representation and compression of multi-dimensional piecewise functions
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Representation and Compression of Multi-Dimensional Piecewise Functions
Dror BaronSignal Processing and Systems (SP&S) SeminarJune 2009
Joint work with:Venkat ChandrasekaranMichael WakinRichard Baraniuk
The Challenge of Multi-D Horizon Functions• Signals have edges
– images (2D) – video (3D)– light field imaging (4D, 5D)
• Horizon class model– multidimensional– discontinuities– smooth areas
• Main challenge: sparse representation• Related applications: approximation, compression,
denoising, classification, segmentation…
N = 2 N = 3
Existing tool: 1D Wavelets
• Advantages for 1D signals:– efficient filter bank implementation– multiresolution framework– sparse representation for smooth signals
• Success motivates application to 2D, but…
2D Signal Representations
• Challenge: geometry - discontinuities along 1D contours – separable 2D wavelets (squares) fail to capture
geometric structure
• Response:– tight frames: curvelets [Candés & Donoho],
contourlets [Do & Vetterli], bandelets [Mallat]
– geometric tilings: wedgelets [Donoho], wedgeprints [Wakin et al.]
13 26 52
Wedgelet Dictionary [Donoho]
wedgelet decomposition
• Piecewise linear, multiscale representation
– supported over a square dyadic block
• Tree-structured approximation
• Intended for C2 discontinuities
Non-Separable Representations have Potential to be Sparse
Non-separable geometric tiling
13 26 52
Separable wavelets
Signal Representations in Higher Dimensions
• Failure of separable wavelets more pronounced in N>2 dimensions
• Similar problems exist– smooth regions separated by discontinuities– discontinuities often smooth functions in N-1
dimensions
• Shortcomings of existing work– not yet extended to higher dimensions– intended for efficient (sparse) representations for
C2 discontinuities
Goals
• Develop representation for higher-dimensional data containing discontinuities– smooth N-dimensional function– (N-1)-dimensional smooth discontinuity
• Optimal rate-distortion (RD) performance– metric entropy – order of RD function
• Flow of research:– From N=2 dimensions, C2-smooth
discontinuities– To N¸2 dimensions, arbitrary smoothness
Piecewise Constant Horizon Functions [Donoho]
• f: binary function in N dimensions
• b: CK smooth (N-1)-dimensional horizon/boundary discontinuity
• Let x 2 [0,1]N and y = {x1,…,xN-1} 2 [0,1](N-1)
Example Horizon Class Functions
N = 2 N = 3
Compression Problem
• Approximate f with R bits !
• Squared L2 error metric (energy)
• Need optimal tradeoff between rate and L2 distortion
Compression via Implicit Approximation
• Edge detection: – estimate horizon discontinuity b – encode using (N-1)-dimensional
wavelets [Cohen et al.]
• Implicitly approximate f from b
• Theorem [Kolmogorov & Tihomirov; Clements]: Metric entropy for CK smooth (N-1)-D function:
L1 distortionO(¢) lower bound
Metric Entropy for Horizon Functions• Problems with edge detection:
– edge detection often impractical– want to approximate f (not b) require solution that provides estimate in N-
dimensions, without explicit knowledge of b
• Theorem: Metric entropy for N-D horizon function f with CK smooth (N-1)-D discontinuity:
• Converse result – our algorithms achieve this RD performance
Motivation for Solution: Taylor’s Theorem
• For a CK function b in (N-1) dimensions,
• Key idea: order (K-1) polynomial approximation on small regions
• Challenge: organize tractable discrete dictionary for piecewise polynomial approximation
derivatives
Surflets: Piecewise Polynomial Approximations on Dyadic Hypercubes
• Surflet at scale j– N-dimensional atom
– defined on hypercube Xj of size 2-j£2-j££2-j
– horizon function with order K-1 polynomial discontinuity (“surface”-let)
• Tile to form multiscale approximation to f
K = 2 K = 3 K = 4
Wedgelet
3D Surflets
K = 2
K = 3
Discrete Surflet Dictionary
• Describe surflet using polynomial coefficients
K = 2 K = 3 K = 4K = 2 K = 3
Wedgelet
Quantization• Challenge: with naïve quantization of coefficients, dictionary size blows up with K and N• Surflet coefficients approximate Taylor coefficients
• Higher-order coefficients quantized with lesser precision same order error for all coefficients
• Response: for order-l coefficient, use step-size
~ O(2-(K-2)j)
~ O(2-2j)
~ O(2-Kj)
~ O(2-Kj)
Approximation without Edge Detection
• “Taylor surflets” – obtained by quantizing derivatives of b – requires knowledge/estimation of b
• “L2-best surflets”
– obtained by searching dictionary for best fit– requires no explicit knowledge of b– fast search algorithm via manifolds
• Theorem: Taylor or L2–best surflets have same asymptotic performance
Tree-structured Surflet Approximation
• Arrange surflets on 2N-tree– each node is either a leaf or has 2N children– all nodes labeled with surflets– leaf nodes provide approximation– interior nodes useful for predictive coding
Tree-structured Surflet Encoder
• Surflet leaf encoder achieves near-optimal RD performance
• Top-down predictive encoder– code all nodes in surflet tree– use parent surflets to predict children – constant # bits per surflet regardless of scale– layered coarse-scale approximation in early bits
• Theorem: Top-down predictive encoder achieves
Discretization
• Signals often acquired discretely (pixels/voxels) Pixelization artifacts at fine scales
• Approach to discrete data– discretize continuous surflet dictionary– coarse scales: use regular dictionary– smaller dictionary at fine scales
• Theorem: Predictive encoder achieves same RD performance at low rate with discretized dictionary
Numerical Example
• N=2,K=3 • 1024£1024 pixels• Scale-adaptive dictionaries
Wedgelets: 482 bits, 29.9 dB Surflets: 275 bits, 30.2 dB
RD Results
• Dictionary 1: fixed-scale wedgelets• Dictionary 2: wedgelets + scale-adaptive• Dictionary 3: surflets + scale-adaptive
Piecewise Smooth Horizon Functions
• g1,g2: real-valued smooth functions
– N dimensional– CKs smooth
• b: CKd smooth (N-1)-dimensional horizon/boundary discontinuity
• Theorem: Metric entropy for CKs smooth N-D horizon function f with CKd smooth discontinuity:
b(x1)
g1([x1, x2])
g2([x1, x2])
Surfprints
• Challenge: – wavelets good in smooth regions– wavelets wasteful near
discontinuity
• Surflets good near edges
• Response: surfprints project surflets to wavelet subspace
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Tree-structured Surprint Encoder
• Discontinuity information needed at finer scales• Top-down encoder
• Prediction not used
• Theorem: Top-down encoder achieves near-optimal
w
w w
w w w w
w w
surfprint
w
coarse
intermediate
maximal
– coarse: keep wavelet nodes– intermediate: nodes with
discontinuity– maximal depth: surfprints
Conclusions and Future Work
• Metric entropy (converse) – piecewise constant/smooth horizon functions– arbitrary dimension & arbitrary smoothness
• Multiresolution compression framework (achievable)– quantization scheme tractable dictionary size – predictive top-down coding optimal performance– scale-adaptive approach to discretization– surfprints at maximal depth near-optimal
• Future research: algorithms
THE END
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