Improved Imaging by Invex Regularizers with Global Optima Guarantees

TL;DR

This paper introduces invex regularizers for image reconstruction, providing the first guarantees of convergence to global optima in non-convex regularization, leading to improved imaging results.

Contribution

It proposes the use of invex regularizers in image reconstruction, offering the first theoretical guarantees of convergence to global optima for such non-convex methods.

Findings

Invex regularizers improve reconstruction quality in various imaging tasks.

Theoretical guarantees of convergence to global optima are established.

Numerical experiments demonstrate the effectiveness of invex regularization.

Abstract

Image reconstruction enhanced by regularizers, e.g., to enforce sparsity, low rank or smoothness priors on images, has many successful applications in vision tasks such as computer photography, biomedical and spectral imaging. It has been well accepted that non-convex regularizers normally perform better than convex ones in terms of the reconstruction quality. But their convergence analysis is only established to a critical point, rather than the global optima. To mitigate the loss of guarantees for global optima, we propose to apply the concept of \textit{invexity} and provide the first list of proved invex regularizers for improving image reconstruction. Moreover, we establish convergence guarantees to global optima for various advanced image reconstruction techniques after being improved by such invex regularization. To the best of our knowledge, this is the first practical work applying invex regularization to improve imaging with global optima guarantees. To demonstrate the effectiveness of invex regularization, numerical experiments are conducted for various imaging tasks using benchmark datasets.