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We consider a compressed sensing problem in which both the measurement andthe sparsifying systems are assumed to be frames (not necessarily tight) of theunderlying Hilbert space of signals, which may be finite or infinitedimensional. The main result gives explicit bounds on the number ofmeasurements in order to achieve stable recovery, which depends on the mutualcoherence of the two systems. As a simple corollary, we prove the efficiency ofnonuniform sampling strategies in cases when the two systems are notincoherent, but only asymptotically incoherent, as with the recovery of waveletcoefficients from Fourier samples. This general framework finds applications toinverse problems in partial differential equations, where the standardassumptions of compressed sensing are often not satisfied. Several examples arediscussed, with a special focus on electrical impedance tomography.

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