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  • Page 1 of 14

    Compressed Sensing… M.E. Davies

    Sparse representations and compressed sensing

    Mike Davies Institute 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?

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    “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 sensing Traditionally 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

  • Page 9 of 14

    Compressed Sensing… M.E. Davies

    Compressed sensing Compressed 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”

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    Invert transform

    X =

    2

    Observed data

    roughly equivalent

    Wavelet image

    Nonlinear Reconstruction

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    original “Tom”

    Sparsifying transform

    Wavelet image

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    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…

    original Linear 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 Radar Compressed 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)

    Sparse representations and compressed sensing Sparsity: formal definition Why Sparsity? Sparse Representations and Generalized Sampling: compressed sensing Compressed sensing Compressed sensing Compressed sensing Potential applications…