Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
Page 1 of 14
Compressed Sensing… M.E. Davies
Sparse representations and compressed sensing
Mike DaviesInstitute for Digital Communications (IDCOM) &
Joint Research Institute for Signal and Image Processing University of Edinburgh
with thanks to
Thomas Blumensath, Gabriel Rilling
Page 2 of 14
Compressed Sensing… M.E. Davies
Sparsity: formal definition
A vector x is K-sparse, if only K of its elements are non-zero.
In the real world “exact” sparseness is uncommon, however, many signals are “approximately” K-sparse. That is, there is a K-sparse vector xK, such that the error k x − x Kk2 is small.
Page 3 of 14
Compressed Sensing… M.E. Davies
Why Sparsity?Why does sparsity make for a good transform?
50 100 150 200 250
50
100
150
200
250
“TOM” image Wavelet Domain
Good representations are efficient - Sparse!
Page 4 of 14
Compressed Sensing… M.E. Davies
Efficient transform domain representations imply that our signals of interest live in a very small set.
Signals of interest
Sparse signal modelL2 ball (not sparse)
Page 5 of 14
Compressed Sensing… M.E. Davies
Sparse signal models can be used to help solve various ill-posed linear inverse problems:
Sparsity and ill-posed inverse problems
Missing Data RecoveryImage De-blurring
Page 6 of 14
Compressed Sensing… M.E. Davies
Sparse Representations and Generalized Sampling:compressed sensing
Page 7 of 14
Compressed Sensing… M.E. Davies
Compressed sensingTraditionally when compressing a signal we take lots of samples move to a transform domain and then throw most of the coefficients away!
Why can’t we just sample signals at the “Information Rate”?
E. Candès, J. Romberg, and T. Tao, “Robust Uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,”IEEE Trans. Information Theory, 2006
D. Donoho, “Compressed sensing,” IEEE Trans. Information Theory, 2006
This is the philosophy of Compressed Sensing…
Page 8 of 14
Compressed Sensing… M.E. Davies
Signal space ~ RN
Set of signals of interest
Observation space ~ RM
M<<N
Linear projection (observation)
Nonlinear Approximation (reconstruction)
Compressed sensing
Compressed Sensing uses nonlinear reconstruction to invert the linear projection operator
Page 9 of 14
Compressed Sensing… M.E. Davies
Compressed sensingCompressed Sensing Challenges:
• Question 1: In which domain is the signal sparse (if at all)?Many natural signals are sparse in some time-frequency or space scale representation
• Question 2: How should we take good measurements?Current theory suggests that a random element in the sampling process is important
• Question 3: How many measurements do we need?Compressed sensing theory provides strong bounds on this as a function of the sparsity
• Question 4: How can we reconstruct the original signal from the measurements?
Compressed sensing concentrates of algorithmic solutions with provable performance and provably good complexity
Page 10 of 14
Compressed Sensing… M.E. Davies
Compressed sensing principle
sparse “Tom”
4
Invert transform
X =
2
Observed data
roughly equivalent
Wavelet image
Nonlinear Reconstruction
3
50 100 150 200 250
50
100
150
200
250
original “Tom”
Sparsifyingtransform
Wavelet image
50 100 150 200 250
50
100
150
200
250
1
Note that 1 is a redundant representation of 2
Page 11 of 14
Compressed Sensing… M.E. Davies
MRI acquisition
Logan-Shepp phantom We sample in this domain
Spatial Fourier Transform
Haar Wavelet Transform
Logan-Shepp phantom Sparse in this domain
Haar Wavelet Transform
Logan-Shepp phantom
Sub-sampled Fourier Transform
≈ 7 x down sampled (no longer invertible)
…but we wish to sample here
Compressed Sensing ideas can be applied to reduced sampling in Magnetic Resonance Imaging:
• MRI samples lines of spatial frequency
• Each line takes time & heats up the patient!
The Logan-Shepp phantom image illustrates this:
Page 12 of 14
Compressed Sensing… M.E. Davies
However what we really want to tackle problems like this…
originalLinear reconstruction (5x under-sampled)
Nonlinear reconstruction (5x under-sampled)
Rapid dynamic MRI acquisition in practice
(data courtesy of Ian Marshall & Terry Tao, SFC Brain Imaging Centre)
Page 13 of 14
Compressed Sensing… M.E. Davies
Synthetic Aperture RadarCompressed Sensing ideas can also be applied to reduced sampling in Synthetic Aperture Radar: Samples lines from spatial Fourier transform. Sub-sampling lines allows adaptive antennas to multi-task (interrupted SAR)
Page 14 of 14
Compressed Sensing… M.E. Davies
Potential applications…
Compressed Sensing provides a new way of thinking about signal acquisition. Potential applications areas include:
• Medical imaging
• Distributed sensing
• Seismic imaging
• Remote sensing (e.g. Synthetic Aperture Radar)
• Acoustic array processing (source separation)
• High rate analogue-to-digital conversion (DARPA A2I research program)
• The single pixel camera (novel Terahertz imaging)