1. 4C8
Image Compression

2. Lossy Compression Effective bit rate = 8 bits/pixel
Effective bit rate = 1 bit/pixel (approx)

3. Transform Coding
In the last set of slides we showed that transforming the image into a difference image reduces the entropy of image.
G(x,y) = I(x,y) – I(x-1,y)

4. Transform Coding This is because entropy is greatest when uniform
Histogram of the original image
Histogram of the difference image

5. Signal Energy

6. Lossy Transform Coding
Lossless
Lossy
Lossless
Lossless

7. Energy Compaction with Xforms

8. The Haar Xform
LoLo
Hi-Lo
Lo-Hi
Hi-Hi

10. Implementation Details
When calculating the haar transform for the image the mid gray value is typical.
Colour Images are processed by treating each colour channel as separate gray scale images.
If YUV colourspace is used subsampling of the U and V channels is probable.

11. Quantisation
After we create the image we quantise the transform coefficients.
Step size is shown by perceptual evaluation
We can assign different step sizes to the different bands.
We can use different step sizes for the different colour channels.
We will consider a uniform step size, Qstep, for each band for now.

12. Entropy
Qstep = 15

13. Entropy Calculating the overall entropy is trickier
Qstep = 15
Calculating the overall entropy is trickier
Each coefficient in a band represents 4 pixel locations in the original image.
So bits/pixel = (bits/coefficient)/4
So the entropy of the transformed and quantised lenna is

14. Mistake in Fig. 5 of handout
Red Dashed Line is the Histogram. Blue bars represent the “entropies” (ie. - p * log2(p) ) and not vice versa

15. Multilevel Haar Xform

16. Calculating the Entropy for Level 2 of the transform
One Level 1 coefficient represents 4 pixels
One level 2 coefficient represents 16 pixels
Bands
Entropy/Coeff
Entropy/pixel
Level 2
LoLo
5.58
0.34
LoHi
2.22
0.14
HiLo
2.99
0.19
HiHi
1.75
0.11
Level 1
1.15
0.29
1.70
0.43
0.80
Total Entropy =
1.70 bits/pixel
Qstep = 15

17. Multilevel Haar Xform

18. Calculating the Entropy for Level 3 of the transform
One Level 1 coefficient represents 4 pixels
One level 2 coefficient represents 16 pixels
One level 3 coefficient represents 64 pixels
Qstep = 15

19. Calculating the Entropy for Level 3 of the transform
One Level 1 coefficient represents 4 pixels
One level 2 coefficient represents 16 pixels
One level 3 coefficient represents 64 pixels
Bands
Entropy/Coeff
Entropy/pixel
Level 3
LoLo
6.42
0.10
LoHi
3.55
0.06
HiLo
4.52
0.07
HiHi
3.05
0.05
Level 2
2.22
0.14
2.99
0.19
1.75
0.11
Level 1
1.15
0.29
1.70
0.43
0.80
Total Entropy =
1.62 bits/pixel
Qstep = 15

20. Multilevel Haar Xform
Qstep = 15

21. Measuring Performance
Compression Efficiency - Entropy
Reconstruction Quality – Subjective Analysis
Haar Transform
Quantisation
Quantisation

22. Reconstruction Qstep = 15

23. Reconstruction Qstep = 30

24. Reconstruction Qstep = 30
Original
Quantised
Haar Transform + Quantisation

25. Laplacian Pdfs

26. GOAL – estimate a theoretical value for the entropy of one of the subbands
So we can estimate x0 for the band by finding the standard deviation of the coefficient values.

27. x1 = 0, x2 = Q/2
x1 = (k-1/2)Q, x2 = (k-1/2)Q

28. See Handout for Missing Steps Here

29. Measured Entropy is less than what we would expect for a laplacian pdf
Measured Entropy is less than what we would expect for a laplacian pdf. This is because the actual decay of the histogram is greater than an exponential decay.

30. Practical Entropy Coding

31. Huffman Coding

32. Practical Results

33. Remember the ideal codelength So if pk = 0.8, then
The code is inefficient because level 0 as a probability >>0.5 (0.8 approx)
Remember the ideal codelength
So if pk = 0.8, then
However, the minimum code length we can use for a symbol is 1 bit.
Therefore, we need to find a new way of coding level 0 – use run length coding

34. RLC

35. RLC coding to create “events”13 -5 1 -1
Define max run of zeros as 8, and we are coding runs of 1, 2, 4 and 8 zeros
Here we have
4 non-zero “events”
1 x Run-of-4-Zeros event
2 x Run-of-2 zeros event
1x Run-of-8-zeros event
1 x Run-of-1-zero event

36. Practical Results

38. Synchronisation
Say we have a source with symbols A, B and C. Say we wish to encode the message ABBCCBCABAA using the following code table
Symbol
Code A B 10 C 11
The Coded message is therefore
Q. What is the decoded message if the 6th bit in the stream is corrupted?
Ie. We receive

39. Synchronisation 010100111101101000 The decoded stream is ABBACCACABA
Symbol
Code A B 10 C 11
The decoded stream is ABBACCACABA
The problem is that 1 bit error causes subsequent symbols to be decoded incorrectly as well.
The stream is said to have lost synchronisation.
A solution is to periodically insert synchronisation symbols into the stream (eg. One at the start of each row). This limits how far errors can propagate.

40. Summary