5.7. Fractal compression Overview

Transcription

1 5.7. Fractal compression Overview 1. Introduction 2. Principles 3. Encoding 4. Decoding 5. Example 6. Evaluation 7. Comparison 8. Literature References 1

2 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 2

3 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 3

4 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. 4

5 Introduction (4) - MRCM Samples after 0 iteration after 1 iteration after 2 iterations after 3 iterations after 4 iterations after 5 iterations 5

6 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 6

7 Principles (1) - Partitions 2 Partitions Range blocks R i (often uniform size, non-overlapping) Domain blocks D i (optional size, overlapping) surjective function D i R i 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 7

8 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: 8

9 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: 9

10 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" 10

11 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) 11

12 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 12

13 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-x f Minimization is achieved by minimizing the error of each partial transformation (also called Collage Coding) 13

14 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 14

15 Encoding (2) - Range block partition example 15

16 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) 16

17 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 17

18 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. 18

19 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. 19

20 Example (1) - Block coding Picture: Lenna, 256x256 pixel, 256 grayscales ( 512x512 = byte) Partitions: Range Blocks - uniform partition (8x8 pixel = 1.024) Domain blocks - uniform, overlapping (16x16 pixel = , optimal but extremely complex choice) Transformations: Rotations: 0,90,180,270 Mirroring Diminution factor = 25% (fixed by partition choices) Brightness transformation 20

21 Example (2) - Complexity and size Complexity Geometrical Adaptation: # range blocks (1.024) x # of domain blocks (58.081) x # geom. transformations (8) = operations Brightness- and contrast adaptation: # range blocks (1.024) x # domain blocks (58.081) = operations Size # transformations (1.024) x [ # bits for source coding (16) + # bits for geometrical transformations (3) + # bits for Brightness- and Contrast adaptation (12) ] = bytes + control data total ~ byte Compression rate of 16,5 21

22 Example (3) - Sample pictures Original (0 iteration) after 1st iteration after 2nd iteration after 10th iteration 22

23 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 23

24 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 seconds 24

25 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) 25

26 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) 26

27 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

28 Comparison (3) - Samples 1 DCT (1:10) Fractal (1:10) Wavelet (1:10) 28

29 Comparison (4) - Samples 2 DCT (1:25) Fractal (1:25) Wavelet (1:25) 29

30 Comparison (5) - Samples 3 DCT (1:50) Fractal (1:50) Wavelet (1:50) 30

31 Comparison (6) - Samples 4 DCT (1:100) Fractal (1:100) Wavelet (1:100) 31

32 Literature References Literature Digitale Bildcodierung, Jens-Rainer Ohm - Springer ISBN: Demo Videokompressionsverfahren im Vergleich, Tosten Milde dpunkt, ISBN: Comparison of image compression methods 32