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Toward 0-norm Reconstruction, and A Nullspace Technique for Compressive Sampling Christine Law Gary Glover Dept. of EE, Dept. of Radiology Stanford University

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Outline 0-norm Magnetic Resonance Imaging (MRI) reconstruction. 0-norm Magnetic Resonance Imaging (MRI) reconstruction. Homotopic Homotopic Convex iteration Convex iteration Signal separation example Signal separation example 1-norm deconvolution in fMRI. 1-norm deconvolution in fMRI. Improvement in cardinality constraint problem with nullspace technique. Improvement in cardinality constraint problem with nullspace technique.

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Shannon vs. Sparse Sampling Nyquist/Shannon : sampling rate ≥ 2*max freq Nyquist/Shannon : sampling rate ≥ 2*max freq Sparse Sampling Theorem: (Candes/Donoho 2004) Sparse Sampling Theorem: (Candes/Donoho 2004) Suppose x in R n is k -sparse and we are given m Fourier coefficients with frequencies selected uniformly at random. If Suppose x in R n is k -sparse and we are given m Fourier coefficients with frequencies selected uniformly at random. If m ≥ k log 2 (1 + n / k) m ≥ k log 2 (1 + n / k) then then reconstructs x exactly with overwhelming probability. n k = 2

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Rat dies 1 week after drinking poisoned wine Example by Anna Gilbert

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x1 x2 x3 x4 x5 x6 x7 y1 y2 y3

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x1 x2 x3 x4 x5 x6 x7 y1 y2 y3 n=7 m=3 k=1

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Reconstruction by Optimization Compressed Sensing theory (2004 Donoho, Candes): under certain conditions, Candes et al. IEEE Trans. Information Theory 2006 52(2):489 Donoho. IEEE Trans. Information Theory 2006 52(4):1289 y are measurements (rats) x are sensors (wine)

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0-norm reconstruction Try to solve 0-norm directly. Try to solve 0-norm directly. For p-norm, where 0 < p < 1 For p-norm, where 0 < p < 1 Chartrand (2006) demonstrated fewer samples y required than 1-norm formulation. Chartrand. IEEE Signal Processing Letters. 2007: 14(10) 707-710. Chartrand. IEEE Signal Processing Letters. 2007: 14(10) 707-710.

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Trzasko (2007): Rewrite the problem Trzasko et al. IEEE SP 14th workshop on statistical signal processing. 2007. 176-180. Trzasko et al. IEEE SP 14th workshop on statistical signal processing. 2007. 176-180. where is tanh, laplace, log, etc. such that

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Homotopic function in 1D Start as 1-norm problem, then reduce slowly and approach 0-norm function.

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Homotopic method

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Demonstration when is big (1 st iteration), solving 1-norm problem. when is big (1 st iteration), solving 1-norm problem. reduce to approach 0-norm solution. reduce to approach 0-norm solution. original x∆x∆

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Example 1 original subsampled

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Zero-filled Reconstruction Fourier sample mask Homotopic Homotopic result: use 4% Fourier data error: -66.2 dB 85 seconds 1-norm result: use 4% Fourier dataerror: -11.4 dB542 seconds 1-norm recon homotopic recon

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Example 2 Angiography Angiography 360x360, 27.5% radial samples 360x360, 27.5% radial samples original

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reconstruction 1-norm method: error: -24.7 dB, 1151 seconds 360x360 reconstruction homotopic method: error: -26.5 dB, 101 seconds original 27.5% samples

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Chretien, An Alternating l1 approach to the compressed sensing problem, arXiv.org. Dattorro, Convex Optimization & Euclidean Distance Geometry, Meboo. Convex Iteration

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Convex Iteration demo zero-filled reconstruction Fourier samp l e mask use 4% Fourier dataerror: -104 dB96 seconds reconstruction

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Signal Separation by Convex Iteration

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1-norm formulation as convex iteration

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Signal Construction = + Cardinality(steps) = 7 Cardinality(cosine) = 4 u u∆u∆ uDuD

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Donoho, Tanner, 2005. Baron, Wakin et al., 2005 Minimum Sampling Rate k/n m/k m measurements k cardinality n record length

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m = 28 measurements m / n = 0.1 subsample m / k = 2.5 sample rate k / n = 0.04 sparsity Cardinality(steps) = 7 Cardinality(cosine) = 4 Signal Separation by Convex Iteration

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1-norm Convex iteration TV only DCT only -241dB reconstruction error u∆u∆ uDuD

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functional Magnetic Resonance Imaging (fMRI) Haemodynamic Response Function (HRF)

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How to conduct fMRI? Huettel, Song, Gregory. Functional Magnetic Resonance Imaging. Sinauer. Stimulus Timing

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How to conduct fMRI? Stimulus Timing

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What does fMRI measure? Neural activitySignallingVascular response Vascular tone (reactivity) Autoregulation Metabolic signalling BOLD signal glia arteriole venule B 0 field Synaptic signalling Blood flow, oxygenation and volume dendrite End bouton

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fMRI signal origin rest Oxygenated Hb Deoxygenated Hb task Oxygenated Hb Deoxygenated Hb

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Haemodynamic Response Function (HRF) Stimulus Timing Canonical HRF Predicted Data = = Time

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Which part of the brain is activated? http://www.fmrib.ox.ac.uk/ Time Actual measurement Prediction Signal Intensity

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= Stimulus TimingMeasurement Actual HRF HRF calibration Variability of HRF

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Deconvolve HRF h 10 0 -10 1010 y Dh D: convolution matrix D(:, 1) W: Coiflet E: monotone cone (d) Wh Discrete wavelet transform

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Deconvolve HRF h 10 0 -10 1010 y stimulus timing W: Coiflet wavelet E: monotone cone D: convolution matrix measurement smoothness Dh D( :, 1)

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Deconvolution results

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in vivo deconvolution results … HRF calibration … HRF deconvolution … Time (s)

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Cardinality Constraint Problem 3 5 Ab 3 1 x 5 1 A = b = A(:,2) * rand(1) + A(:,5) * rand(1)

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Find x with desired cardinality e.g. k = 2, want =

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m=3 n=5 Ab 3 1 x 5 1 Check every pair for k = 2 Possible # solution: In general, From the Range perspective …

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3 5 A 2 5 = 0 Perspective Nullspace Perspective Z Particular soln. General soln.

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Z = 0.570.34 -0.03 -0.26 -0.160.02 -0.68 -0.63 0.220.47 xp = 0.34 1.11 -0.30 0

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How to find intersection of lines? Sum of normalized wedges

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x = 0.34 1.11 -0.30 0 x = 0 0.68 0 0.51 Z = -0.59-0.36 0.06-0.55 0.15 0.36 0.74-0.08 -0.27 0.66 Znew = -0.41 -0.18 -0.80 -0.30 0.480.19 -0.26 -0.07 1.000.37 Znew = Z * randn(2);

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Summary Ways to find 0-norm solutions other than Ways to find 0-norm solutions other than 1-norm (homotopic, convex iteration) 1-norm (homotopic, convex iteration) fewer measurements fewer measurements faster faster In cardinality constraint problem, convex iteration and nullspace technique success more often than 1-norm. In cardinality constraint problem, convex iteration and nullspace technique success more often than 1-norm.

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