quality essentially depends on the complexity of the

algorithms that search appropriate transformation

parameters.


Example (1) - Block coding

Picture: Lenna, 256x256 pixel, 256 grayscales

(→ 512x512 = 262.122 byte)

Partitions:

Range Blocks - uniform partition (8x8 pixel = 1.024)

Domain blocks - uniform, overlapping (16x16 pixel =

58.081, optimal but extremely complex choice)

Transformations:

Rotations: 0°,90°,180°,270°

Mirroring

Diminution factor = 25% (fixed by partition choices)

Brightness transformation


Example (2) - Complexity and size

Complexity Geometrical Adaptation:

• # range blocks (1.024) x # of domain blocks (58.081) x # geom. transformations (8) = 475.799.552 operations

Brightness- and contrast adaptation: • # range blocks (1.024) x # domain blocks (58.081)

= 59.474.944 operations

Size # transformations (1.024) x

[ # bits for source coding (16) +

# bits for geometrical transformations (3) +

# bits for Brightness- and Contrast adaptation (12) ]

= 3.986 bytes

+ control data

→ total ~ 8.192 byte

→ Compression rate of 16,5


Example (3) - Sample pictures

Original (0 iteration) after 1st iteration

after 2nd iteration after 10th iteration


Evaluation (1) - General

Powerful alternative to block orientated methods such as

e.g. DCT

Remarkable compression rates

Excellent zoom function / presentation of natural photos

and sharp edges

Negative effect at too high compression rates (see sample

pictures)

Complexity still too high to be considered a reasonable

alternative to Wavelets (even though special HW support

is applied)

Popularity and use furthermore reduced by license fees


Evaluation (2) - Performance

Performance and complexity mainly depend on implemented

search strategies

The following three statements might give an impression of the

present situation and the development of the performance:

1988 – Barnsley about encoding a "complex" photo:

Complexity of about 100 hours on 2 Workstations

1997 – Diploma Thesis: Fractal Image compression / adaptive

algorithms

Complexity of only the range block partition of a 512x 512 BW

photo still varies between 1,5 – 57 minutes

2002 – Class: Digital Image Processing (University of

Dortmund, Germany)

Complexity of a "regular" photo still ranges from 50 – 147

seconds


Evaluation (3) - Fields of application

Fractal compression is mainly used for picture and video

files

Nevertheless, good results can also be achieved for audio

files

Application only seems reasonable when consumer/user

does not get involved with coding complexity

2 commercial examples:

Star Trek II (1982): Background landscapes were created by IFS

Encarta (1992): fractal encoded images of a multimedia

encyclopedia from Microsoft

Hybrid coding methods gain interest (e.g. the additional

use of DCT to obtain a scalable overall quality level for

any picture)


Comparison (1) - Experiment

4 compression methods have been compared based on

their compression rate and the corresponding PSNR

Experiment has been conducted by Y. Fisher, D. Rogovin,

and T. P. Shen (motive: Lenna with 512x512 pixels)

Compared methods

Wavelets

JPG (based on DCT)

Fractal (HV partition)

Fractal (Quadtree partition)


Comparison (2) - Evaluation

Compression Rates 1 – 5:

Best results obtained through wavelet transformation

Application of the complex fractal compression must be

reasonably questioned for efficiency reasons

Compression Rates 5 – 30:

All methods provide similar good results (excluding quadtree)

Compression Rates 30+:

Cleary demonstrates the limited use of DCT whose resulting

quality strongly decreases

Resulting quality of all 3 further methods decreases proportionally

to increasing compression rate

Generally the fractal compression provides the best

results at compression rates from 10-30


Comparison (3) - Samples 1

DCT (1:10) Fractal (1:10) Wavelet (1:10)











Literature References

Literature

Digitale Bildcodierung, Jens-Rainer Ohm - Springer ISBN: 3-540-

58579-6

Videokompressionsverfahren im Vergleich, Tosten Milde –d-

punkt, ISBN:3-920993-23-3

Demo

http://www.eurecom.fr/~image/DEMOS/FRACTAL/

Comparison of image compression methods

http://archiv.tu-

chemnitz.de/pub/1998/0010/data/vortrag/dth/pictures.html

http://www.eurecom.fr/~image/DEMOS/FRACTAL/

http://archiv.tu-chemnitz.de/pub/1998/0010/data/vortrag/dth/pictures.html



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5.7. Fractal compression Overview