1. Image Transforms for Robust Coding
Javier Pinilla-Dutoit
Educational Technology Research Group
03/01/2019
Javier University of Birmingham

2. Javier Pinilla-Dutoit@The University of Birmingham
Introduction
“The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point.” (Claude Shannon, 1948)
What information should be transmitted?
How should it be transmitted?
03/01/2019
Javier University of Birmingham

3. Javier Pinilla-Dutoit@The University of Birmingham
Communication System
03/01/2019
Javier University of Birmingham

4. Javier Pinilla-Dutoit@The University of Birmingham
Source Coding
Goals
Reduction of redundancy
Removal of irrelevancy (irreversible)
Stages
Transformation
Quantization (lossy)
Entropy coding
Image samples
Transform coefficients
Symbol stream
Bit stream
03/01/2019
Javier University of Birmingham

5. Javier Pinilla-Dutoit@The University of Birmingham
Lossless and Lossy
Lossless (2:1 to 8:1)
Limited by the entropy of the message (Claude E. Shannon)
Lossy (100:1)
Entropy is a measure of information content: the more probable the message, the lower its information content, the lower its entropy
03/01/2019
Javier University of Birmingham

6. Javier Pinilla-Dutoit@The University of Birmingham
Transformation
Example: “rotation of coordinate axis”
Improves statistical distribution of image elements
Decomposition into components of differing variance
03/01/2019
Javier University of Birmingham

7. Transformations for Image Coding
Block transforms (blocks in the spatial domain)
Fixed size
Quadtrees
Sub-band decompositions (blocks in the frequency domain)
Uniform
Logarithm
Pyramids
Wavelet
Wavelet packets
03/01/2019
Javier University of Birmingham

8. Javier Pinilla-Dutoit@The University of Birmingham
Quantization
Four-level digital
Eight-level digital
representation
representation
Two-bit resolution
Three-bit resolution
Can be used to exploit features of the human visual system
03/01/2019
Javier University of Birmingham

9. Quantization (bit allocation)
Example: "rounding to the nearest integer"
Non-uniform (variable bit allocation)
Based on the statistics of the source (Laplacian quantizer)
Based on the human visual system (perceptually-tuned quantization)
03/01/2019
Javier University of Birmingham

10. Javier Pinilla-Dutoit@The University of Birmingham
Entropy Coding
Methods based on repeated characters: run-length encoding
The repeated character is replaced by the number of occurrences and by the character itself
e.g. AAAABBBCCCCC (12 symbols) is coded as 4A3B5C (6 symbols) [2:1]
Methods based on probability of occurrence: Huffman coding, arithmetic coding
Symbols that occur more often are assigned shorter codes
e.g. natural languages: ‘a’, ‘is’, ‘the’ (shorter words) are those with higher probability
Dictionary-based methods: Lempel-Zip coding
Words and phrases within a text stream are likely to be repeated
e.g. acronyms: The Lempel-Zip encoding methods (LZ) are string-matching techniques. LZ were invented in 1977 and 1978
03/01/2019
Javier University of Birmingham

11. Javier Pinilla-Dutoit@The University of Birmingham
Fourier Analysis
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Javier University of Birmingham

12. Fourier Transform Bases
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Javier University of Birmingham

13. Javier Pinilla-Dutoit@The University of Birmingham
Fourier Transform
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Javier University of Birmingham

14. Short-time Fourier Analysis
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Javier University of Birmingham

15. Short-time Fourier Transform Bases
-400
-200
200
400 -1 1
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Javier University of Birmingham

16. Wavelet Transform Bases
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Javier University of Birmingham

17. Javier Pinilla-Dutoit@The University of Birmingham
Time vs. Frequency
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Javier University of Birmingham

18. Application: Local Analysis
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Javier University of Birmingham

19. Javier Pinilla-Dutoit@The University of Birmingham
Wavelet Transform
03/01/2019
Javier University of Birmingham

20. Javier Pinilla-Dutoit@The University of Birmingham
2-D Wavelet Transform
03/01/2019
Javier University of Birmingham

21. Discrete Wavelet Transform and Filter Banks
03/01/2019
Javier University of Birmingham

22. Haar Wavelet Compression1
8 x
6 x
9 x
4 x
2 x
7 x
1 x
1 x
3 x
-1 x
-1 x
5 x 9 8 8 7 8 8 5 4 4 3 4 2 1 -1
Original
Transformed
Compressed
Reconstructed
03/01/2019
Javier University of Birmingham

23. Cortex Transform (Watson, 1987)
03/01/2019
Cortex Transform (Watson, 1987)
Radial and angular frequency bands
Simulated response of neurons in the human visual cortex
Invertible
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Javier University of Birmingham

24. Modified Cortex Transform (Daly, 1993)
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Javier University of Birmingham

25. Javier Pinilla-Dutoit@The University of Birmingham
Cortex Filters
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Javier University of Birmingham

26. Javier Pinilla-Dutoit@The University of Birmingham
Cortex Layers
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Javier University of Birmingham

27. Cortex Frequency Spectrum
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Cortex Frequency Spectrum
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Javier University of Birmingham

28. Assignment: Haar Wavelet
Consider the signal x(n)={13, 11, -3, 7, 5, -13, 15, 9}.
Apply the Haar decomposition to these coefficients. The Haar transform is implemented as a filter bank where the output of every lowpass filter is yL (n) = ½ x (n) + ½ x (n+1) and the output of every highpass filter is yH (n) = ½ x (n) – ½ x (n+1). After every filtered operation, the signal is downsampled by a factor of 2.
Extensions:
Reconstruct the original signal from the transformed coefficients.
Make a drawing of the analysis and synthesis filter banks.
Plot the original, transformed and reconstructed signal and the Haar basis functions.
03/01/2019
Javier University of Birmingham

29. Assignment: Haar Wavelet
Solution:
Resolution
Averages
Detail Coefficients 8
13, 11, -3, 7, 5, -13, 15, 9 4
12, 4, -4, 12
1, -5, 9, 3 2
8, 4
2, -8 1 6
03/01/2019
Javier University of Birmingham