反问题的压缩感知与稀疏Radon变换的样本复杂度
Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform
AI总结：
本文为涉及任意前向算子的抽象反问题发展了无限维压缩感知的一般理论，并首次为建模计算机断层扫描的稀疏Radon变换提供了严格的恢复估计，证明了在平行束和扇形束设置下，当信号关于紧支撑小波基是$s$-稀疏时，仅需$m\gtrsim s$（受对数因子影响）的测量即可实现稳定恢复。
摘要：
压缩感知允许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，并受对数因子影响。经典理论通常考虑随机线性测量或子采样等距变换，并已找到许多应用，包括加速的磁共振成像，该成像由子采样傅里叶变换建模。在这项工作中，我们为涉及任意前向算子的抽象反问题（可能是不适定的）发展了无限维压缩感知的一般理论。这是通过考虑广义限制等距性质和前向映射的准对角化性质来实现的。作为一个显著的应用，我们首次为稀疏Radon变换（即具有有限数量的角度$θ_1,\dots,θ_m$）获得了严格的恢复估计，该变换为计算机断层扫描建模，涵盖平行束和扇形束两种设置。在未知信号关于紧支撑小波的标准正交基是$s$-稀疏的情况下，我们证明了在条件\\[ m\gtrsim s, \\]下（受对数因子影响）的稳定恢复。
英文摘要：
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 $θ_1,\dots,θ_m$), 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
