1. DCT based simple classification scheme for fractal image compression
Author: D.J. Duh, J.H. Jeng and S.Y. Chen
Source: Image and Vision Computing , vol. 23,
no. 13, Nov 29, 2005, pp
Reporter: Yi-Hui Chen
Date:

2. Fractal Image compression (1/2)
Self-similarity
Reduce the image redundancies
Self-affine transformations
input
Ex1:
Ex2:
Ex3:

3. Fractal Image compression (2/2)
Y. Fisher, Fractal Image Compression, Theory and Application, Springer, New York, 1994.

4. Motivation (1/3) Non-overlapping Range block Overlapping Domain block
The range pool is the set of the 1024 non-overlapping blocks of size 8x8
1024 = [(256)/8]2
Overlapping Domain block
The domain pool is the set of the overlapping blocks of size 16x16 (58081 = ( )2 )
The size is 256x256

5. Motivation (2/3) … 1024*58081*5 58081 5 58081*5 Range pool Domain pool
Get the minimize Similarity measure 5
The past fractal image compression required large amount of computation.
58081
58081*5
1024*58081*5
Range pool
All
transformation1 …
transformation2
Domain pool
transformation3
transformation4
transformation5

6. Motivation (3/3) Classification
Reduce the search space for the range and the domain pools to speedup the encoding.
DCT-based classification
Smooth type (S)
Diagonal/sub-diagonal type (D)
Horizontal/vertical type (H) S D H
Range pools
Domain pools

7. The proposed model (1/5) Σ
Lower coefficients possess most of the energy of the block.
Two DCT coefficients F(1,0) and F(0,1)
If two blocks are similar, they have roughly the same lower DCT coefficients B1 8
F1(1,0) =(√2)/8 * Σ
Σ(50, 30, 20, 20, 30, 40, 10, 15) * cos(π/16)
Σ(60, 70, 30, 20, 40, 50, 10, 25) * cos(3π/16)
Σ(80, 40, 20, 10, 20, 30, 10, 35) * cos(5π/16)
Σ(50, 40, 20, 10, 30, 50, 10, 10) * cos(7π/16)
Σ(10, 20, 30, 40, 50, 60, 40, 35) * cos(9π/16)
Σ(10, 20, 30, 40, 45, 40, 30, 45) * cos(11π/16)
Σ(20, 30, 10, 10, 10, 10, 10, 15) * cos(13π/16)
Σ(30, 30, 30, 20, 30, 20, 10, 15) * cos(15π/16)

8. The proposed model (2/5) Σ F1(0,1) =(√2)/8 * Σ *
cos(π/16)
cos(5π/16)
cos(9π/16)
cos(13π/16) *
cos(3π/16)
cos(7π/16)
cos(11π/16)
cos(15π/16)
F1(0,1) =(√2)/8 * Σ
(50, 30, 20, 20, 30, 40, 10, 15)
(60, 70, 30, 20, 40, 50, 10, 25)
(80, 40, 20, 10, 20, 30, 10, 35)
(50, 40, 20, 10, 30, 50, 10, 10)
(10, 20, 30, 40, 50, 60, 40, 35)
(10, 20, 30, 40, 45, 40, 30, 45)
(20, 30, 10, 10, 10, 10, 10, 15)
(30, 30, 30, 20, 30, 20, 10, 15)

9. The proposed model (3/5) Three classifications
If F(0,1)<Ts and F(1,0)<Ts, the type is S
Smooth type
Else if F(1,0)-F(0,1)<TD, the type is D
Diagonal/sub-diagonal type
Else the type is H
Horizontal/vertical type S D H
Range pools
Domain pools S D H

10. The proposed model (4/5) Find the threshold Ts
F*1 = max{|F1(0,1)|, |F1(1,0)|} =max{10,15}=15
F*2 = max{|F2(0,1)|, |F2(1,0)|}=max{17,25}=25
F*3 = max{|F3(0,1)|, |F3(1,0)|}=max{20,5}=20
F*4 = max{|F4(0,1)|, |F4(1,0)|}=max{12,10}=12
F*5 = max{|F5(0,1)|, |F5(1,0)|}=max{15,6}=6
F*6 = max{|F6(0,1)|, |F6(1,0)|}=max{32,35}=35 B1 B2 B3 B4 B5 B6
2. sorting
F*5, F*4, F*1, F*3, F*2, F*6
Sorting results:
3. Find Ts located at the first one-third of range blocks
Ts = F*4=12
4. For the rest F* to compute TD

11. The proposed model (5/5) Find the threshold TD F’6, F’1, F’2, F’3
F’1 = abs|F1(0,1) - F1(1,0)| abs|10-15|=5
F’2 = abs|F2(0,1)- F2(1,0)|=abs|17-25|=8
F’3 = abs|F3(0,1)- F3(1,0)|=abs|20-5|=15
F’6 = abs|F6(0,1)- F6(1,0)|=abs|32-35|=3
5. sorting
F’6, F’1, F’2, F’3
Sorting results:
6. Find TD located at the middle of range blocks
TD = F’1+ F’2 = (5+8)/2 = 6.5

12. Experimental results

13. Conclusions
A simple classification scheme based on the two lowest DCT coefficients is equipped into the fractal image compression to speedup the encoder.
The image quality can be well preserved due to the classification is performed according to the edge properties.

14. Merry Christmas