Improved recovery guarantees and sampling strategies for TV minimization in compressive imaging
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In this paper, we consider the use of Total Variation (TV) minimization for compressive imaging; that is, image reconstruction from subsampled measurements. Focusing on two important imaging modalities – namely, Fourier imaging and structured binary imaging via the Walsh–Hadamard transform – we derive uniform recovery guarantees asserting stable and robust recovery for arbitrary random sampling strategies. Using this, we then derive a class of theoretically-optimal sampling strategies. For Fourier sampling, we show recovery of an image with approximately s-sparse gradient from m ≳_d s ·log^2(s) ·log^4(N) measurements, in d ≥ 1 dimensions. When d = 2, this improves the current state-of-the-art result by a factor of log(s) ·log(N). It also extends it to arbitrary dimensions d ≥ 2. For Walsh sampling, we prove that m ≳_d s ·log^2(s) ·log^2(N/s) ·log^3(N) measurements suffice in d ≥ 2 dimensions. To the best of our knowledge, this is the first recovery guarantee for structured binary sampling with TV minimization.
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