Post on 01-Aug-2020
transcript
Signal Processing with Side InformationA Geometric Approach via Sparsity
João F. C. Mota
Heriot-Watt University, Edinburgh, UK
2/23
Side Information
Medical imaging
MRI PET
Consumer electronics Robotics
multi-modal
prior information
heterogeneous
Signal processing tasks
Denoising
Reconstruction
Demixing (source separation)
Compression
Inpainting, super-resolution, …
Recommender systems
How to represent multi-modal or heterogeneous data ?
How to process it ?
3/23
Outline
Compressed Sensing with Prior Information
Application: Video Background Subtraction
X-ray Image Separation
Conclusions
N DeligiannisVUB-Belgium
M RodriguesUCL
4/23
Compressed Sensing (CS)
How do we integrate in the problem?
Reconstruction guarantees?
Compressed SensingSucess rate (50 trials)
number of measurements
What if we know ? prior information
medical images, video, …
CS boundOur bound
CS performanceCS + PI
sparse
iid Gaussian
Basis pursuit
5/23
Intuition
measurements
Tangent cone of at
solutions of
Our approach
prior information (PI)
model for PI small
random orientation
6/23
Good components
Bad components
7/23
parameter-freeTheorem (L1-L1 minimization) [M, Deligiannis, Rodrigues, 2017]
i.i.d.
L1-L1 minimization
Theorem (BP) [Chandrasekaran, Recht, Parrilo, Willsky, 2012]
sparse
support overestimation
8/23
Experimental Results
Sucess rate (50 trials)
number of measurements
L1-L1 L1-L2
BP BP bound
Mod-CSMod-CS
[Vaswani and Lu, 2010]
Gaussian
9/23
Prior Information can help, but can also hinder
L1-L1 works better than L1-L2 (theory and practice)
(Computable) bounds are tight for L1-L1, but not for L1-L2
Theory predicts optimal ; indicates how to improve
Limitations: Gaussian matrices; bounds depend on unknown parameters
Summarizing
10/23
Outline
Compressed Sensing with Prior Information
Application: Video Background Subtraction
X-ray Image Separation
Conclusions
N DeligiannisVUB-Belgium
M RodriguesUCL
A SankaranarayananCMU-USA
V CevherEPFL-CH
11/23
observed
Compressive Background Subtraction
How to recover from online ?
How many measurements from frame ?
linear operation
CS camera
12/23
Our Approach
Estimate from past frames:
Integrate into BP
Assumption: background is static & known fg measurements
Basis Pursuit
sparse
foregroundCompressive sensing for background subtraction [Cevher, Sankaranarayanan, Duarte, et al, 2008]
Problems
Prior frames are ignored
fixed; depends on foreground area
background
via minimization
13/23
Problem Statement
sparse arbitrary function sparsetime
measurements
Model
Problem
Compute a minimal # of measurements
Reconstruct perfectlyonline algorithm w/ adaptive rate
14/23
Algorithm: computed at iteration
Gaussian
parameters of
L1-L1 minimization
oversampling factor
and repeat ...
# measurements of
Set Estimate
Acquire with
15/23
Estimating a Frame
estimation extrapolation
linear motion
overlap: take average; gaps: fill w/ average of neighbors
state-of-the-art in video coding
16/23
Experimental Results280 frames
Number of measurements
Frame
CS oracle prior state-of-the-art [Warnell et al, 2014]
L1-L1 oracle
Oursreduction of 67%
modified CS (nonadaptive)
17/23
Experimental Results280 frames
Relative error
Frame
reconstruction
estimation
determined by solver
modified CS (nonadaptive)
18/23
Outline
Compressed Sensing with Prior Information
Application: Video Background Subtraction
X-ray Image Separation
Conclusions
N DeligiannisVUB-Belgium
M RodriguesUCL
B CornelisVUB-Belgium
I DaubechiesDuke-USA
19/23
Motivation: X-Ray of Ghent Altarpiece
Mixed X-Ray
Can we use the visual images to separate the x-rays?
20/23
Approach: Coupled Dictionary LearningTraining step VisibleX-Ray
coupling
Demixing step
w/ sparse columns
learn dictionaries by alternating minimization
mixed x-ray visual front visual back
21/23
Results
mixed x-ray
visuals in grayscale
Ours
multiscale MCA w/KSVD
MCA [Bobin et al, 07’]
reconstructed x-rays
22/23
Summary / Conclusionsdata
Low-rank modelobservations
multi-modal features
dictionaries sparse columns
sparse sparse
prior information
measurements
Better models?
Guarantees?
Scalable algorithms?
X-ray separation
Reconstruction w/ PI
Applications
medical imaging (MRI + PET + ECG) SAR + microwave imaging
super-resolution (depth + visual) robotics (laser + sonar)
23/23
References
J. F. C. Mota, N. Deligiannis, M. R. D. Rodrigues Compressed Sensing with Prior Information: Optimal Strategies, Geometries, and BoundsIEEE Transactions on Information Theory, Vol 63, No 7, 2017
J. F. C. Mota, N. Deligiannis, A. C. Sankaranarayanan, V. Cevher, M. R. D. Rodrigues Adaptive-Rate Reconstruction of Time-Varying Signals with Application in Compressive Foreground ExtractionIEEE Transactions on Signal Processing, Vol 64, No 14, 2016
N. Deligiannis, J. F. C. Mota, B. Cornelis, M. R. D. Rodrigues, I. DaubechiesMulti-Modal Dictionary Learning For Image Separation With Application in Art InvestigationIEEE Transactions on Image Processing, Vol 26, No 2, 2017