Infinite dimensional compressed sensing from anisotropic measurements and applications to inverse problems in PDE

Abstract

We consider a compressed sensing problem in which both the measurement and the sparsifying systems are assumed to be frames (not necessarily tight) of the underlying Hilbert space of signals, which may be finite or infinite dimensional. The main result gives explicit bounds on the number of measurements in order to achieve stable recovery, which depends on the mutual coherence of the two systems. As a simple corollary, we prove the efficiency of nonuniform sampling strategies in cases when the two systems are not incoherent, but only asymptotically incoherent, as with the recovery of wavelet coefficients from Fourier samples. This general framework finds applications to inverse problems in partial differential equations, where the standard assumptions of compressed sensing are often not satisfied. Several examples are discussed, with a special focus on electrical impedance tomography.

42 pages

References in corpus (6)

• Sparsity and Incoherence in Compressive Sampling
• Accelerated High-Resolution Photoacoustic Tomography via Compressed Sensing
• Compressed sensing and sparsity in photoacoustic tomography
• Multiwave imaging in an enclosure with variable wave speed
• Time reversal method with stabilizing boundary conditions for Photoacoustic tomography
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Cited by in corpus (7)

• Infinite-dimensional inverse problems with finite measurements
• Calderón's Inverse Problem with a Finite Number of Measurements
• Inverse problems on low-dimensional manifolds
• Calderón's Inverse Problem with a Finite Number of Measurements II: Independent Data
• Compressed sensing photoacoustic tomography reduces to compressed sensing for undersampled Fourier measurements
• Sampling theorems for inverse problems on Riemannian manifolds
• Series reversion in Calderón's problem