This work is concerned with the recovery of piecewise constant images from noisy linear measurements. We study the noise robustness of a variational reconstruction method, which is based on total (gradient) variation regularization. We show that, if the unknown image is the superposition of a few simple shapes, and if a non-degenerate source condition holds, then, in the low noise regime, the reconstructed images have the same structure: they are the superposition of the same number of shapes, each a smooth deformation of one of the unknown shapes. Moreover, the reconstructed shapes and the associated intensities converge to the unknown ones as the noise goes to zero.

翻译：本文关注从含噪线性测量中恢复分段常数图像的问题。我们研究了一种基于全变差正则化的变分重构方法的噪声鲁棒性。研究表明，若未知图像是少数简单形状的叠加，且满足非退化源条件，则在低噪声条件下，重构图像具有相同结构：它们由相同数量的形状叠加而成，每个形状均为未知形状的光滑变形。此外，随着噪声趋近于零，重构形状及其关联强度将收敛至真实值。