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 both the parallel-beam and the fan-beam settings. In the case when the unknown signal is $s$-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove stable 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，所需测量次数与未知信号的稀疏度成正比（仅含对数因子）。经典理论通常考虑随机线性测量或子采样等距变换，并已在诸多领域得到应用，例如以子采样傅里叶变换建模的加速磁共振成像。本文针对抽象逆问题（可能为不适定问题），发展了无限维压缩感知的通用理论，涉及任意前向算子。该理论通过引入广义约束等距性质及前向映射的拟对角化性质得以实现。作为重要应用，我们首次获得了稀疏Radon变换（即有限角度$\theta_1,\dots,\theta_m$情形下的Radon变换，用于建模计算机断层扫描）的严格恢复估计，涵盖平行束与扇束两种设置。当未知信号关于紧支撑小波正交基为$s$-稀疏时，我们证明了在条件 \[ m\gtrsim s \]（忽略对数因子）下的稳定恢复。