# CSEP 521 Applied Algorithms Spring Lossy Image Compression

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1 CSEP 521 Applied Algorithms Spring 2005 Lossy Image Compression

2 Lossy Image Compression Methods Scalar quantization (SQ). Vector quantization (VQ). DCT Compression JPEG Wavelet Compression SPIHT UWIC (University of Washington Image Coder) EBCOT (JPEG 2000) Lecture 7 - Lossy Image Compression 2

3 JPEG Standard JPEG - Joint Photographic Experts Group Current image compression standard. Uses discrete cosine transform, scalar quantization, and Huffman coding. JPEG 2000 uses to wavelet compression. Lecture 7 - Lossy Image Compression 3

4 Barbara 32:1 compression ratio.25 bits/pixel (8 bits) original JPEG VQ Wavelet-SPIHT UWIC Lecture 7 - Lossy Image Compression 4

5 JPEG Lecture 7 - Lossy Image Compression 5

6 VQ Lecture 7 - Lossy Image Compression 6

7 SPIHT Lecture 7 - Lossy Image Compression 7

8 UWIC Lecture 7 - Lossy Image Compression 8

9 Original Lecture 7 - Lossy Image Compression 9

10 Images and the Eye Images are meant to be viewed by the human eye (usually). The eye is very good at interpolation, that is, the eye can tolerate some distortion. So lossy compression is not necessarily bad. The eye has more acuity for luminance (gray scale) than chrominance (color). Gray scale is more important than color. Compression is usually done in the YUV color coordinates, Y for luminance and U,V for color. U and V should be compressed more than Y This is why we will concentrate on compressing gray scale (8 bits per pixel) images. Lecture 7 - Lossy Image Compression 10

11 Distortion original Encoder compressed decompressed x y xˆ Decoder Lossy compression: Measure of distortion is commonly mean squared error (MSE). Assume x has n real components (pixels). x i= 1 xˆ n 1 MSE = (x ˆ n 2 i xi) Lecture 7 - Lossy Image Compression 11

12 PSNR Peak Signal to Noise Ratio (PSNR) is the standard way to measure fidelity. PSNR = 10log 10 2 m ( ) MSE where m is the maximum value of a pixel possible. For gray scale images (8 bits per pixel) m = 255. PSNR is measured in decibels (db)..5 to 1 db is said to be a perceptible difference. Decent images start at about 30 db Lecture 7 - Lossy Image Compression 12

13 Rate-Fidelity Curve PSNR bits per pixel SPIHT coded Barbara Properties: - Increasing - Slope decreasing Lecture 7 - Lossy Image Compression 13

14 PSNR is not Everything VQ PSNR = 25.8 db PSNR = 25.8 db Lecture 7 - Lossy Image Compression 14

15 Wavelet Transform Wavelet Transform A family of transformations that filters the data into low resolution data plus detail data. image low pass filter L L wavelet transformed image H H high pass filter detail subbands Lecture 7 - Lossy Image Compression 15

16 Wavelet Transformed Barbara (Enhanced) Low resolution subband Detail subbands Lecture 7 - Lossy Image Compression 16

17 Wavelet Transformed Barbara (Actual) most of the details are small so they are very dark. Lecture 7 - Lossy Image Compression 17

18 Wavelet Transform Compression image (pixels) Encoder wavelet transform transformed image (coefficients) wavelet coding bit stream Decoder wavelet decoding transformed image (approx coefficients) inverse wavelet transform distorted image Wavelet coder transmits wavelet transformed image in bit plane order with the most significant bits first. Lecture 7 - Lossy Image Compression 18

19 sign plane Bit Planes of Coefficients Coefficients are normalized between 1 and 1 Lecture 7 - Lossy Image Compression 19

20 Why Wavelet Compression Works Wavelet coefficients are transmitted in bit-plane order. In most significant bit planes most coefficients are 0 so they can be coded efficiently. Only some of the bit planes are transmitted. This is where fidelity is lost when compression is gained. Natural progressive transmission compressed bit planes... truncated compressed bit planes Lecture 7 - Lossy Image Compression 20

21 Rate-Fidelity Curve PSNR bits per pixel More bit planes of the wavelet transformed image that is sent the higher the fidelity. SPIHT coded Barbara Lecture 7 - Lossy Image Compression 21

22 Wavelet Coding Methods EZW - Shapiro, 1993 Embedded Zerotree coding. SPIHT - Said and Pearlman, 1996 Set Partitioning in Hierarchical Trees coding. Also uses zerotrees. ECECOW - Wu, 1997 Uses arithmetic coding with context. EBCOT Taubman, 2000 Uses arithmetic coding with different context. JPEG 2000 new standard based largely on EBCOT GTW Hong, Ladner 2000 Uses group testing which is closely related to Golomb codes UWIC - Ladner, Askew, Barney 2003 Like GTW but uses arithmetic coding Lecture 7 - Lossy Image Compression 22

23 Wavelet Transform image (pixels) Encoder wavelet transform transformed image (coefficients) wavelet coding bit stream Decoder wavelet decoding transformed image (approx coefficients) inverse wavelet transform distorted image A wavelet transform decomposes the image into a low resolution version and details. The details are typically very small so they can be coded in very few bits. Lecture 7 - Lossy Image Compression 23

24 One-Dimensional Average Transform (1) x y average (x+y)/2 x y detail (y-x)/2 How do we represent two data points at lower resolution? Lecture 7 - Lossy Image Compression 24

25 One-Dimensional Average Transform (2) x y (x+y)/2 = L low pass filter L x y high pass filter (y-x)/2 = H H x = L - H y = L + H Transform detail Inverse Transform Lecture 7 - Lossy Image Compression 25

26 One-Dimensional Average Transform (3) L Low Resolution Version H Note that the low resolution version and the detail together have the same number of values as the original. Detail Lecture 7 - Lossy Image Compression 26

27 One-Dimensional Average Transform (4) A B L H B[i] = 1 2 A[2i] + B[n/2 + i] = A[2i + 1], A[2i] i < A[2i + 1], n 2 0 i < n 2 L = B[0..n/2-1] H = B[n/2..n-1] Lecture 7 - Lossy Image Compression 27

28 One-Dimensional Average Inverse Transform B L H A A[2i] = B[i] B[n/2 + i], 0 A[2i + 1] = B[i] + B[n/2 + i], i < n 2 0 i < n 2 Lecture 7 - Lossy Image Compression 28

29 Two Dimensional Transform (1) low resolution subband horizontal transform L H vertical transform LL LH HL HH Transform each row Transform each column in L and H 3 detail subbands Lecture 7 - Lossy Image Compression 29

30 Two Dimensional Transform (1) LL LH horizontal transform LLL HLL LH vertical transform LLLL LHLL HLLL HHLL LH HL HH HL HH HL HH Transform each row in LL Transform each column in LLL and HLL 2 levels of transform gives 7 subbands. k levels of transform gives 3k + 1 subbands. Lecture 7 - Lossy Image Compression 30

31 Two Dimensional Average Transform horizontal transform negative value vertical transform Lecture 7 - Lossy Image Compression 31

32 Wavelet Transformed Image 2 levels of wavelet transform 1 low resolution subband 6 detail subbands Lecture 7 - Lossy Image Compression 32

33 Wavelet Transform Details Conversion to reals. Convert gray scale to floating point. Convert color to Y U V and then convert each to band to floating point. Compress separately. After several levels (3-8) of transform we have a matrix of floating point numbers called the wavelet transformed image (coefficients). Lecture 7 - Lossy Image Compression 33

34 Wavelet Transforms Technically wavelet transforms are special kinds of linear transformations. Easiest to think of them as filters. The filters depend only on a constant number of values. (bounded support) Preserve energy (norm of the pixels = norm of the coefficients) Inverse filters also have bounded support. Well-known wavelet transforms Haar like the average but orthogonal to preserve energy. Not used in practice. Daubechies 9/7 biorthogonal (inverse is not the transpose). Most commonly used in practice. Lecture 7 - Lossy Image Compression 34

35 Haar Filters 1 1 low pass =, high pass = , low pass high pass B[i] = B[n/2 1 A[2i] i] = 2 1 A[2i + 1], 0 i < 2 1 A[2i] + A[2i + 1], 2 n 2 0 i < n 2 Want the sum of squares of the filter coefficients = 1 Lecture 7 - Lossy Image Compression 35

36 Daubechies 9/7 Filters low pass filter low pass high pass B[i] = h j 4 j= 4 B[n/2 + i] = h A[2 i + j 3 j= 3-1 j], g A[2 i + j high pass filter i < j], n 2 0 i < n 2 g j reflection used near boundaries Lecture 7 - Lossy Image Compression 36

37 Linear Time Complexity of 2D Wavelet Transform Let n = number of pixels and let b be the number of coefficients in the filters. One level of transform takes time O(bn) k levels of transform takes time proportional to bn + bn/ bn/4 k-1 < (4/3)bn. The wavelet transform is linear time when the filters have constant size. The point of wavelets is to use constant size filters unlike many other transforms. Lecture 7 - Lossy Image Compression 37

38 Wavelet Transform image (pixels) Encoder wavelet transform transformed image (coefficients) wavelet coding bit stream Decoder wavelet decoding transformed image (approx coefficients) inverse wavelet transform distorted image Wavelet coder transmits wavelet transformed image in bit plane order with the most significant bits first. Lecture 7 - Lossy Image Compression 38

39 Wavelet Coding Normalize the coefficients to be between 1 and 1 Transmit one bit-plane at a time For each bit-plane Significance pass: Find the newly significant coefficients, transmit their signs. Refinement pass: transmit the bits of the known significant coefficients. Lecture 7 - Lossy Image Compression 39

40 Significant Coefficients magnitude coefficients bit-plane 1 threshold Lecture 7 - Lossy Image Compression 40

41 Significant Coefficients magnitude coefficients bit-plane 2 threshold Lecture 7 - Lossy Image Compression 41

42 Significance & Refinement Passes Code a bit-plane in two passes Significance pass codes previously insignificant coefficients also codes sign bit Refinement pass refines values for previously significant coefficients Main idea: Significance-pass bits likely to be 0; Refinement-pass bit are not Coefficient List # value refinement bits Lecture 7 - Lossy Image Compression 42 : : Bit-plane 3

43 bit plane 1 Compressed size bpp.0014 PSNR 15.3 Lecture 7 - Lossy Image Compression 43

44 bit planes 1 2 Compressed size bpp.0033 PSNR 16.8 Lecture 7 - Lossy Image Compression 44

45 bit planes 1 3 Compressed size bpp.0072 PSNR 18.8 Lecture 7 - Lossy Image Compression 45

46 bit planes 1 4 Compressed size bpp : 1 PSNR 20.5 Lecture 7 - Lossy Image Compression 46

47 bit planes 1 5 Compressed size bpp.035 ratio 229 : 1 PSNR 22.2 Lecture 7 - Lossy Image Compression 47

48 bit planes 1 6 Compressed size bpp.118 ratio 68 : 1 PSNR 24.8 Lecture 7 - Lossy Image Compression 48

49 bit planes 1 7 Compressed size bpp.303 ratio 26 : 1 PSNR 28.7 Lecture 7 - Lossy Image Compression 49

50 bit planes 1 8 Compressed size bpp.619 ratio 13 : 1 PSNR 32.9 Lecture 7 - Lossy Image Compression 50

51 bit planes 1 9 bpp ratio 7 : 1 PSNR 37.5 Lecture 7 - Lossy Image Compression 51

52 UWIC A simple image coder based on Bit-plane coding Significance pass Refinement pass Arithmetic coding Careful selection of contexts based on statistical studies Priority queue for selecting contexts to code Implemented by undergraduates Amanda Askew and Dane Barney in Summer Lecture 7 - Lossy Image Compression 52

53 UWIC Block Diagram Encoder image (pixels) wavelet transform transformed image (coefficients) subtract LL Avg Divide into bit-planes Bitplanes bit plane encoding using AC bit stream bit plane decoding using AC Decoder Bitplanes Recombine bit-planes add LL Avg transformed image (approx coefficients) inverse wavelet transform distorted image Lecture 7 - Lossy Image Compression 53

54 Arithmetic Coding in UWIC Performed on each individual bit plane. Alphabet is ={0,1} Uses integer implementation with 32-bit integers. (Initialize L = 0, R = ) Uses scaling and adaptation. Uses contexts based on statistical studies. Lecture 7 - Lossy Image Compression 54

55 Coding the Bit-Planes Code most significant bit-planes first Significance pass for a bit-plane First code those coefficients that were insignificant in the previous bit-plane. Code coefficients most likely to be significant first (priority queue). If a coefficient becomes significant then code its sign. Refinement pass for a bit-plane Code the refinement bit for each coefficient that is significant in a previous bit-plane Lecture 7 - Lossy Image Compression 55

56 Contexts (per bit plane) Significance pass contexts: Contexts based on Subband level Number of significant neighbors Sign context Refinement contexts 1st refinement bit has a context All other refinement bits have one context Context Principles Bits in a given context have a probability distribution Bits in different contexts have different probability distributions Lecture 7 - Lossy Image Compression 56

57 Subband Level Image is divided into subbands until LL band (subband level 0) is less than 16x16 Barbara image has 7 subband levels Lecture 7 - Lossy Image Compression 57

58 Statistics for Subband Levels Barbara (8bpp) Subband Level # significant # insignificant % significant % % % % % % % Lecture 7 - Lossy Image Compression 58

59 Significant Neighbor Metric Count # of significant neighbors children count for at most 1 0,1,2,3+ Neighbors of : p a a a p a parent spatially adjacent a a a a a c c i c spatially identical child i i c c Lecture 7 - Lossy Image Compression 59

60 Number of Significant Neighbors Barbara (8bpp) Significant neighbors # significant # insignificant % significant % % % % % % % % Lecture 7 - Lossy Image Compression 60

61 Refinement Bit Context Statistics Barbara (8bpp) 0 s 1 s % 0 s 2 nd Refinement Bits 146, , % Other Refinement Bits 475, , % Sign Bits 128, , % Barbara at 2bpp: 2 nd Refinement bit % 0 s = 65.8% Lecture 7 - Lossy Image Compression 61

62 Context Details Significance pass contexts per bit-plane: Max neighbors* num subband levels contexts For Barbara: contexts for sig neighbor counts of 0-3 and subband levels of 0-6 = 4*7 = 28 contexts Index of a context. Max neighbors * subband level + num sig neighbors Example num sig neighbors = 2, subband level = 3, index = 4 * = 14 Sign context 1 contexts 2 Refinement contexts 1st refinement bit is always 1 not transmitted 2nd refinement bit has a context all other refinement bits have a context Number of contexts per bit-plane for Barbara = = 31 Lecture 7 - Lossy Image Compression 62

63 Priority Queue Used in significance pass to decide which coefficient to code next Goal code coefficients most likely to become significant All non-empty contexts are kept in a max heap Priority is determined by: # sig coefficients coded / total coefficients coded Lecture 7 - Lossy Image Compression 63

64 Reconstruction of Coefficients Coefficients are decoded to a certain number of bit planes xxxxx What should X s be? < XXXXX < is half-way Handled the same as SPIHT and GTW if coefficient is still insignificant, do no interpolation if newly significant, add on.38 to scale if significant, add on.5 to scale 2 -k A = k.a A111.A01100.A100 Lecture 7 - Lossy Image Compression 64

65 Original Barbara Image Lecture 7 - Lossy Image Compression 65

66 Barbara at.5 bpp (PSNR = 31.68) Lecture 7 - Lossy Image Compression 66

67 Barbara at.25 bpp (PSNR = 27.75) Lecture 7 - Lossy Image Compression 67

68 Barbara at.1 bpp (PSNR = 24.53) Lecture 7 - Lossy Image Compression 68

69 Results PSNR (db) Compression of Barbara Bit rate (bits/pixel) JPEG2000 UWIC GTW SPIHT JPEG Lecture 7 - Lossy Image Compression 69

70 Results PSNR (db) Compression of Lena Bit rate (bits/pixel) UWIC GTW SPIHT JPEG2000 JPEG Lecture 7 - Lossy Image Compression 70

71 Results Compression of RoughWall PSNR (db) UWIC GTW SPIHT JPEG2000 JPEG Bit rate (bits/pixel) Lecture 7 - Lossy Image Compression 71

72 UWIC Notes UWIC competitive with JPEG 2000, SPIHT- AC, and GTW. Developed in Java from scratch by two undergraduates in 2 months. Lecture 7 - Lossy Image Compression 72

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