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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 (2):489 Donoho. IEEE Trans. Information Theory (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) Chartrand. IEEE Signal Processing Letters. 2007: 14(10)

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Trzasko (2007): Rewrite the problem Trzasko et al. IEEE SP 14th workshop on statistical signal processing Trzasko et al. IEEE SP 14th workshop on statistical signal processing 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: dB 85 seconds 1-norm result: use 4% Fourier dataerror: 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: dB, 1151 seconds 360x360 reconstruction homotopic method: error: 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, 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? 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 y Dh D: convolution matrix D(:, 1) W: Coiflet E: monotone cone (d) Wh Discrete wavelet transform

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Deconvolve HRF h 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 = xp =

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How to find intersection of lines? Sum of normalized wedges

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x = x = Z = Znew = 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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