Image Compression: Coding and Decoding

1. EE4328, Section 005 Introduction to Digital Image ProcessingImage CompressionZhou WangDept. of Electrical EngineeringThe Univ. of Texas at ArlingtonFall 2006

2. Image Compression: Coding and Decoding original image 262144 Bytes From [Gonzalez & Woods] From [Gonzalez & Woods] compressed bitstream 00111000001001101… (2428 Bytes) image encoder image decoder compression ratio (CR) = 108:1

3. Some General Concepts • How Can an Image be Compressed, AT ALL!? • If images are random matrices, better not to try any compression • Image pixels are highly correlated  redundant information • Information, Uncertainty and Redundancy • Information is uncertainty (to be) resolved • Redundancy is repetition of the information we have • Compression is about removing redundancy because • Entropy and Entropy Coding • Entropy is a statistical measure of uncertainty, or a measure of the amount of information to be resolved • Entropy coding: approaching the entropy (no-redundancy) limit

4. More Concepts • Bit Rate and Compression Ratio • Bit rate: bits/pixel, sometimes written as bpp • Compression ratio (CR): • Binary, Gray-Scale, Color Image Compression • Original binary image: 1 bit/pixel • Original gray-scale image: typically 8bits/pixel • Original Color image: typically 24bits/pixel • Lossless, Nearly lossless and Lossy Compression • Lossless: original image can be exactly reconstructed • Nearly lossless: reconstructed image nearly (visually) lossless • Lossy: reconstructed image with loss of quality (but higher CR) number of bits to represent the original image CR = number of bits in compressed bit stream

5. Binary Image Compression

6. Run Length Coding • Run Length • The length of consecutively identical symbols • Run length Coding Example • When Does it Work? • Images containing many runs of 1’s and 0’s • When Does it Not Work?

7. Run Length Coding CCITT test image No. 1 Size: 17282376 1 bit/pixel (bpp) original: 513216 bytes compressed: 37588 bytes CR = 13.65

8. Run Length Coding • Decoding Example A binary image is encoded using run length code row by row, with “0” represents white, and “1” represents black. The code is given by Row 1: “0”, 16 Row 2: “0”, 16 Row 3: “0”, 7, 2, 7 Row 4: “0”, 4, 8, 4 Row 5: “0”, 3, 2, 6, 3, 2 Row 6: “0”, 2, 2, 8, 2, 2 Row 7: “0”, 2, 1, 10, 1, 2 Row 8: “1”, 3, 10, 3 Row 9: “1”, 3, 10, 3 Row 10: “0”, 2, 1, 10, 1, 2 Row 11: “0”, 2, 2, 8, 2, 2 Row 12: “0”, 3, 2, 6, 3, 2 Row 13: “0”, 4, 8, 4 Row 14: “0”, 7, 2, 7 Row 15: “0”, 16 Row 16: “0”, 16 decode Decode the image

9. Run Length Coding • Decoding Example A binary image is encoded using run length code row by row, with “0” represents white, and “1” represents black. The code is given by Row 1: “0”, 16 Row 2: “0”, 16 Row 3: “0”, 7, 2, 7 Row 4: “0”, 4, 8, 4 Row 5: “0”, 3, 2, 6, 3, 2 Row 6: “0”, 2, 2, 8, 2, 2 Row 7: “0”, 2, 1, 10, 1, 2 Row 8: “1”, 3, 10, 3 Row 9: “1”, 3, 10, 3 Row 10: “0”, 2, 1, 10, 1, 2 Row 11: “0”, 2, 2, 8, 2, 2 Row 12: “0”, 3, 2, 6, 3, 2 Row 13: “0”, 4, 8, 4 Row 14: “0”, 7, 2, 7 Row 15: “0”, 16 Row 16: “0”, 16 decode

10. contour image region image n m Chain Coding Assume the image contains only single-pixel-wide contours, like this, not this From Prof. Al Bovik After the initial point position, code direction only (3bits/step) Code Stream: (3, 2), 1, 0, 1, 1, 1, 1, 3, 3, 3, 4, 4, 5, 4 initial point position chain code

11. Chain Coding • Decoding Example The chain code for a 8x8 binary image is given by: column row (1, 6), 7, 7, 0, 1, 1, 3, 3, 3, 1, 1, 0, 7, 7 decode Decode the image

12. Chain Coding • Decoding Example The chain code for a 8x8 binary image is given by: column row (1, 6), 7, 7, 0, 1, 1, 3, 3, 3, 1, 1, 0, 7, 7 decode

13. Lossless Gray-Scale Image Compression

14. Variable Word Length Coding • Intuitive Idea • Assign short words to gray levels that occur frequently • Assign long words to gray levels that occur infrequently • How Much Can Be Compressed? • Theoretical limit: entropy of the histogram • Practical algorithms (approach entropy): Huffman coding, arithmetic coding Maximum entropy: Uniform distribution typically in-between Minimum entropy: Impulse (delta) distribution From Prof. Al Bovik

15. Variable Word Length Coding: Example • A 4x4 4bits/pixel original image is given by Default Code Book 0: 0000 1: 0001 2: 0010 3: 0011 4: 0100 5: 0101 6: 0110 7: 0111 8: 1000 9: 1001 10: 1010 11: 1011 12: 1100 13: 1101 14: 1110 15: 1111 Bit rate = 4bits/pixel Total # of bits used to represent the image: 4x16 = 64 bits encode

16. Variable Word Length Coding: Example • Encode the original image with a CODE BOOK given left Huffman Code Book 0: 0000000 1: 0000001 2: 0001 3: 0000010 4: 0000011 5: 0000100 6: 01 7: 0000101 8: 10 9: 00100 10: 11 11: 0000110 12: 0000111 13: 001010 14: 0011 15: 001011 Total # of bits used to represent the image: 4+2+2+2+2+2+2+2+2+2+2+2+5+2+2+4 = 39 bits encode Bit rate = 39/16 = 2.4375 bits/pixel CR = 64/39 = 1.6410

17. image histogram (high entropy) DPCM histogram (low entropy) Predictive Coding • Intuitive Idea • Image pixels are highly correlated (dependent) • Predict the image pixels to be coded from those already coded • Differential Pulse-Code Modulation (DPCM) • Simplest form: code the difference between pixels • Key features: Invertible, and lower entropy (why?) DPCM: 82, 1, 3, 2, -32, -1, 1, 4, -2, -3, -5, …… Original pixels: 82, 83, 86, 88, 56, 55, 56, 60, 58, 55, 50, …… From Prof. Al Bovik

18. Advanced Predictive Coding • Higher Order (Pattern) Prediction • Use both 1D and 2D patterns for prediction • Apply Image Transforms before Predictive Coding • Decouple dependencies between image pixels • Use Advanced Statistical Image Models • Better understanding of “the nature” of image structures implies potentials of better prediction 1D Causal: 2D Causal: 1D Non-causal: 2D Non-Causal:

19. Lossy Gray-Scale Image Compression

20. Quantization • Quantization: Widely Used in Lossy Compression • Represent certain image components with fewer bits (compression) • With unavoidable distortions (lossy) • Quantizer Design • Find the best tradeoff between maximal compression  minimal distortion • Scalar Quantization     Uniform scalar quantization: 248 8 40 ... 24 1 2 3 4 Non-uniform scalar quantization:

21. Quantization • Vector Quantization • Group multiple image components together  form a vector • Quantize the vector in a higher dimensional space • More efficient than scalar quantization (in terms of compression) image component 2 Vector quantization: image component 1 From Prof. Al Bovik

22. Ideas on Lossy Image Compression code each block independently • Block-Based Image Compression • Transform-Domain Compression • Scalar or vector quantization of transform coefficients (instead of image pixels) Partition image From Prof. Al Bovik

23. Discrete Cosine Transform (DCT) • 2D-DCT: • Inverse 2D-DCT: where discontinuities: high frequencies continuous • DFT vs. DCT periodic extension by DFT reflected periodic extension by DFT From Prof. Al Bovik

24. 2D-DCT image block DC component low frequency high frequency 2D-DCT low frequency high frequency DCT block

25. JPEG Compression • Partition the image into 8x8 blocks, for each block - 128 DCT scalar quantization zig-zag scan

26. JPEG Compression • Adjust Quantization Step to Achieve Tradeoff between CR and distortion JPEG: 5KB Original: 100KB JPEG: 9KB • Artifacts: Inside blocks: blurring (why?); Across blocks: blocking (why?)

27. Wavelet and JPEG2000 Compression • Wavelet Transform Energy Compaction Lower Entropy

28. Wavelet and JPEG2000 Compression • Bitplane Coding • Scan bitplanes from MSB to LSB • Progressive (scalable) JPEG2000 (64:1) JPEG (64:1)