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Image Transforms for Robust Coding

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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 03/01/2019 Javier University of Birmingham

12 Fourier Transform Bases
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13 Javier Pinilla-Dutoit@The University of Birmingham
Fourier Transform 03/01/2019 Javier University of Birmingham

14 Short-time Fourier Analysis
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15 Short-time Fourier Transform Bases
-400 -200 200 400 -1 1 03/01/2019 Javier University of Birmingham

16 Wavelet Transform Bases
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17 Javier Pinilla-Dutoit@The University of Birmingham
Time vs. Frequency 03/01/2019 Javier University of Birmingham

18 Application: Local Analysis
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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 Compression
1 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 03/01/2019 Javier University of Birmingham

24 Modified Cortex Transform (Daly, 1993)
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25 Javier Pinilla-Dutoit@The University of Birmingham
Cortex Filters 03/01/2019 Javier University of Birmingham

26 Javier Pinilla-Dutoit@The University of Birmingham
Cortex Layers 03/01/2019 Javier University of Birmingham

27 Cortex Frequency Spectrum
03/01/2019 Cortex Frequency Spectrum 03/01/2019 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


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