Wedgelets and Image Compression

Transcription

1 Wedgelets and Image Compression Laurent Demaret, Mattia Fedrigo, Hartmut Führ Summer school: New Trends and Directions in Harmonic Analysis, Approximation Theory, and Image Analysis, Inzell, Germany, 20 September 2007

2 Contents 1. Image Compression: Basis 2. Wedgelet Segmentations 3. Data structure and compression 4. first results

3 Classical Compression Methods Original Image JPEG (6.8 KB) JPEG2000 (6.5 KB) DCT FWT + Contexts

4 Zoom Original Image JPEG (6.8 KB) JPEG2000 (6.5 KB) DCT FWT + Contexts

5 Mathematical Framework Justification Approximation Theory Image: real-valued function, continuous domain Ansatz natural images have some regularity f X (quasi?)-banach space, X L 2 (Ω) Approximation f n = i α iϕ i,ϕ A, n-approximation We look for A, such that ( ) f f 1 n 2 2 = O, for some α > 0, and f X n α Critics Asymptotic results Continuous vs Discrete

6 Orthogonal Transforms Old and new Ansätze FOURIER: non optimal (bad for local singularities) WAVELETS : optimal Non Linear Approximation rates for Besov spaces and Bounded Variation + in 2D Isotropic vs Anisotropic Methods = Structure of the set of singularities Geometrical Methods TRIANGULATIONS: good theoretical Approximation rates (Mallat2004, Demaret-Iske 2006) CURVELETS: not adaptive but quasi-optimal (contains flexible geometrical features) (Candès2004) BANDELETS (Mallat-LePennec1999) WEDGELETS (Donoho1999), (Lissowska)

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8 Wavelets and Contours Wedge (left) and its Wavelet coefficients (right)

9 Geometrical Segmentations S Z 2 set of pixels f R S image P family of partitions P 2 S of S f P R S best constant approximation with f P r constant, r P S segmentations (P, f P ) γ 0 penalisation parameter Goal: Efficient Minimisation of the penalised Functional Result H f,γ : S R, (P, f P ) γ P + f f P 2 2 (γ 0). ( ˆP, ˆf ˆP) argmin (P,f P ) H z,γ optimal tradeoff between penalisation and reconstruction quality

10 Wedgelet Segmentations H f,γ : S R, (P, f P ) γ P + f f P 2 2 (γ 0). Problem Size of the search space :( P > 2 S!) MCMC: slow and not exact Restriction of the search space discrete wedges nested Quadtree structure fast moment computation: Green-like formula

11 Representation Elements DCT basis (JPEG) (Haar) Wavelet basis Wedgelet partitions

12 Data Structure Quadtree Partition dyadic Wedge Partition

13 Example f (W, f W ) f W

14 Compression: Algorithm Idee Wedgelet representation contains too much redundancies = Correlation Model between neighbours ALGORITHM Tree Coding Model Coding IF (Model = constant over square) (quantised) mean value encoded IF (Modell = constant over each Wedge) Angle Encoding and relative position Coding of the (quantised) mean values

15 Compression: Features mixed Models (e.g. square constant, wedge constant, wedge linear...) corresponding penalisation : estimation of the coding costs H γ : (f, (P, f P )) γ( i C(W i ) + j C(Q j ) )+ f f P 2 2, γ 0, W i : wedge, Q j : square, C estimator for the coding costs Coding combinatorial encoding angle coding : resolution-adaptive Prediction Method

16 Prediction Observation Representation still strongly redundant "not natural", arbitrary quadtree structure Main idea Multiresolution differential coding only "Brotherhood" correlations Extraction of spatial correlation between quadtree "cousins" Predictability of the current piece from the causal (i.e. already coded) information

17 How to Code the Leaves? Tree Levels of the leaves

18 Predictive Coding: an Illustration Binary Tree : 45 bits Context = 0 > Bottom to Top Non-Predictive: log 2 (64) + log 2 ( ( 64) 8 ) = 39 bits Bottom to Top Predictive: log 2 (64) + log 2 ( ( 42) 1 ) + log2 ( ( 22) 7 ) = 31 bits

19 First results (1) Comparison between "pure Wedge" and "Wedge+Constant" Models with higher penalisation for Wedges versus Squares C(W i ) = 3.5 C(Q i ) (a) Original Image (b) Squares: b, 30,54 db(c) only Wedges: b, 30,42 db (d) Wedges + Squares: b, 30,60 db Model Tree Models Const. Angles Line Wedge Total values number values pure squares bits symb pure wedges bits symb wedges + squares bits symb

20 First Results (2) Circles, WC, 533 B, PSNR: db Peppers, WC, B, PSNR: db

21 Work in Progress systematic investigation of the penalisation functional rate-distortion Optimisation Depends on the resolution Contexts change penalty Contextual Encoding Compression with richer regression models (e.g. linear) aim: avoid bloc artefacts Correct theoretical framework for discrete Data non asymptotical results