Conditions for well-posed color recovery in scattering media

2026-05-05

AI Summary

Color recovery in scattering media constitutes an ill-posed inverse problem due to spectral projection and unknown medium parameters, leading to non-unique solutions. This work establishes, for the first time from first principles, a sufficient condition for well-posed color recovery with an ideal hyperspectral camera by analyzing naturally occurring cross-pixel structural relationships—termed “recovery patterns”—present in images. Theoretical analysis demonstrates that sensor enhancements alone are insufficient to overcome medium-induced color distortion; instead, structural priors are essential to constrain the solution to a single candidate. By formalizing these intrinsic image structures as a principled prior, this study lays a new algorithmic foundation for quantitative hyperspectral imaging in scattering environments.

Abstract

Recovering scene color from images captured in scattering media is a fundamental inverse problem in optical imaging. Yet the problem is intrinsically ill-posed as multiple solutions can explain the same observation, and prediction error cannot be controlled without understanding the space of candidate solutions. Here, we present sufficient conditions under which color recovery in a scattering medium becomes well-posed. Observing that ill-posedness stems from (i) projection of spectral signals onto pixel intensities, and (ii) unknown medium parameters, we demonstrate that sensor improvements alone cannot resolve medium-induced distortions without additional constraints. We identify recovery patterns, cross-pixel relationships that naturally occur in images, and prove, for an ideal hyperspectral camera, that they restrict the solution to a unique candidate. This opens the door to a new class of vision algorithms grounded in first principles, enabling quantitative analysis of images in scattering environments.

Problem

• color recovery
• scattering media
• ill-posed inverse problem
• spectral imaging
• hyperspectral camera

Innovation

• well-posed color recovery
• scattering media
• recovery patterns
• hyperspectral imaging
• inverse problems