Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear measurements or subsampled isometries and has found many applications, including accelerated magnetic resonance imaging, which is modeled by the subsampled Fourier transform. In this work, we develop a general theory of infinite-dimensional compressed sensing for abstract inverse problems, possibly ill-posed, involving an arbitrary forward operator. This is achieved by considering a generalized restricted isometry property, and a quasi-diagonalization property of the forward map. As a notable application, for the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles $\theta_1,\dots,\theta_m$), which models computed tomography. In the case when the unknown signal is $s$-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove exact recovery under the condition \[ m\gtrsim s, \] up to logarithmic factors.

翻译：压缩感知能够从少量测量中恢复稀疏信号，其测量数量与未知信号的稀疏度成正比，仅需考虑对数因子。经典理论通常考虑随机线性测量或次采样等距变换，并已在诸多领域得到应用，包括以次采样傅里叶变换建模的加速磁共振成像。本研究针对抽象逆问题（可能是不适定的）发展了无限维压缩感知的通用理论，该理论可适用于任意前向算子。我们通过建立广义限制等距性质和前向映射的准对角化性质来实现这一目标。作为重要应用，首次获得了稀疏Radon变换（即具有有限角度$\theta_1,\dots,\theta_m$）的严格恢复估计，该变换可建模计算机断层扫描。当未知信号相对于紧支撑小波正交基为$s$-稀疏时，我们证明了在条件\[ m\gtrsim s \]（仅需考虑对数因子）下的精确恢复。