# Tutorial on Image Compression

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1 Tutorial on Image Compression Richard Baraniuk Rice University dsp.rice.edu

2 Agenda Image compression problem Transform coding (lossy) Approximation linear, nonlinear DCT-based compression JPEG Wavelet-based compression EZW, SFQ, EQ, JPEG2000 Open issues

3 Image Compression Problem

4 Images 2-D function Idealized view some function space defined over In practice ie: an matrix

5 Images 2-D function Idealized view some function space defined over In practice ie: an matrix (pixel average)

6 Quantization Approximate each continuous-valued pixel value with a discrete-valued variable Ex: 3-bit quantization = 8-level approximation Human eye blind to 8-bit quantization = 256 levels

7 From Images to Bits

8 The Need for Compression Modern digital camera megapixels

9 How Much Can We Compress? [M. Vetterli +] 2 (256x256x8) possible images ~500,000 bits [David Field] Dennis Gabor, September 1959 (Editorial IRE) the 20 bits per second which, the psychologists assure us, the human eye is capable of taking in Index all pictures ever taken in the history of mankind 100 years x ~44 bits Search the Web google.com: 5-50 billion images online ~33-36 bits JPEG on Mona Lisa ~200,000 bits JPEG2000 takes a few less, thanks to wavelets

10 Lossy Image Compression Given image approximate using bits Error incurred = distortion Example: squared error

11 Rate-Distortion Analysis better compression schemes push the R-D curve down

12 Rate-Distortion Analysis better compression schemes push the R-PSNR curve up

13 Lossy Transform Coding

14 Image Compression Space-domain coding techniques perform poorly Why? smoothness strong correlations redundancies too many bits

15 Transform Coding Quantize coefficients of an image expansion coefficients basis, frame quantize to total bits

16 Wavelet Transform Standard 2-D tensor product wavelet transform

17 Transform Coding Transform sparse set of coefficients (many 0)

18 Transform Coding Quantize approximate real-valued coefficients using bits sets small coefficients = 0

19 Quantization and Thresholding Quantization thresholds small coefficients to zero

20 Quantization and Thresholding Quantization thresholds small coefficients to zero deadzone 111

21 Transform Coding Quantize approximate real-valued coefficients using bits sets small coefficients = 0

22 Transform Coding bits Entropy code reduce excess redundancy in the bitstream Ex: Huffman coding, arithmetic coding, gzip,

23 Transform Coding/Decoding bits bits

24 Sparse Approximation

25 Computational Harmonic Analysis Representation coefficients basis, frame Analysis study through structure of should extract features of interest Approximation uses just a few terms exploit sparsity of

26 Wavelet Transform Sparsity Many (blue)

27 Nonlinear Approximation -term approximation: use largest independently Greedy / thresholding few big sorted index

28 Linear Approximation -term approximation: use first index

29 Error Approximation Rates as Optimize asymptotic error decay rate Nonlinear approximation works better than linear

30 Compression is Approximation Lossy compression of an image creates an approximation coefficients basis, frame quantize to total bits

31 NL Approximation is not Compression Nonlinear approximation chooses coefficients but does not worry about their locations threshold

32 Location, Location, Location Nonlinear approximation selects largest to minimize error (easy threshold) Compression algorithm must encode both a set of and their locations (harder)

33 Local Fourier Compression JPEG

34 JPEG Motivation Image model: images are piecewise smooth Transform: Fourier representation sparse for smooth signals edge texture texture

35 JPEG Motivation Image model: images are piecewise smooth Transform: Fourier representation sparse for smooth signals Deal with edges: local Fourier representation (DCT on 8x8 blocks)

36 JPEG and DCT Local DCT (Gabor transform with square window or wavelet packets) Divide image into 8x8 blocks Take Discrete Cosine Transform (DCT) of each block

37 Discrete Cosine Transform (DCT) 8x8 block Project onto 64 different basis functions (tensor products of 1-D DCT) Real valued Orthobasis

38 Discrete Cosine Transform (DCT) 8x8 block Project onto 64 different basis functions (tensor products of 1-D DCT) Real valued increasing frequency Orthobasis

39 Discrete Cosine Transform (DCT) 8x8 block Project onto 64 different basis functions (tensor products of 1-D DCT) Real valued rapid coefficient decay for smooth block Orthobasis

40 JPEG Quantization 8 more bits fewer bits

41 JPEG Quantization 8 more bits zero fewer bits Quasi-linear approximation in each block (fixed scheme)

42 JPEG Compression 256x256 pixels, 12,500 total bits, 0.19 bits/pixel

43 JPEG Compression Worldwide coding standard Problems local Fourier representation not sparse for edges so poor approximation at low rates blocking artifacts (discontinuities between 8x8 blocks)

44 Wavelet Compression

45 Enter Wavelets Standard 2-D tensor product wavelet transform

46 Location, Location, Location Nonlinear approximation selects largest to minimize error (easy threshold) Compression algorithm must encode both a set of and their locations (harder)

47 2-D Dyadic Partition Multiscale analysis Zoom in by factor of 2 each scale

48 2-D Dyadic Partition = Quadtree Multiscale analysis Zoom in by factor of 2 each scale Each parent node has 4 children at next finer scale

49 Wavelet Quadtrees Wavelet coefficients structured on quadtree each parent has 4 children at next finer scale

50 Wavelet Persistence Smooth region Singularity/texture - small values down tree - large values down tree

51 Zero Tree Approximation Idea: Prune wavelet subtrees in smooth regions tree-structured thresholding

52 Zero Tree Approximation Prune wavelet quadtree in smooth regions zero-tree - smooth region (prune) significant - edge/texture region (keep) Z: all wc s below=0 ie: wc s of S Z Z smooth smooth WT Z S S S S S smooth not smooth

53 Set threshold EZW Compression [Shapiro 92] Iterate: +S 1. Reduce Z Z 2. Threshold Z +S 3. Assign labels +S, S, Z, I Z Z Z Z Encode symbols with arithmetic coder

54 Set threshold EZW Compression [Shapiro 92] Iterate: +S+S 1. Reduce Z Z 2. Threshold Z +S+S 3. Assign labels +S, S, Z, I Encode symbols with arithmetic coder Z Z-S Z ZI Z Z Z Z Z Z Z Z

55 EZW Compression [Shapiro 92] Greedy algorithm based on persistence heuristic Encodes larger coefficients with more bits Progressive encoding (embedded) adds one bit of information to each significant coefficient per iteration SPIHT similar

56 EZW Compression 256x256 pixels, 9,800 total bits, 0.15 bits/pixel

57 SFQ Compression [Orchard, Ramchandran, Xiong] Space Frequency Quantization S EZW is a greedy algorithm SFQ optimize placement of S and Z symbols by dynamic programming Rate-distortion optimal Not progressive Z S Z Z Z S Z Z Z Z Z S Z Z Z Z

58 SFQ Compression 256x256 pixels, 9,500 total bits, bits/pixel

59 EQ Compression Estimation Quantization [Orchard, Ramchandran, LoPresto] Not tree-based Scans thru each wavelet subband and estimates variance of each wc from its neighbors Quantize wc as a Gaussian rv with this variance Not progressive

60 EQ Compression 256x256 pixels, 10,100 total bits, bits/pixel

61 JPEG2000 Compression Not tree-based Similar to JPEG applied to wavelet transform Can be progressive

62 JPEG2000 Compression 256x256 pixels, 9,400 total bits, bits/pixel

63 Discussion and Conclusions

64 Summary Compression is approximation, but approximation is not (quite) compression Modern image compression techniques exploit piecewise smooth image model smooth regions yield small transform coefficients and sparse representation

65 Issues Why L 2 distortion metric? Pixelization at fine scales

66 Issues Current wavelet methods do not improve on decay rate of JPEG! WHY? neither DCT nor wavelets are the right transform JPEG (DCT) JPEG2000 (wavelets)

67 1-D Piecewise Smooth Signals smooth except for singularities at a finite number of 0-D points Fourier sinusoids: suboptimal greedy approximation and extraction wavelets: optimal greedy approximation extract singularity structure

68 2-D Piecewise Smooth Signals smooth except for singularities along a finite number of smooth 1-D curves geometry texture texture Challenge: analyze/approximate geometric structure

69 Inefficient - large number of significant WCs cluster around edge contours, no matter how smooth

70 2-D Wavelets: Poor Approximation Even for a smooth C 2 contour, which straightens at fine scales Too many wavelets required! -term wavelet approximation not

71 Solution 1: Upgrade the Transform Introduce anisotropic transform curvelets, ridgelets, contourlets, Optimal error decay rates for cartoons

72 Solution 2: Upgrade the Processing Replace coefficient thresholding by a new wavelet coefficient model that captures anisotropic spatial correlations of wavelet coefficients

73 Richard Baraniuk Mike Wakin Hyeokho Choi Justin Romberg Web: dsp.rice.edu

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