SLIDE 1 A Sample Distortion Analysis for Compressed Imaging
Chunli Guo, Mike E. Davies
Institute for Digital Communications University of Edinburgh, UK
1
SLIDE 2 Talk Outline
- Introduction
- Compressed Sensing: from Sparse to Compressible, from
Deterministic to Stochastic
- Sample Distortion (SD) framework
- definition, examples, lower bounds and convexity
- Multi-resolution CS
- Wavelet Statistical Image Model
- SD and Optimal Bandwise Sampling
- Oracle Bounds
- Sample Allocation with tree structure
- Natural Image Examples
- Conclusions
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SLIDE 3 Sparsity and Compressed Sensing
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SLIDE 4 Compressed sensing
Compressed Sensing assumes a sparse/compressible set of signals Uses random projections for
Signal reconstruction by a nonlinear mapping. Compressed sensing provides practical algorithms with guaranteed performance e.g. L1 min., OMP, CoSaMP, IHT. Closely linked with theory of n- widths
Set of signals of interest random projection (observation) nonlinear approximation (reconstruction)
SLIDE 5
The L1 solution guarantee
SLIDE 6
Compressible vectors (deterministic)
SLIDE 7
Compressible Distributions
SLIDE 8
A Sample-Distortion framework for CS
SLIDE 9
Sample Distortion Framework
[Guo & D. 2011/2012]
SLIDE 10 Sample Distortion Framework
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SLIDE 11 SD Functions for 2-state GSM
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( ) 0.38 (0,1.198) 0.62 (0,0.0044) p x N N
BAMP SD fun Convexity implies achievable (by zeroing) MBB EBB BAMP SD fun
( ) 0.38 (0,1.198) 0.62 (0,0.0044) p x N N
SLIDE 12 SD Lower Bounds
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SLIDE 13 Example: Generalized Gaussian
Laplace EBB Gaussian EBB GGD, α=0.4 EBB
Wavelet coefficients of natural images are often modelled as GGD with α ≈ 0.4-1.0
SLIDE 14 SD Lower Bounds
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SLIDE 15 Convexity of D(δ)
Theorem: The SD function, D(δ), is convex
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1 1
,
2 2
,
Convex hull achievable by combinations of and
1 1
,
2 2
,
1
D
2
D
3
D
1
3
2
SLIDE 16 Gaussian encoders are not optimal!
Folk theorem - Gaussian encoders are optimal. False! If Gaussian-specific SD function is not convex we can do better
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SLIDE 17 SD Functions for 2-state GSM
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BAMP SD fun Convexity implies achievable (by zeroing) MBB EBB
( ) 0.38 (0,1.198) 0.62 (0,0.0044) p x N N
SLIDE 18
Multi-resolution Compressive Imaging
SLIDE 19 Statistical image model
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SLIDE 20
Test Images
SLIDE 21
Image model example
cameraman
SLIDE 22
Image model example
GGD representation with Haar wavelets... Estimated shape parameters for each level cameraman
SLIDE 23
Image model example
e.g. cameraman GGD wavelet representation Estimation of the variance for each level
SLIDE 24
Bandwise Compressive Imaging
SLIDE 25
Bandwise sampling
SLIDE 26
Optimal Bandwise sample allocation
SLIDE 27 Bandwise Sampling
Convexified MMSE AMP distortion reduction function (band 1 for cameraman image model)
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distortion distortion reduction
SLIDE 28 Bandwise Sample Allocation
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DR fun for cameraman image
SLIDE 29
Bandwise CS sample allocation
Sample allocation (% of full sampling) per band for m = 170, 600, 2000 and 10000 measurements. There are typically no more than 2-3 partially sampled bands
SLIDE 30 Bandwise CS Performance
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e.g. cameraman
SLIDE 31
adding Tree Structure
SLIDE 32 Incorporating Tree Structure
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We can add tree-based priors on coefficients and decode using Turbo AMP scheme [Som, Schniter 2012]: This calculates marginal probabilities for hidden states and incorporate into MMSE AMP
SLIDE 33 Bandwise CS Sample Allocation
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Sample allocation (% of full sampling) per band for δ= 10%, 15.26%, 25% and 30%
m=6554
m=1000 m=16384 m=19661
SA for cvx SD fun Empirical best SA with tree info SA for SD fun with
SLIDE 34 Bandwise CS Performance
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e.g. cameraman
SLIDE 35 Reconstructed I
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Image reconstructions from 10000 measurements (15%)
SLIDE 36 SA for General Image Statistics
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SLIDE 37 Open questions
- How to derive sample allocations for more
sophisticated models? – analysis representations, tree structured model, etc.
- How to allocate samples within constrained
sampling schemes (e.g. partial Fourier)?
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SLIDE 38 References
- R. Gribonval, V. Cevher, and M. E. Davies. Compressible Distributions for
High-dimensional Statistics. Preprint, 2011, available at arXiv:1102.1249v2.
- M. E. Davies and C. Guo, Sample-Distortion Functions for Compressed
Sensing, 49th Allerton Conf. on Communication, Control, and Computing, 2011.
- C. Guo and M. E. Davies , Sample-Distortion for Compressed Imaging,
- n IEEE TSP, 2013.
