Stable Image Reconstruction via Two-Parameter Power-Scale Variation Minimization

Abstract

In this article, we introduce a power-scale variation (PSV${}_{a,p}$) with two tunable parameters: the sparsity-inducing exponent $p\in (0,1]$ and the scaling factor $a\in (0,\infty )$. By minimizing the PSV${}_{a,p}$, we establish stable reconstructions in both the gradient and the image domains under the restricted isometry property (RIP) framework. Furthermore, we design an iteratively re-weighted least squares algorithm IRLSPSV to solve the unconstrained PSV${}_{a,p}$ minimization. Numerical experiments demonstrate its superior performance and broad applicability. The main novelties are: (i) the PSV${}_{a,p}$ minimization enjoys great flexibility and wide applicability due to its two tunable parameters $a$ and $p$, (ii) as $a\to \infty$, the PSV${}_{a,p}$ minimization reduces to the $p$-th power total variation (TV${}_p$) minimization and, even in this limiting case, the established RIP condition for image reconstruction is also new, (iii) the derived RIP upper bound $\overline{\hat{I}}´$ is proved to be asymptotically optimal in $a$ for gradient recovery, (iv) sensitivity analysis confirms the distinct roles of $a$ and $p$, thereby motivating a practical parameter tuning scheme for the proposed model.