Underwater Color Restoration with Vanishing Uncertainty

Grigory Solomatov; Derya Akkaynak

eess.IVAug 16, 2026

Underwater Color Restoration with Vanishing Uncertainty

Authors: Grigory Solomatov, Derya Akkaynak

Organizations: Hatter Department of Marine Technologies, University of Haifa, Haifa, Israel · Interuniversity Institute for Marine Sciences, Eilat, Israel

Abstract

Underwater color restoration promises to unlock color as a reliable signal for aquatic sciences, but achieving this with scientific confidence remains out of reach. Current methods are validated almost exclusively on an empirical basis, which provides confidence only to the extent that the vast diversity of possible visibility conditions is covered with end-to-end testing using a known ground truth. This is exacerbated by color restoration being a fatally ill-posed problem when considered in full mathematical generality, requiring additional constraints to narrow the solution to a finite uncertainty interval. The gap between which constraints suffice in theory and which constraints are satisfied by real-world data is poorly understood, making it unclear whether existing methods are solving a problem that is actually solvable. In this article, we investigate the theoretical side of this gap, identifying idealized conditions which guarantee bounded uncertainty that converges to zero as the spatial resolution of the camera increases.

Figures & tables

Fig. 1: Underwater image of a coral reef. Left: original. Right: SeaThru [ 6 ] .

Fig. 2: Extrapolation of

P=P[q,Q]

, where

q(\lambda)=-\lambda

Q(\lambda)=1

and

\Lambda=[0,1]

. The black curve describes

P(z)

, which is treated as unknown except for the points

z_{i}

(black dots) that are linearly spaced between

z=10

and

z=15

. The green curve describes the extrapolation

\sum_{i}\alpha_{i}(z)P(z_{i})\approx P(z)

, where the

\alpha_{i}

were optimized numerically using

u_{1}=1/e

and

u_{2}=1

. The gray region describes the uncertainty bounds as given by Lemma IV.4 . Dotted blue denotes log-linear regression for comparison.

Fig. 3: With the context of Figure 2 , the black curves in the left column describe the coefficients

\alpha_{i}(z)

. In the right column, magenta is

\sum_{i}\alpha_{i}(0)u^{z_{i}}

for

u_{1}\leq u\leq u_{2}

and black is

u^{0}=1

, while their difference is

\Psi_{\mathcal{Z},\mathcal{A}}(0,u)

Fig. 4: Inherent radiance segmentation in an idealized setting. Top left: depth map $z(x,y)=y+x/\sqrt{2}$ visualized as a heat map. Top right: pixel intensities $p(x,y)=\int_{\Lambda}e^{qz(x,y)}Q[x,y]\mathrm{d}\lambda$ , where $\Lambda=[0,1],q(\lambda)=-\lambda$ , and $Q[x,y](\lambda)=\binom{3}{k}\lambda^{k}(1-\lambda)^{3-k}$ for $0\leq k\leq 3$ are the 4 Bernstein polynomials of degree 3. Bottom left: pairs $g(x,y)=(z(x,y),p(x,y))$ , with the black points coming from the pixels and the gray curves coming from the regions $\mathcal{H}(x,y)$ . Bottom right: The Bernstein polynomials used for the functions $Q[x,y]=(L[x,y]-B)S_{j}$ . As the image resolution increases, the black points in the bottom left subfigure become increasingly concentrated within their associated gray curves, making segmentation easier.

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Conditions for well-posed color recovery in scattering media

Grigory Solomatov, Derya Akkaynak

Hatter Department of Marine Technologies, University of Haifa, Haifa, Israel · Interuniversity Institute for Marine Sciences, Eilat, Israel

Image Restoration Inverse Problems Image Inverse Problems

Open paper AskPDF arXiv

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.

Abstract

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