Conditions for well-posed color recovery in scattering media

TL;DR

This paper establishes conditions under which color recovery in scattering media becomes well-posed, enabling more accurate optical imaging by leveraging intrinsic image patterns and ideal sensor capabilities.

Contribution

It identifies key constraints and recovery patterns that ensure unique color reconstruction in scattering media, advancing the theoretical foundation of optical imaging.

Findings

Identifies spectral projection and unknown medium parameters as sources of ill-posedness.

Demonstrates that ideal hyperspectral cameras can leverage natural image patterns for unique solutions.

Provides theoretical conditions for well-posed color recovery 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.