DCT, Wavelets and X-lets: The Quest for Image Representation, Approximation and Compression Martin Vetterli, EPFL & UC Berkeley Joint work with B.Beferull-Lozano, A.Cunha, M.Do, P.L.Dragotti, R.Shukla, V.Velisavljevic and C.Weidmann PCS-07, Lisbon, Nov.7 2007 Audiovisual Communications Laboratory
Outline 1. Introduction: Fourier, Wavelets and Image Compression 2. Representation, Approximation and Compression or 1D Linear versus non-linear approximation Piecewise smooth functions: approximation and compression 3. Going to Two Dimensions: Non-Separable Constructions Why 2D is hard: Geometry 4. QuadTree Structures: Taking advantage of geometrical smoothness 5. Directionlets: Taking advantage of directionality 6. Contourlets: Taking advantage of directions and scale 7. Going to Seven Dimensions: The Plenoptic Function Taking advantage of the structure of the world A video model and the value of memory 8. Conclusions and Outlook PCS07-2
What are we trying to accomplish? 1. Review impact of wavelets on compression Approximation rates for piecewise smooth functions Rate-distortion behaviors 2. Point out limits of this theory and practice Only one-dimensional, hard to get tight upper/lower bounds 3. Explore the image compression problem 2D, piecewise smooth, contours, no wavelets What is possible, impossible: limits on approximation rates 4. Give constructions that use directionality and scale in 2D Reap some of the potential benefits Give practical algorithms and achievable bounds Three examples: Quadtrees, Directionlets and Contourlets 5. Look at new, higher dimensional datasets The Plenoptic Function The Plenoptic Function A video model from a cut of the the plenoptic function PCS07-3
Acknowledgements Sponsors: NSF Switzerland Collaborations: B.Beferull-Lozano, UV A.Cunha, UIUC M.Do, UIUC P.L.Dragotti, Imperial College R.Shukla, HPLabs V.Velisavljevic, DTelekomLabs C.Weidmann, TRC Vienna Discussions and Interactions: I. Daubechies, Princeton R.DeVore, Carolina D. Donoho, Stanford V.Goyal, MIT J. Kovacevic, CMU PCS07-4
Compression: How many bits for Mona Lisa? PCS07-5
A few numbers D. 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 Huffman code Mona Lisa index A few bits (Lena Y/N?, Mona Lisa ), what about R(D). Search the Web! http://www.google.com, 5-50 billion images online, or 33-36 bits JPEG 186K There is plenty of room at the bottom! JPEG2000 takes a few less, thanks to wavelets Note: 2 (256x256x8) possible images (D.Field) Homework in Cover-Thomas, Kolmogorov, MDL, Occam etc PCS07-6
Source Coding: some background Exchanging description complexity for distortion: rate-distortion theory [Shannon:58, Berger:71] known in few cases...like i.i.d. Gaussians (but tight: no better way!) or -6dB/bit typically: difficult, simple models, high complexity (e.g. VQ) high rate results, low rate often unknown PCS07-7
New image coding standard JPEG 2000 Old versus new JPEG: D(R) on log scale Main points: improvement by a few db s lot more functionalities (e.g. progressive download on internet) at high rate ~ -6db per bit: KLT behavior low rate behavior: much steeper: NL approximation effect? is this the limit? PCS07-8
New image coding standard JPEG 2000 From the comparison, JPEG fails above 40:1 compression JPEG2000 survives Note: images courtesy of www.dspworx.com PCS07-9
Signal Representations 1807: Joseph Fourier upsets the French Academy... Fourier Series: Harmonic series, freq. changes, f 0, 2f 0, 3f 0,... 1910: Alfred Haar discovers the Haar wavelet dual to the Fourier, scale changes S 0, 2S 0, 4S 0, 8S 0... PCS07-10
Representations, Bases and Frames Ingredients: as set of vectors, or atoms, an inner product, e.g. a series expansion Many possibilities: orthonormal bases (e.g. Fourier series, wavelet series) biorthogonal bases overcomplete systems or frames Note: no transforms, uncountable PCS07-11
Outline 1. Introduction: Fourier, Wavelets and Image Compression 2. Representation, Approximation and Compression or 1D Linear approximation Non-linear approximation Approximation power of a basis on a class of functions Piecewise smooth functions: approximation Compression is harder than approximation Some R(D) results for piecewise smooth functions 3. Going to Two Dimensions: Non-Separable Constructions 4. QuadTree Structures 5. Directionlets 6. Contourlets 7. Going to Seven Dimensions: The Plenoptic Function 8. Conclusions and Outlook PCS07-12
Local Fourier Basis? The Gabor or Short-time Fourier Transform Time-frequency atoms localized at When small enough Example: Spectogram PCS07-13
Another Good Basis! Replace (shift, modulation) by (shift, scale) or then there exist good localized orthonormal bases, or wavelet bases PCS07-14
Examples of bases Haar Daubechies, D 2 PCS07-15
Wavelets and representation of piecewise smooth functions Goal: efficient representation of signals like where: Wavelet act as singularity detectors Scaling functions catch smooth parts Noise is circularly symmetric Note: Fourier gets all Gibbs-ed up! PCS07-16
Thus: a piecewise smooth signal expands as: phase changes randomize signs, but not decay a singularity influence only L wavelets at each scale wavelet coefficients decay fast PCS07-17
Approximations, aproximation The linear approximation method Given an orthonormal basis for a space S and a signal the best linear approximation is given by the projection onto a fixed sub-space of size M (independent of f!) The error (MSE) is thus Ex: Truncated Fourier series project onto first M vectors corresponding to largest expected inner products, typically LP PCS07-18
The Karhunen-Loeve Transform: The Linear View (1/2) Best Linear Approximation in an MSE sense: Vector processes., i.i.d.: Consider linear approximation in a basis Then: Karhunen-Loeve transform (KLT): For 0<M<N, the expected squared error is minimized for the basis {g n } where g m are the eigenvectors of R x ordered in order of decreasing eigenvalues. Proof: eigenvector argument inductively. Note: Karhunen-47, Loeve-48, Hotelling-33, PCA, KramerM-56, TC PCS07-19
From linear to non-linear approximation theory The non-linear approximation method Given an orthonormal basis for a space S and a signal the best nonlinear approximation is given by the projection onto an adapted subspace of size M (dependent on f!) set of M largest The error (MSE) is thus and Difference: take the first M coeffs (linear) or take the largest M coeffs (non-linear) PCS07-20
From linear to non-linear approximation theory Nonlinear approximation This is a simple but nonlinear scheme Clearly, if is the NL approximation scheme: This could be called adaptive subspace fitting From a compression point of view, you pay for the adaptivity in general, this will cost bits These cannot be spent on coefficient representation anymore LA: pick a subspace a priori NLA: pick after seeing the data PCS07-21
Non-Linear Approximation Example Nonlinear approximation power depends on basis Example: Two different bases for Fourier series Wavelet series: Haar wavelets Linear approximation in Fourier or wavelet bases Nonlinear approximation in a Fourier basis Nonlinear approximation in a wavelet basis PCS07-22
Non-linear Approximation Example Fourier basis: Wavelet basis: N=1024, M=64 N=1024, M= 64, J=6 Nonlinear approximation is not necessarily much better! Depends on the basis, and the class of signals, but it can be big! PCS07-23
Smooth versus piecewise smooth functions: It depends on the basis and on the approximation method s=2, N=2^16, D_3, 6 levels PCS07-24
Wavelets and Compression Compression is just one bit trickier than approximation A small but instructive example: Assume, signal is of length N, k is U[0, N-1] and is This is a Gaussian RV at location k Note: R x = l! Linear approximation: Non-linear approximation, M > 0: PCS07-25
Given budget R for block of size N: 1. Linear approximation and KLT: equal distribution of R/N bits This is the optimal linear approximation and compression! 2. Rate-distortion analysis [Weidmann:99] High rate cases: Obvious scheme: pointer + quantizer This is the R(D) behavior for R >> Log N Much better than linear approximation Low rate case: Hamming case solved, inc. multiple spikes: - there is a linear decay at low rates L2 case: upper bounds that beat linear approx. PCS07-26
But is NLA always better than LA for compression? Counter example: R bits for a jointly gaussian vector source of size N i.i.d. N(0,1) Optimal scheme is LA performance is NLA incurs penalty need log 2 (N K) bits to index significant coeffs, which is O(N) when K is O(N/2) or worst case performance is but c 1, c 2 depends on coding, density, etc Thus, we could loose several db s by going to NLA but this is not the operating range where NLA is interesting! PCS07-27
Piecewise smooth functions: pieces are Lipschitz-α The following D(R) behavior is reachable [CohenDGO:02]: There are 2 modes: corresponding to the Lipschitz- pieces corresponding to the discontinuities: can be improved! PCS07-28
Rate-distortion behavior using an oracle An oracle decides to optimally code a piecewise polynomial by allocating bits where needed : Consider the simplest case Two approximations errors : quantization of step location : quantization of amplitude Rate allocation: versus Result: PCS07-29
Piecewise polynomial, with max degree N A. Nonlinear approximation with wavelets having zero moments B. Oracle-based method Thus wavelets are a generic but suboptimal scheme oracle method asymptotically superior but dependent on the model Conclusion on compression of piecewise smooth functions: D(R) behavior has two modes: 1/polynomial decay: cannot be (substantially) improved exponential mode: can be improved, important at low rates PCS07-30
Can we improve wavelet compression? Key: Remove depencies across scales: dynamic programming: Viterbi-like algorithm tree based algorithms: pruning and joining wavelet footprints: wavelet vector quantization Theorem [DragottiV:03]: Consider a piecewise smooth signal f(t), where pieces are Lipschitz-. There exists a piecewise polynomial p(t) with pieces of maximum degree such that the residual is uniformly Lipschitz-. This is a generic split into piecewise polynomial and smooth residual PCS07-31
Footprint Basis and Frames Suboptimality of wavelets for piecewise polynomials is due to independent coding of dependent wavelet coefficients Compression with wavelet footprints Theorem: [DragottiV:03] Given a bounded piecewise polynomial of deg D with K discontinuities. Then, a footprint based coder achieves This is a computational effective method to get oracle performance What is more, the generic split piecewise smooth into uniformly smooth + piecewise polynomial allows to fix wavelet scenarios, to obtain This can be used for denoising and superresolution as well PCS07-32
Outline 1. Introduction: Fourier, Wavelets and Image Compression 2. Representation, Approximation and Compression or 1D 3. Going to Two Dimensions: Non-Separable Constructions Why 2D is hard: Geometry Most constructions are separable The potential gain of truly non-separable constructions 4. QuadTree Structures 5. Directionlets 6. Contourlets 7. Going to Seven Dimensions: The Plenoptic Function 8. Conclusions and Outlook PCS07-33
Going to Two Dimensions: Non-Separable Constructions Going to two dimensions requires non-separable bases Objects in two dimensions we are interested in textures: per pixel smooth surfaces: per object! PCS07-34
Models of the world: Gauss-Markov Piecewise polynomial the usual suspect Many proposed models: mathematical difficulties one size fits all reality check Lena is not PC, but is she BV? PCS07-35
Current approaches to two dimensions. Mostly separable, direct or tensor products DWT Fourier and wavelets are both direct product constructions Wavelets: good for point singularities but what is needed are sparse coding of edge singularities! 1D: singularity 0-dimensional (e.g. spike, discontinuity) 2D: singularity 1-dimensional (e.g. smooth curve) PCS07-36
Recent work on geometric image processing Long history: compression, vision, filter banks Contour-texture, quadtrees [Kocher, Kunt, Leonardi] Pyramids [Burt-Adelson] Subband coding, nonseparable [Smith etc] Current signal adapted schemes Bandelets [LePennec & Mallat]: wavelet expansions centered at discontinuity as well as along smooth edges Non-linear tilings [Cohen, Mattei]: adaptive segmentation Tree structured approaches [Shukla et al, Baraniuk et al] Current bases and frames Wedgelets [Donoho]: Basic element is a wedge Ridgelets [Candes, Donoho]: Basic element is a ridge Curvelets [Candes, Donoho] Scaling law: width ~length 2, L(R 2 ) set up Multidirectional pyramids and contourlets [Do et al] Discrete-space set-up, l(z 2 ), tight frame with small redundancy Computational framework PCS07-37
Nonseparable schemes and approximation Approximation properties: wavelets good for point singularities ridgelets good for ridges curvelets good for curves Consider c 2 boundary between two csts Rate of approximation, M-term NLA in bases, c 2 boundary Fourier: O(M -1/2 ) Wavelets: O(M -1 ) Curvelets: O(M -2 ) Note: adaptive schemes, Bandelets: O(M -α ) PCS07-38
Compression of non-separable objects Object we know how to compress. Basis element Approximation Wavelets Ridgelets Rate/distortion Oracle Wavelets.poor Ridgelets.suboptimal adaptive schemes: close to oracle fixed basis: under investigation PCS07-39
Outline 1. Introduction: Fourier, Wavelets and Image Compression 2. Representation, Approximation and Compression or 1D 3. Going to Two Dimensions: Non-Separable Constructions 4. QuadTree Structures: Quadtree coding has a long history Geometrical smoothness can be taken advantage of Efficient coding on quadtrees for smooth 2D functions with smooth boundaries Algorithms and R(D) behavior 5. Directionlets 6. Contourlets 7. Going to Seven Dimensions: The Plenoptic Function 8. Conclusions and Outlook PCS07-40
Tree Based Geometric Compression [ShuklaDDV[ ShuklaDDV:03] Idea tree and quadtree algorithms popular, many pruning algorithms optimality proofs for wedgelets [Donoho:99] new pruning and joining algorithm Intuition: full tree dyadic tree pruned & joined tree Results: Rate-distortion optimal for piecewise polynomials that is, like an oracle method (up to constants) PCS07-41
Extension to Quadtree: Example Results: consider a piecewise polynomial 2D signal, with polynomial boundaries, the following rate-distortion behavior is achieved this is like an oracle method, and much better than prune algorithms which have a penalty complexity: polynomial PCS07-42
The prune-join quadtree algorithm polynomial fit to surface and to boundary on a quadtree rate-distortion optimal tree pruning and joining quadtree with R(D) pruning R(D) Joining of similar leaves Note: careful R(D) optimization! PCS07-43
Geometric Compression versus JPEG2000 at 0.11 bits/pixel, PSNR: 28.95 27.75 30.01 29.22 Pruned-joined quadtree JPEG2000 PCS07-44
Upperbounds on R-D Behaviors A. Boundary is piecewise polynomial: Oracle: The prune quadtree algorithm (independent coding): The prune-join quadtree algorithm with joint coding: B. Boundary is piecewise smooth: The prune quadtree algorithm (code blocks independently) achieves the oracle performance (up to log factor): C. Computational complexity of quadtree algorithms: PCS07-45
Behavior of tree algorithms on piecewise smooth fcts ppf: piecewise polynomial functions psf: piecewise smooth functions Signal Class Oracle Coder Wavelet Coder Prune tree Coder Prune-join tree Coder 1-D PPF 2-D PPF 1-D PSF 2-D PSF At most log penalty with polynomial complexity (and a bit more work gets rid of logs ) PCS07-46
Outline 1. Introduction: Fourier, Wavelets and Image Compression 2. Representation, Approximation and Compression or 1D 3. Going to Two Dimensions: Non-Separable Constructions 4. QuadTree Structures 5. Directionlets: Wavelet transform and fully separable wavelet transform Skewed fully separable wavelet transform Approximation power Compression results 6. Contourlets 7. Going to Seven Dimensions: The Plenoptic Function 8. Conclusions and Outlook PCS07-47
Directionlets [V.Velisavlevic Velisavlevic,, BBL, MV 06-07] PCS07-48
Standard versus fully separable wavelet transform Difference between usual and fully separable wavelet transform Note: fully separable wavelet transform was proposed in 1989 by Westerink, reappeared in 1999 by several authors PCS07-49
Directionlets: More directions, anisotropy Lattice-based filtering and subsampling PCS07-50
Directionlets: More directions and anisotropy Skewed anisotropic wavelet transforms Directional vanishing moments leads to approximation power PCS07-51
Directionlets: Approximation power, compression Main result This not optimal (1/N 2 would be), but computationally easy PCS07-52
Directionlets: Approximation results PCS07-53
Directionlets: Compression results PCS07-54
Outline 1. Introduction: Fourier, Wavelets and Image Compression 2. Representation, Approximation and Compression or 1D 3. Going to Two Dimensions: Non-Separable Constructions 4. QuadTree Structures 5. Directionlets 6. Contourlets: Taking advantage of directions and scale Directional filter banks Pyramid Approximation power Compression and denoising 7. Going to Seven Dimensions: The Plenoptic Function 8. Conclusions and Outlook PCS07-55
Multiresolution directional filterbanks and contourlets [M.Do] Idea: find a direct discrete-space construction that has good approximation properties for smooth functions with smooth boundaries directional analysis as in a local Radon transform multiresolution as in wavelets and pyramids computationally easy bases or low redundancy frame Background: curvelets [Candes-Donoho] indicate that good fixed bases do exist for approximation of piecewise smooth 2D functions a frequency-direction relationship indicates a scaling law an effective compression algorithm requires - close to a basis (e.g. tight frame with low redundancy) - discrete-space set up and computationally efficient Question: can we go from l(z 2 ) to L(R 2 ), just like filter banks lead to wavelets? Answer: contourlets! PCS07-56
Directional Filter Banks [BambergerS[ BambergerS:92, DoV:02] divide 2-D spectrum into slices with iterated tree-structured f-banks fan filters quincunx sampling shearing PCS07-57
Iterated directional filter banks: efficient directional analysis Example: PCS07-58
Example of basis functions 6 levels of iteration, or 64 channels elementary filters are Haar filters orthonormal directional basis 64 equivalent filters, below the 32 mostly horizontal ones are shown This ressembles a local Radon transform, or radonlets! changes of sign (for orthonormality) approximate lines (discretizations) PCS07-59
Adding multiresolution: : use a pyramid! Result: tight pyramid and orthogonal directional channels => tight frame low redundancy < 4/3, computationally efficient PCS07-60
Frequency split flexible tilings: d ~ j 1/2 or width ~ length 2 usual dyadic tiling It just becomes a MD filter design problem Equivalent basis functions: Contourlets PCS07-61
A directional multiresolution analysis Theorem [Do:01] For a finite number of directions, this generates a tight frame for L 2 (R 2 ) with frame bound equal 1 Method: Define embedded lowpass directional spaces V j,k and directional bandpass spaces W j,k This defines contourlets: how do they compare to wavelets? PCS07-62
Basis functions: Wavelets versus contourlets Wavelets Contourlets PCS07-63
Pyramidal directional filter bank expansion: Example Pepper image and its expansion Compression, denoising, inverse problems: if it is sparse, it is going to work! PCS07-64
Approximation properties Wavelets Contourlets PCS07-65
An approximation theorem Curvelets lead to optimal approximation, what about contourlets? Result [M.Do:03] Simple B/W image model with c 2 boundary Contourlet with scaling w ~ l 2 and 1 directional vanishing moment Then the M-term NLA satisfies Proof (very sketchy...): - Amplitude of contourlets ~ 2-3j/4 and coeffs ~2-3j/4 l 3 jkn - Three types of coefficients (significant which match direction insignificant that overlap, and zero) - levels 3J and J, respectively, leading to M ~ 2 3J/2 - squared error can be shown to be ~ 2-3J PCS07-66
Example: denoising with contourlets original noisy wavelet 13.8 db countourlets 15.4 db PCS07-67
Outline 1. Introduction: Fourier, Wavelets and Image Compression 2. Representation, Approximation and Compression or 1D 3. Going to Two Dimensions: Non-Separable Constructions 4. QuadTree Structures 5. Directionlets 6. Contourlets 7. Going to Seven Dimensions: The Plenoptic Function The plenoptic function and the structure of the world Plenoptic sampling, bandlimitdness Taking advantage of the structure of the world A video model based on the plenoptic function The value of memory 8. Conclusions and Outlook PCS07-68
The Plenoptic Function [Adelson[ Adelson] Multiple camera systems physical world (e.g. landscape, room) distributed signal acquisition possible images: plenoptic function, 7-dim! Background: pinhole camera epipolar geometry multidimensional sampling Implications on communications camera sources are correlated in a particular way limits on number on independent cameras different BW requirements at different locations PCS07-69
Examples 3D 3D 2D 4D 5D [Stanford multi-camera array] [Imperial College multi-camera array] PCS07-70
On Plenoptic Sampling [Shum et al] Model Epipolar geometry points become lines slope depends on depth of field Plenoptic function a collection of lines (modulo covering/uncovering) slopes bounded by (min, max) depth Fourier transform...a pie slice... can be sampled PCS07-71
Plenoptic function Structured multidimensional images compression Denoising Superresolution One can use all the tricks we learned for non-separable 2D processing! PCS07-72
The POF as a generator of video Many practical scenarios can be represented in this setting Two sources of complexity: the camera trajectory the reality around it Suggests a model of POF(W(t),t). PCS07-73
POF and Rate-Distortion Problem [CunhaDV06] Question: what are the information rates of the source? Two coding problems: reconstruct the reality, reconstruct sequence of sample realities (e.g. video) PCS07-74
A Simplified Model: Bernoulli Camera + iid Painted Wall An ensemble of trajectories modeled with a random walk. An ensemble of realities that are modeled with a strip/wall of stationary pixels drawn from an alphabet. The pixels are assumed to be independent of trajectory. Simplest example: X n iid, and N i ~ Bern(p W ). PCS07-75
Video: The vector process Model leads to a vector process V over time indexed by t: V t := (X Wt ; X Wt + 1 ; ; X Wt + L 1 ) Can show process is stationary and mean-ergodic if X is stationary and mean-ergodic. We are thus ready to ask information-theoretic questions about the process V. PCS07-76
Video Coding versus Light Field Coding Two coding problems with similar input data: 1. The video coding case: code and transmit the vector process V (Eg: as in video). 2. The recording reality case: recover the reality process X (Eg: as in light-field coding). In the video coding case ordering is important. High signal correlation over time. The light-field case ordering is not as important. We will derive information rates for both cases. We look first at the lossless case. Some of it can be extended to the lossy case. PCS07-77
The Video Coding Problem (lossless case) Lossless Case: we seek to compute the entropy rate in bits/block: If the trajectory W is available, then we only need pixels from new sites to reproduce the process. For a Bernoulli random walk, new sites occur with average of new sites per time unit. This gives a sufficient rate of bits/block. Because the new sites need to be coded in any case, a necessary rate is therefore This is lead to a theorem bounding the bit-rate, which is tight for large blocks and large alphabet sizes PCS07-78
Memory and Recurrence Amount of bits saved with memory depends on rate of recurrence When p W =0.5, we have a recurrent trajectory --- results in lower bitrate By contrast, p W =0 means camera panning --- scene changes increase bitrate Eg: Hollywood sitcom highly recurrent (same background over and over again). PCS07-79
Experiments Video coding case: we use a sequence of frames extracted from a large panorama. A simple coder employing disparity compensation, DPCM, and JPEG coding of the residual. The scene is recurrent. For recurrent scenes, finite memory coder underperforms considerably. Reconstructing reality case: simple code for trajectories Coding trajectory information: with large memory, average length converges to zero. Very simple VLC can optimally code trajectory information. PCS07-80
The Video Coding Problem: Dynamic Reality PCS07-81
Dynamic Binary Wall: Effect of Memory Memory and innovations. Shown is the difference between the conditional entropy and the true entropy for the binary innovations with p X =0.5, p W =0.5, and L=8. The curves show the intuitive fact that when the background changes too rapidly, there is little to be gained in bitrate by utilizing more memory. PCS07-82
Conclusions: What have we accomplished so far? Wavelets and their secrets Good for 1D piecewise smooth objects Strong results, works in practice, but only in 1D! Sparsity and non-linear approximation LA versus NLA: approximation rates can be vastly different! Good to have in the tool box Sparse sampling & Compressed sensing Compression: Operational, high rate, D(R) improvements still possible low rate analysis difficult Two-dimensions: really harder! and none used in JPEG2000... approximation starts to be understood, compression mostly open Many interesting constructions, we saw three! More work needed to make it real PCS07-83
Outlook New images plenoptic functions (set of all possible images) non bandlimited images (FRI?) manifolds, structure of natural images Distributed images interactive approximation/compression SW, WZ, DKLT... PCS07-84
Why Image Representation Remains a Fascinating Topic PCS07-85
Publications For overviews: D.Donoho, M.Vetterli, R.DeVore and I.Daubechies, Data Compression and Harmonic Analysis, IEEE Tr. on IT, Oct.1998. M. Vetterli, Wavelets, approximation and compression, IEEE Signal Processing Magazine, Sept. 2001 M.Vetterli, J.Kovacevic and V.Goyal, Fourier and Wavelets: Theory, Algorithms and Applications, 200X ;) For more details, Theses C.Weidmann, Oligoquantization in low-rate lossy source coding, PhD Thesis, EPFL, 2000. M. N. Do, Directional Multiresolution Image Representations, Ph.D. Thesis, EPFL, 2001. P. L. Dragotti, Wavelet Footprints and Frames for Signal Processing and Communications, PhD Thesis, EPFL, 2002. R.Shukla, Rate-distortion optimized geometrical image processing, PhD Thesis, EPFL, 2004. V.Velisavljevic, Directionlets: Anisotropic multidirectional representation with separable filtering, PhD Thesis, EPFL, 2005 A.Cunha, PhD Thesis, UIUC, 2006. PCS07-86
Papers: A.Cohen, I.Daubechies, O.Gulieruz and M.Orchard, On the importance of combining wavelet-based non-linear approximation with coding strategies, IEEE Tr. on IT, 2002 P. L. Dragotti, M. Vetterli. Wavelets footprints: theory, algorithms and applications, IEEE Transactions on Signal Processing, May 2003. R. Shukla, P. L. Dragotti, M. N. Do and M. Vetterli, Rate-distortion optimized tree structured compression algorithms for piecewise smooth images, IEEE Transactions Image Processing, 2004. M. N. Do and M. Vetterli, Framing pyramids. IEEE Transactions on Signal Processing, Sept. 2003. M.N.Do and M.Vetterli, Contourlets: A computational framework for directional multiresolution image representation, submitted, 2003. V. Velisavljevic, B. Beferull-Lozano, M. Vetterli and P.L. Dragotti, Directionlets: Anisotropic Multidirectional representation with separable filtering, IEEE Transactions on Image Processing, 2006. C.Weidmann, M.Vetterli, Rate-distortion behavior of sparse sources, IEEE Tr. on IT, under revision. A.Cunha, M.Do, M.Vetterli, On the information rate of the plenoptic function, to be submitted. PCS07-87