Wavelet-based image compression

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1 Institut Mines-Telecom Wavelet-based image compression Marco Cagnazzo Multimedia Compression

2 Outline Introduction Discrete wavelet transform and multiresolution analysis Filter banks and DWT Multiresolution analysis 2/ Institut Mines-Telecom Wavelet-based image compression

3 Outline Introduction Discrete wavelet transform and multiresolution analysis 3/ Institut Mines-Telecom Wavelet-based image compression

4 Wavelets and images : Motivations Image model : trends + anomalies 4/ Institut Mines-Telecom Wavelet-based image compression

5 Wavelets and images : Motivations Image model : trends + anomalies 4/ Institut Mines-Telecom Wavelet-based image compression

6 Wavelets and images : Motivations Image model : trends + anomalies 4/ Institut Mines-Telecom Wavelet-based image compression

7 Wavelets and images : Motivations Anomalies : Abrupt variations of the signal High frequency contributions Objects contours Good spatial resolution Rough frequency resolution Trends : Slow variations of the signal Low frequency contributions Objects texture Rough spatial resolution Good frequency resolution 5/ Institut Mines-Telecom Wavelet-based image compression

8 Wavelets and images : Motivations Time-frequency boxes of basis signals Time analysis: θ γ(t) = δ(t t 0 ) Frequency analysis: θ γ(t) = e 2iπf 0t Short Time Fourier Transform (STFT): F F θ γ(t) = g(t t 0 )e 2iπf 0t σ t,σ ξ independent from γ T Time Analysis T Frequency Analysis Wavelet Transform: θ γ(t) = 1 ( ) t b ψ a a F F σ t(a,b) = aσ t,(1,0) σ ξ(a,b) = a 1 σ ξ,(1,0) T STFT T Wavelet Transform 6/ Institut Mines-Telecom Wavelet-based image compression

9 Wavelets and images : Motivations Signal model: an image row 7/ Institut Mines-Telecom Wavelet-based image compression

10 Wavelets and images : Motivations Signal model: an image row 7/ Institut Mines-Telecom Wavelet-based image compression

11 Wavelets and images : Motivations Signal model: an image row 7/ Institut Mines-Telecom Wavelet-based image compression

12 Wavelets and Multiple resolution analysis Approximation: low resolution version Details : zeros when the signal is polynomial x[k] = c 0 [k] c 1 [k] d 1 [k] / Institut Mines-Telecom Wavelet-based image compression

13 Outline Filter banks and DWT Multiresolution analysis Introduction Discrete wavelet transform and multiresolution analysis Filter banks and DWT Multiresolution analysis 9/ Institut Mines-Telecom Wavelet-based image compression

14 1D filter banks Filter banks and DWT Multiresolution analysis Decomposition č[k] h 2 c[k] x[k] g ď[k] 2 d[k] Analysis filter bank 2 : decimation : c[k] = č[2k] 10/ Institut Mines-Telecom Wavelet-based image compression

15 Reconstruction Filter banks and DWT Multiresolution analysis c[k] 2 ĉ[k] h x[k] d[k] 2 ˆd[k] g Synthesis filter bank 2 : interpolation operator, doubles the sample number { c[k/2] if k is even ĉ[k] = 0 if k is odd 11/ Institut Mines-Telecom Wavelet-based image compression

16 Filter properties Filter banks and DWT Multiresolution analysis Perfect reconstruction (PR) FIR Orthogonality Vanishing moments Symmetry 12/ Institut Mines-Telecom Wavelet-based image compression

17 Filter banks and DWT Multiresolution analysis Perfect reconstruction conditions We want PR after synthesis and analysis filter banks : k Z, x k = x k+l X (z) = z l X (z) h č[k] 2 c[k] 2 ĉ[k] h x[k] x[k] g ď[k] 2 d[k] 2 ˆd[k] g 13/ Institut Mines-Telecom Wavelet-based image compression

18 Z-domain relationships Filter banks and DWT Multiresolution analysis filter Č(z) = n= č n z n = H(z) X (z) ( z 1/2)] decimation C(z) = 1 [Č( z 1/2) + 2 Č interpolation Ĉ(z) = C (z 2) output X (z) = H(z) C (z 2) + G(z) D (z 2) X (z) = 1 ] [ H(z) H(z)+ G(z) G(z) X (z) ] [ H(z) H( z)+ G(z) G( z) X ( z) 2 14/ Institut Mines-Telecom Wavelet-based image compression

19 PR conditions in Z Filter banks and DWT Multiresolution analysis k Z, x k = x k+l X (z) = z l X (z) H(z) H(z)+ G(z) G(z) = 2z l H(z) H( z)+ G(z) G( z) = 0 Non distortion Non aliasing 15/ Institut Mines-Telecom Wavelet-based image compression

20 Filter banks and DWT Multiresolution analysis Perfect reconstruction conditions Matrix form For simplicity, we ignore the delay, l = 0 If the analysis filter bank is given, the synthesis one is determined by: [ ] H(z) G(z) H( z) G( z) ] [ H(z) = G(z) [ ] 2 0 We assume that the modulation matrix is invertible. 16/ Institut Mines-Telecom Wavelet-based image compression

21 Filter banks and DWT Multiresolution analysis Perfect reconstruction conditions Synthesis filter bank Modulation matrix determinant : (z) = H(z) G( z) G(z) H( z) H(z) = 2 (z) G( z) G(z) = 2 (z) H( z) 17/ Institut Mines-Telecom Wavelet-based image compression

22 Filter banks and DWT Multiresolution analysis Perfect reconstruction with FIR filters Finite impulse response filters: It can be shown that in this case the PR condition is equivalent to the alterning signs condition. Example: h(k) = a b c h(k) = p -q r -s t g(k) = p q r s t g(k) = -a b -c 18/ Institut Mines-Telecom Wavelet-based image compression

23 Orthogonality Filter banks and DWT Multiresolution analysis Orthogonality assures energy conservation: k= (x k ) 2 = k= (c k ) 2 + k= (d k ) 2 reconstruction error = quantization error on DWT coefficients For non orthogonal filters, the reconstruction errors is a weighted sum of the quantization errors on the DWT subbands, with suitable weights ω i 19/ Institut Mines-Telecom Wavelet-based image compression

24 Vanishing moments Filter banks and DWT Multiresolution analysis Vanishing moments (VM) represent filter ability to reproduce polynomials: a filter with p VM can represent polynomials with degree < p The High-pass filter will not respond to a polynomial input with degree < p In this case all the signal information is preserved in the approximation signal (half the samples) A filter with p VM has at least 2p taps 20/ Institut Mines-Telecom Wavelet-based image compression

25 Borders problem Filter banks and DWT Multiresolution analysis Filterbank properties such as we saw, are valid for infinite-size signals We are interested in finite support signals How to interpret the previous results for finite support signals? 21/ Institut Mines-Telecom Wavelet-based image compression

26 Borders problem Filter banks and DWT Multiresolution analysis Filterbank properties such as we saw, are valid for infinite-size signals We are interested in finite support signals How to interpret the previous results for finite support signals? Zero padding would introduce a coefficient expansion Filtering an N-size signal with an M-size produces a signal with size N + M 1 21/ Institut Mines-Telecom Wavelet-based image compression

27 Borders problem Filter banks and DWT Multiresolution analysis Filterbank properties such as we saw, are valid for infinite-size signals We are interested in finite support signals How to interpret the previous results for finite support signals? Zero padding would introduce a coefficient expansion Filtering an N-size signal with an M-size produces a signal with size N + M 1 Periodization? Symmetrization? 21/ Institut Mines-Telecom Wavelet-based image compression

28 Introduction Filter banks and DWT Multiresolution analysis Borders problem: Coefficient expansion x y = h x h 22/ Institut Mines-Telecom Wavelet-based image compression

29 Filter banks and DWT Multiresolution analysis Borders problem: Periodization A signal x of support N is considered as a periodic signal x of period N Filtering x with h results into a periodic output ỹ ỹ has the same period N as x So we need to compute just N samples of ỹ However, periodization introduces jumps in a regular signal 23/ Institut Mines-Telecom Wavelet-based image compression

30 Filter banks and DWT Multiresolution analysis Borders problem: Periodization x ỹ / Institut Mines-Telecom Wavelet-based image compression

31 Filter banks and DWT Multiresolution analysis Borders problem: Symmetry Symmetrization before periodization reduces the impact on signal regularity But it doubles the number of coefficients... 25/ Institut Mines-Telecom Wavelet-based image compression

32 Filter banks and DWT Multiresolution analysis Borders problem: Symmetry Symmetrization before periodization reduces the impact on signal regularity But it doubles the number of coefficients... Unless the filters are symmetric, too We use x as half-period of xs x s has a period of 2N samples Filtering xs with h, produces ỹ s If h is symmetric, ỹs is periodic and symmetric, with period 2N: we only need to compute N samples 25/ Institut Mines-Telecom Wavelet-based image compression

33 Filter banks and DWT Multiresolution analysis Borders problem: Symmetry x s ỹ s / Institut Mines-Telecom Wavelet-based image compression

34 Haar filter Filter banks and DWT Multiresolution analysis Symmetric Orthogonal VM = 1 h(k) = 1 1 h(k) = 1 1 g(k) = 1-1 g(k) = -1 1 Only capable to represent piecewise constant signals 27/ Institut Mines-Telecom Wavelet-based image compression

35 Filter banks and DWT Multiresolution analysis Summary: perfect reconstruction and borders Convolution involves coefficient expansion Solution: circular convolution Circular convolution allows to reconstruct an N-samples signal with N wavelet coefficients The periodization generates borders discontinuities, i.e. spurious high frequencies coefficients that demand a lot of coding resources Solution: Symmetric periodization No discontinuities Does it double the coefficient number? No, if the filter is symmetric! Bad news: the only orthogonal symmetric FIR filter is Haar! 28/ Institut Mines-Telecom Wavelet-based image compression

36 Biorthogonal filters Filter banks and DWT Multiresolution analysis Cohen-Daubechies-Fauveau filters With biorthogonal (i.e. PR) filters, if h has p VM and h has p VM, the filter has at least p+ p 1 taps. The CDF filters have the following properties: They are symmetric (linear phase) They maximize the VM for a given filter length They are close to orthogonality (weights ω i are close to one) They are by far the most popular in image compression 29/ Institut Mines-Telecom Wavelet-based image compression

37 9/7 biorthogonal filters Filter banks and DWT Multiresolution analysis Filter coefficients: n 0 ±1 ±2 ±3 ±4 h[l] h[l] Impulse response of low-pass filters For high pass filters, we have (alterning sign condition): g[l] = ( 1) l+1 h[l 1] and g[l] = ( 1) l 1 h[l + 1]. 30/ Institut Mines-Telecom Wavelet-based image compression

38 Filter banks and DWT Multiresolution analysis 1D Multiresolution analysis Decomposition h 2 c3[k] h 2 c2[k] h 2 c1[k] g 2 d3[k] c0[k] g 2 d2[k] g 2 d1[k] Three level wavelet decomposition structure 31/ Institut Mines-Telecom Wavelet-based image compression

39 Reconstruction Filter banks and DWT Multiresolution analysis c3[k] 2 h c 2 [k] 2 h d3[k] 2 g c1[k] 2 h d2[k] 2 g c0[k] d1[k] 2 g Reconstruction from wavelet coefficients 32/ Institut Mines-Telecom Wavelet-based image compression

40 2D AMR Filter banks and DWT Multiresolution analysis 2D Filter banks for separable transform One decomposition level h[l] (1, 2) a j+1 [n, m] h[k] (2, 1) a j [n, m] g[l] (1, 2) dj+1 H [n, m] h[l] (1, 2) dj+1 V [n, m] g[k] (2, 1) g[l] (1, 2) dj+1 D [n, m] 33/ Institut Mines-Telecom Wavelet-based image compression

41 Filter banks and DWT Multiresolution analysis 2D-DWT subbands: orientations (A) (V) f hor (H) (D) f ver (A), (H), (V) and (D) respectively correspond to approximation coefficients, horizontal, vertical and diagonal detail coefficients. 34/ Institut Mines-Telecom Wavelet-based image compression

42 2D AMR: multiple levels Filter banks and DWT Multiresolution analysis a 0 a 1 d H 1 d V 1 d D 1 a 2 d H 2 d V 2 d D 2 a 3 d H 3 d V 3 d D 3 Three levels of separable 2D-AMR. 35/ Institut Mines-Telecom Wavelet-based image compression

43 Filter banks and DWT Multiresolution analysis 2D-DWT subbands: orientations (A) (V3) (H3) (D3) (V2) (V1) f hor (H2) (D2) (H1) (D1) f ver 36/ Institut Mines-Telecom Wavelet-based image compression

44 Example Filter banks and DWT Multiresolution analysis 37/ Institut Mines-Telecom Wavelet-based image compression

45 Example Filter banks and DWT Multiresolution analysis 38/ Institut Mines-Telecom Wavelet-based image compression

46 Example Filter banks and DWT Multiresolution analysis 39/ Institut Mines-Telecom Wavelet-based image compression

47 Example Filter banks and DWT Multiresolution analysis 40/ Institut Mines-Telecom Wavelet-based image compression

48 Example Filter banks and DWT Multiresolution analysis 41/ Institut Mines-Telecom Wavelet-based image compression

49 Example Filter banks and DWT Multiresolution analysis 42/ Institut Mines-Telecom Wavelet-based image compression

50 Example Filter banks and DWT Multiresolution analysis 43/ Institut Mines-Telecom Wavelet-based image compression

51 Outline Introduction Discrete wavelet transform and multiresolution analysis 44/ Institut Mines-Telecom Wavelet-based image compression

52 Compression with DWT Methods based on inter-scale dependencies: (Embedded Zerotrees of Wavelet coefficients), SPIHT (Set Partitioning in Hierarchical Trees) Tree-based representation of dependencies Advantages: good exploitation of inter-scale dependencies, low complexity Disadvantage: no resolution scalability Methods not based on inter-scale dependencies Explicit bit-rate allocation among subbands Entropy coding of coefficients Advantages: Good exploitation of intra-scale dependencies, random access, resolution scalability Disadvantage: no exploitation of inter-scale dependencies 45/ Institut Mines-Telecom Wavelet-based image compression

53 Embedded Zerotrees of Wavelet coefficients Main characteristics Quality scalability (i.e. progressive representation) Lossy-to-lossless coding Small complexity Rate-distortion performance much better than JPEG above all at small rates 46/ Institut Mines-Telecom Wavelet-based image compression

54 Progressive representation of DWT coefficients Each new coding bit must convey the maximum of information Each new coding bit must reduce as much as possible distortion We first send the largest coefficients Problem: localization overhead 47/ Institut Mines-Telecom Wavelet-based image compression

55 Example: an image and its wavelet coefficients LL Approximation HL Vertical details Approximation Vertical details DWT 2 further levels LH Horizontal details HH Diagonal details Horizontal details Diagonal details 48/ Institut Mines-Telecom Wavelet-based image compression

56 Progressive representation: subband order 49/ Institut Mines-Telecom Wavelet-based image compression

57 Algorithm The subband scan order alone is not enough to assure that largest coefficients are sent first We need to localize the largest coefficients Without having to send explicit localization information Idea: to exploit the inter-band correlation to predict the position of non-significant coefficients If the prediction is correct we save many coding bits (for all the predicted coefficients) 50/ Institut Mines-Telecom Wavelet-based image compression

58 Zero-tree of wavelet coefficients 51/ Institut Mines-Telecom Wavelet-based image compression

59 idea Auto-similarity : When a coefficient is small (below a threshold) it is probable that its descendants are small as well In this case we use a single coding symbol to represent the coefficient and all its descendants. If c and all its descendants are smaller than the threshold, c is called a zero-tree root With just one symbol, (ZT) we code ( N n ) coefficients The localization information is implicit in the significance information 52/ Institut Mines-Telecom Wavelet-based image compression

60 algorithm 1. k = 0 2. n = log 2 ( c max ) 3. T k = 2 n 4. while (rate < available rate) Dominant pass Refining pass Tk+1 T k /2 k k end while 53/ Institut Mines-Telecom Wavelet-based image compression

61 Dominant pass For each coefficient c (in the scan order) If c T n, the coefficient is significant If c > 0 we encode SP (Significant Positive) If c < 0 we encode SN (Significant Negative) If c < T n, we compare all its descendants with the threshold If no descendant is significant, c is coded as a zero-tree root (ZT) Otherwise the coefficient is coded as Isolated Zero (IZ) 54/ Institut Mines-Telecom Wavelet-based image compression

62 Refining pass We encode a further bit for all significant coefficients This is equivalent to halve the quantization step 55/ Institut Mines-Telecom Wavelet-based image compression

63 Iteration and termination The k-th dominant pass allows to encode the k-th bit-plane A significant coefficient c is such that 2 k < c < 2 k+1 For the next step we halve the threshold: it is equivalent to pass to the next bitplane Algorithm stops when the bit budget is exhausted; or when all the bitplanes have been coded 56/ Institut Mines-Telecom Wavelet-based image compression

64 Algorithm: summary Bitplane coding: at the k-th pass, we encode the bitplane log 2 T k Progressive coding: each new bitplane allows refining the coefficients quantization Lossless coding of significance symbols Lossless-to-lossy coding: When an integer transform is used, and all the bitplanes are coded, the original image can be restored with zero distortion 57/ Institut Mines-Telecom Wavelet-based image compression

65 Algorithm: Example T 0 = 2 log 2 26 = 16 58/ Institut Mines-Telecom Wavelet-based image compression

66 Algorithm: Example T 0 = 2 log 2 26 = 16 Bitstream: SP ZR ZR ZR 1 IZ ZR ZR ZR SP SP IZ IZ SP IZ SP SI SP SP SN IZ IZ SP IZ IZ IZ 58/ Institut Mines-Telecom Wavelet-based image compression

67 JPEG2000 JPEG2000 aims at challenges unresolved by previous standards: Low bit-rate coding: JPEG has low quality for R < 0.25 bpp Synthetic images compression Random access to image parts Quality and resolution scalability 59/ Institut Mines-Telecom Wavelet-based image compression

68 New functionalities Region-of-interest (ROI) coding Quality and resolution scalability Tiling Exact coding rate Lossy-to-lossless coding 60/ Institut Mines-Telecom Wavelet-based image compression

69 Algorithm JPEG2000 is made up of two tiers First tier DWT and quantization Lossless coding of codeblocks Second tier EBCOT: embedded block coding with optimized truncation Scalability (quality, resolution) and ROI management 61/ Institut Mines-Telecom Wavelet-based image compression

70 Quantization in JPEG2000 DWT coefficients are encoded with a very fine quantization step For the lossless coding case, DWT coefficients are integers, and they are not quantified In summary, it is not in the quantization step that the really lossy operations are performed The lossy coding is performed by the bitstream truncation of Tier 2 62/ Institut Mines-Telecom Wavelet-based image compression

71 EBCOT Embedded Block Coding with Optimized Truncation Each subband is split in equally sized blocks of coefficients, called codeblocks The codeblocks are losslessly and independently coded with an arithmetic coder We generate as much bitstreams as codeblocks in the image 63/ Institut Mines-Telecom Wavelet-based image compression

72 Bitplane coding Most significant bitplane LL2 HL2 HL1 LH2 HH2 LH1 HH1 64/ Institut Mines-Telecom Wavelet-based image compression

73 Bitplane coding Second bitplane LL2 HL2 HL1 LH2 HH2 LH1 HH1 65/ Institut Mines-Telecom Wavelet-based image compression

74 Bitplane coding Third bitplane LL2 HL2 HL1 LH2 HH2 LH1 HH1 66/ Institut Mines-Telecom Wavelet-based image compression

75 Bitplane coding Fourth bitplane LL2 HL2 HL1 LH2 HH2 LH1 HH1 67/ Institut Mines-Telecom Wavelet-based image compression

76 Bitplane coding Fifth bitplane LL2 HL2 HL1 LH2 HH2 LH1 HH1 68/ Institut Mines-Telecom Wavelet-based image compression

77 Example of bitstreams associated to codeblocks BP1 BP2 BP3 BP4 BP5 BP6 BP7 BP8 LL2 LH2 HL2 HH2 LH1 HL1 HH1 69/ Institut Mines-Telecom Wavelet-based image compression

78 Optimization Introduction EBCOT If we keep all the bitstreams of all the codeblocks, we end up with a huge bitrate We have to truncate the bitstream to attain the target bit-rate Problem: how to truncate the bitstreams with a minimum resulting distortion? min D i subject to R i R tot i Solution : Lagrange multiplier J = ( ) D i +λ R i R i i i 70/ Institut Mines-Telecom Wavelet-based image compression

79 EBCOT Rate-distortion curve per each codeblock 71/ Institut Mines-Telecom Wavelet-based image compression

80 EBCOT Rate-distortion curve per each codeblock 71/ Institut Mines-Telecom Wavelet-based image compression

81 EBCOT Embedded block coding with optimized truncation Optimal truncation point: D i R i = λ The value of the Lagrange multiplier can be find by an iterative algorithm. We can have several truncations for several target rates (quality scalability) 72/ Institut Mines-Telecom Wavelet-based image compression

82 Example of bit-rate allocation with EBCOT Allocation for maximal quality and minimal resolution BP1 BP2 BP3 BP4 BP5 BP6 BP7 BP8 LL2 LH2 HL2 HH2 LH1 HL1 HH1 73/ Institut Mines-Telecom Wavelet-based image compression

83 Example of bit-rate allocation with EBCOT Allocation for maximal quality and medium resolution BP1 BP2 BP3 BP4 BP5 BP6 BP7 BP8 LL2 LH2 HL2 HH2 LH1 HL1 HH1 74/ Institut Mines-Telecom Wavelet-based image compression

84 Example of bit-rate allocation with EBCOT Allocation for maximal quality and maximal resolution BP1 BP2 BP3 BP4 BP5 BP6 BP7 BP8 LL2 LH2 HL2 HH2 LH1 HL1 HH1 75/ Institut Mines-Telecom Wavelet-based image compression

85 Example of bit-rate allocation with EBCOT Allocation for perceptual quality and maximal resolution BP1 BP2 BP3 BP4 BP5 BP6 BP7 BP8 LL2 LH2 HL2 HH2 LH1 HL1 HH1 76/ Institut Mines-Telecom Wavelet-based image compression

86 Example of bit-rate allocation with EBCOT Allocation for a given bit-rate, maximal quality and resolution BP1 BP2 BP3 BP4 BP5 BP6 BP7 BP8 LL2 LH2 HL2 HH2 LH1 HL1 HH1 77/ Institut Mines-Telecom Wavelet-based image compression

87 Example of bit-rate allocation with EBCOT Allocation pour several layers and maximal resolution BP1 BP2 BP3 BP4 BP5 BP6 BP7 BP8 LL2 LH2 HL2 HH2 LH1 HL1 HH1 78/ Institut Mines-Telecom Wavelet-based image compression

88 JPEG Comparison JPEG / JPEG2000 Image Originale, 24 bpp 79/ Institut Mines-Telecom Wavelet-based image compression

89 Comparison JPEG / JPEG2000 Rate: 1bpp 80/ Institut Mines-Telecom Wavelet-based image compression

90 Comparison JPEG / JPEG2000 Rate: 0.75bpp 81/ Institut Mines-Telecom Wavelet-based image compression

91 Comparison JPEG / JPEG2000 Rate: 0.5bpp 82/ Institut Mines-Telecom Wavelet-based image compression

92 Comparison JPEG / JPEG2000 Rate: 0.3bpp 83/ Institut Mines-Telecom Wavelet-based image compression

93 Comparison JPEG / JPEG2000 Rate: 0.2bpp 84/ Institut Mines-Telecom Wavelet-based image compression

94 Comparison JPEG / JPEG2000 Rate: 0.2bpp pour JPEG, 0.1 pour JPEG / Institut Mines-Telecom Wavelet-based image compression

95 Error effect: JPEG JPEG, p E = 10 4 JPEG, p E = / Institut Mines-Telecom Wavelet-based image compression

96 Error effect: JPEG and JPEG, p E = 10 4, p E = / Institut Mines-Telecom Wavelet-based image compression

97 Error effect: JPEG and JPEG, p E = 10 3, p E = / Institut Mines-Telecom Wavelet-based image compression

98 Image coding and robustness Markers insertion Markers period Marker emulation prevention Trade-off between robustness and rate 89/ Institut Mines-Telecom Wavelet-based image compression

99 Error robustness in JPEG2000 Data priorization is possible No dependency among codeblocks No error propagation No block-based transform No blocking artifacts 90/ Institut Mines-Telecom Wavelet-based image compression

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