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A Perceptually Inspired Variational Framework for Color Enhancement

Rodrigo Palma-Amestoy, Edoardo Provenzi, Marcelo Bertalmío, Vicent Caselles

TL;DR

This work introduces a principled variational framework for color enhancement inspired by human vision, formulating energy functionals that balance local contrast amplification with global dispersion control around gray and the original data. By deriving three explicit inverse-contrast terms and an entropic dispersion term, the authors connect perceptual principles to practical color correction, and show how gradient-descent minimization yields effective algorithms. They establish existence and stability, draw connections to ACE and Retinex, and provide a computational strategy to reduce complexity from O(N^2) to O(N log N) using FFT-based convolutions. Extensive tests on diverse images demonstrate improved contrast and reduced color cast, with a simple noise-control workflow to mitigate noise amplification in dark regions. This framework offers a transparent, perceptually grounded alternative to existing color enhancement methods with clear tunable parameters and theoretical guarantees.

Abstract

Basic phenomenology of human color vision has been widely taken as an inspiration to devise explicit color correction algorithms. The behavior of these models in terms of significative image features (such as contrast and dispersion) can be difficult to characterize. To cope with this, we propose to use a variational formulation of color contrast enhancement that is inspired by the basic phenomenology of color perception. In particular, we devise a set of basic requirements to be fulfilled by an energy to be considered as `perceptually inspired', showing that there is an explicit class of functionals satisfying all of them. We single out three explicit functionals that we consider of basic interest, showing similarities and differences with existing models. The minima of such functionals is computed using a gradient descent approach. We also present a general methodology to reduce the computational cost of the algorithms under analysis from ${\cal O}(N^2)$ to ${\cal O}(N\log N)$, being $N$ the number of input pixels.

A Perceptually Inspired Variational Framework for Color Enhancement

TL;DR

This work introduces a principled variational framework for color enhancement inspired by human vision, formulating energy functionals that balance local contrast amplification with global dispersion control around gray and the original data. By deriving three explicit inverse-contrast terms and an entropic dispersion term, the authors connect perceptual principles to practical color correction, and show how gradient-descent minimization yields effective algorithms. They establish existence and stability, draw connections to ACE and Retinex, and provide a computational strategy to reduce complexity from O(N^2) to O(N log N) using FFT-based convolutions. Extensive tests on diverse images demonstrate improved contrast and reduced color cast, with a simple noise-control workflow to mitigate noise amplification in dark regions. This framework offers a transparent, perceptually grounded alternative to existing color enhancement methods with clear tunable parameters and theoretical guarantees.

Abstract

Basic phenomenology of human color vision has been widely taken as an inspiration to devise explicit color correction algorithms. The behavior of these models in terms of significative image features (such as contrast and dispersion) can be difficult to characterize. To cope with this, we propose to use a variational formulation of color contrast enhancement that is inspired by the basic phenomenology of color perception. In particular, we devise a set of basic requirements to be fulfilled by an energy to be considered as `perceptually inspired', showing that there is an explicit class of functionals satisfying all of them. We single out three explicit functionals that we consider of basic interest, showing similarities and differences with existing models. The minima of such functionals is computed using a gradient descent approach. We also present a general methodology to reduce the computational cost of the algorithms under analysis from to , being the number of input pixels.

Paper Structure

This paper contains 27 sections, 5 theorems, 82 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Basic HVS phenomenological properties
  3. Human color constancy
  4. Local contrast enhancement
  5. Visual adaptation
  6. The basic set of assumptions
  7. The first assumption: General structure of the energy functional
  8. The second assumption: Properties of the contrast function
  9. Contrast functions satisfying the Weber-Fechner's law
  10. Explicit energy functionals complying with our set of assumptions
  11. Three basic inverse contrast terms
  12. Entropic dispersion term
  13. Minimization of the energy functionals $E^{f}_{w,\alpha,\beta}(I)=D_{\alpha,\beta}^{\cal E}(I) + C_w^{f}(I)$
  14. Features of the three computational algorithms and comparison with existing models
  15. Correspondence between $E^{\log}_{w,\epsilon}$ and the ACE model
  16. ...and 12 more sections

Key Result

Proposition 5.1

Let $c:(0,1]\times (0,1] \to \mathbb{R}$ be a continuous function. Let $E(I) := D_{\alpha,\beta}^{\cal E}(I) + C_w(I)$ where $C_w(I)$ is given in (def:contrastterm). There is a minimum of $E(I)$ in the class of functions ${\cal F}_\rho$. An analogous result holds if we use the quadratic dispersion t

Figures (15)

  • Figure 1: A picture with blue cast and its global R,G,B histograms.
  • Figure 2: Dotted line. Graphic of entropic dispersion function around 1/2. Solid line. Graphic of quadratic dispersion function around 1/2.
  • Figure 3: Left: Surface of $r^{\rm id}_\epsilon$. Center: Surface of $r^{\log}_\epsilon$. Right: Surface of $r^{-\cal M}_\epsilon$. Corresponding to the function $s_\epsilon(z) = \arctan(z/\epsilon)/\arctan(1/\epsilon)$ with $\epsilon = 1/20$.
  • Figure 4: Three different original images of our test set showing different features to be enhanced.
  • Figure 5: Outputs of the algorithm corresponding to the energy $E^{\, \rm id}_{w,\epsilon}$.
  • ...and 10 more figures

Theorems & Definitions (5)

  • Proposition 5.1
  • Proposition 5.2
  • Lemma 9.1
  • Lemma 9.2
  • Lemma 9.3