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.

我们考虑一个压缩的传感问题，其中测量系统和稀疏系统都被假定为信号的基础希尔伯特空间的帧（不一定紧密），可以是有限维或无限维。主要结果给出了测量数量的明确界限，以实现稳定的恢复，这取决于两个系统的相互一致性。作为一个简单的推论，我们证明了当两个系统不是不相干，而只是渐近地不相干时，以及从傅立叶样本中恢复小波系数时，非均匀采样策略的效率。该通用框架可用于偏微分方程中的逆问题，在这种情况下，通常无法满足压缩传感的标准假设。讨论了几个例子，