Stable Image Reconstruction via Two-Parameter Power-Scale Variation Minimization
2026-06-22 • Information Theory
Information Theory
AI summaryⓘ
The authors introduce a new mathematical method called power-scale variation (PSV) with two adjustable settings to help improve image reconstruction. They show that minimizing this PSV can stably recover images and their gradients under certain technical conditions (RIP). They also create an algorithm to efficiently solve this problem and test it successfully with experiments. Their work highlights how the two parameters each affect results and provides guidance on how to choose them. Additionally, they demonstrate that their method generalizes existing approaches and offers new theoretical guarantees.
power-scale variationsparsitygradient domainimage domainrestricted isometry propertyiteratively re-weighted least squarestotal variationimage reconstructionparameter tuning
Authors
Ziwei Li, Wengu Chen, Huanmin Ge, Limei Huo, Dachun Yang
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 $\overlineδ$ 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.