1. Last update on June 15, 2010 Doug Young Suh suh@khu.ac.kr
Transformation
Last update on June 15, 2010
Doug Young Suh
7/30/2018

2. Entropy and compression
amount of information = degree of surprise
Entropy and average code length
Information source and coding
Memoryless source : no correlation
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Red blue yellow yellow red black red
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7/30/2018
Media Lab. Kyung Hee University

3. Media Lab. Kyung Hee University
Entropy
Entropy
What if {0.99,0.003,0.003,0.004}? H(X)≈0
Reduce H(X)!!
We need narrower pdf.
1/4
1/2
1/4
1/8
7/30/2018
Media Lab. Kyung Hee University

4. Media Lab. Kyung Hee University
How to get small H(X)?
Transformation
Signals in the frequency domain
or mapping into more probable vectors
Predict to reduce uncertainty
H(X) = “degree of uncertainty”
Prediction by using known information
7/30/2018
Media Lab. Kyung Hee University

5. Set of ortho-normal vectors
Inner product of two orthogonal vectors is 0.
Inner product
Add multiplications at the same positions.
Normal
Normal
7/30/2018
Media Lab. Kyung Hee University

6. Media Lab. Kyung Hee University
Mapping to { , }
Any vector in the 2 dimensional space can be represented by weighted sum of 2 ortho-normal vectors.
                                                                      
In this case, c0 is stronger.
What does it mean “a weight is large”?
weights
7/30/2018
Media Lab. Kyung Hee University

7. Media Lab. Kyung Hee University
How to get the weights?
Inner product for weight
More generally,
7/30/2018
Media Lab. Kyung Hee University

8. Media Lab. Kyung Hee University
2 point DCT/IDCT
2 point DCT/IDCT
DCT
IDCT
7/30/2018
Media Lab. Kyung Hee University

9. 4-pt transform 4 basis vectors (i.e. code) possible C1 =[½ ½ ½ ½ ]
Matrix representation
Networked Video

10. Media Lab. Kyung Hee University
8-pt DCT with 8D vectors
Any set of 8 data can be represented by weighted sum of 8 ortho-normal vectors
8 weights for 8 ortho-normal vectors in the 8 dimensional spaces.
Frequency 0  DC, constant (not varying)
7/30/2018
Media Lab. Kyung Hee University

11. 8X8D ortho-normal vectors
is calculated for u=1 and x=0~7 (cosine 1/2 period)
is calculated for u=2 and x=0~7 (cosine 1 period)
is calculated for u=3 and x=0~7 (cosine 1.5 period)
is calculated for u=7 and x=0~7 (cosine 3.5 period)
7/30/2018
Media Lab. Kyung Hee University

12. 8 8D ortho-normal vectors
DC frequency 0
u=0
From low frequency
u=1
u=2
u=3
To high frequency
n=0  n=  n=7
7/30/2018
Media Lab. Kyung Hee University

13. 8 8D ortho-normal vectors
From low frequency
u=4
u=5
u=6
u=7
To high frequency
n=0  n=  n=7
7/30/2018
Media Lab. Kyung Hee University

14. Media Lab. Kyung Hee University
8-pt DCT/IDCT
IDCT
DCT
7/30/2018
Media Lab. Kyung Hee University

15. Media Lab. Kyung Hee University
8-pt DCT of DC
DC : constant signal
255
255
255
255
255
255
255
255
722
7/30/2018
Media Lab. Kyung Hee University

16. Media Lab. Kyung Hee University
8-pt DCT of step signal
Step signal
255
255
255
255
360
326
-114 77 65
7/30/2018
Media Lab. Kyung Hee University

17. Media Lab. Kyung Hee University
8-pt DCT of step signal
High frequency signal
255
255
255
255
360 65 77
114
327
7/30/2018
Media Lab. Kyung Hee University

18. Media Lab. Kyung Hee University
8x8 DCT
2D DCT of the 8x8 block, 64 pixels
8 point DCT for 8 rows of 8 pixels
 8 point DCT for 8 columns
8x8 DCT
8x8 IDCT
7/30/2018
Media Lab. Kyung Hee University

19. 8x8 DCT 8 pt DCT is to calculate weights for 8 1D basic patterns,
while 8x8 2D DCT is to calculate weights for 64 2D basic patterns.
For example,

20. 8x8 DCT
64 8x8 patterns
U=0
U=7
v=0
v=7

21. Media Lab. Kyung Hee University
2D 8x8 DCT
By using 8X8 DCT, 64 weights are calculated and stored, respectively.
These are 2D 8x8 DCT coefficients.
7/30/2018
Media Lab. Kyung Hee University

22. 2D 8x8 DCT at horizontal edge
F00 and F01 are large.
For u>0, they are almost 0.
DCT
Block with Horizontal edge
DCT of Horizontal edge
7/30/2018
Media Lab. Kyung Hee University

23. Block with vertical edge Media Lab. Kyung Hee University
2D 8x8 DCT at vertical edge
F00 and F10 are large.
For v>0, the values are small.
F10 < 0 ?
DCT
Block with vertical edge
DCT of vertical edge
7/30/2018
Media Lab. Kyung Hee University

24. Low pass filtering (LPF)
Remove weights of higher frequency
LPF
7/30/2018
Media Lab. Kyung Hee University

25. Media Lab. Kyung Hee University
Energy compaction
More compression when energy distribution is focused to a direction.
The simpler an image, the more compression.
DCT is better than DFT for image.  KL transform!
Ortho-normal vector patterns of DCT are better suited to image. 1 2 3 4 5 6 7
Simple image
Complex image
DCT
DFT
7/30/2018
Media Lab. Kyung Hee University

26. Matrix representation
Matrix representation of 4x4-pt transform
where , then
Complexity ~ O(N3)  needs fast algorithm
Networked Video

27. Media Lab. Kyung Hee University
Summary
Transform for compression
Energy compaction
transform
= inner product to ortho-normal vectors
weighted sum
Weights = frequency coefficients
No information loss at all !!
7/30/2018
Media Lab. Kyung Hee University

28. Video encoding 2-1 2-3 2-2 DCT Q VLC IQ IDCT Original Video Encoded+
DCT Q
VLC
Encoded
Bitstream IQ
Motion
Estimation
Motion vector
Frame
Memory
IDCT
Networked Video