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Characterization of the Distortion-Perception Tradeoff for Finite Channels with Arbitrary Metrics

Dror Freirich, Nir Weinberger, Ron Meir

TL;DR

We address the distortion-perception tradeoff in finite-alphabet channels by formulating the problem with a Wasserstein-1 perception index and a general distortion matrix. The DP function $D(P)$ is shown to be computable via linear programming and, in the discrete setting, is necessarily piecewise linear with a finite set of breakpoints. A dual OT-based characterization yields a structural understanding: the DP curve is the upper envelope of a finite family of linear functions, with breakpoints tied to dual-vertex projections. For binary sources we derive a closed-form expression, revealing explicit breakpoints and linear segments, enabling exact, efficient computation of the DP curve. These results unify and extend prior work on rate-distortion-perception and discrete DP, with practical implications for constructing perceptually constrained reconstructions via simple breakpoint-based strategies.

Abstract

Whenever inspected by humans, reconstructed signals should not be distinguished from real ones. Typically, such a high perceptual quality comes at the price of high reconstruction error, and vice versa. We study this distortion-perception (DP) tradeoff over finite-alphabet channels, for the Wasserstein-$1$ distance induced by a general metric as the perception index, and an arbitrary distortion matrix. Under this setting, we show that computing the DP function and the optimal reconstructions is equivalent to solving a set of linear programming problems. We provide a structural characterization of the DP tradeoff, where the DP function is piecewise linear in the perception index. We further derive a closed-form expression for the case of binary sources.

Characterization of the Distortion-Perception Tradeoff for Finite Channels with Arbitrary Metrics

TL;DR

We address the distortion-perception tradeoff in finite-alphabet channels by formulating the problem with a Wasserstein-1 perception index and a general distortion matrix. The DP function is shown to be computable via linear programming and, in the discrete setting, is necessarily piecewise linear with a finite set of breakpoints. A dual OT-based characterization yields a structural understanding: the DP curve is the upper envelope of a finite family of linear functions, with breakpoints tied to dual-vertex projections. For binary sources we derive a closed-form expression, revealing explicit breakpoints and linear segments, enabling exact, efficient computation of the DP curve. These results unify and extend prior work on rate-distortion-perception and discrete DP, with practical implications for constructing perceptually constrained reconstructions via simple breakpoint-based strategies.

Abstract

Whenever inspected by humans, reconstructed signals should not be distinguished from real ones. Typically, such a high perceptual quality comes at the price of high reconstruction error, and vice versa. We study this distortion-perception (DP) tradeoff over finite-alphabet channels, for the Wasserstein- distance induced by a general metric as the perception index, and an arbitrary distortion matrix. Under this setting, we show that computing the DP function and the optimal reconstructions is equivalent to solving a set of linear programming problems. We provide a structural characterization of the DP tradeoff, where the DP function is piecewise linear in the perception index. We further derive a closed-form expression for the case of binary sources.
Paper Structure (15 sections, 10 theorems, 57 equations, 2 figures)

This paper contains 15 sections, 10 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The distortion-perception tradeoff
  4. Related work
  5. Problem formulation
  6. Linear Programming formulation
  7. Total Variation as a Wasserstein distance
  8. The Dual Problem
  9. Main results
  10. Piecewise linearity of DP functions
  11. Full characterization of channels with binary sources
  12. The linear optimization problem and strong duality
  13. A dual form for the TV distance setting
  14. Derivation of Eq. \ref{['eq:DPdiscrete_explicit']}
  15. Full characterization for binary sources (proof of Theorem \ref{['thm::DP:binarysource']})

Key Result

Theorem 1

(The perception-distortion tradeoff). If $d_p(p, q)$ is convex in its second argument, then the distortion-perception function APP_eq:D_P::General_definition is monotonically non-increasing and convex.

Figures (2)

  • Figure 1: The distortion-perception (DP) function.(Left) The minimal distortion possible for a certain level of perceptual quality forms a convex, non-increasing curve. The region below the curve can not be attained by any reconstruction method. (Right) In our discrete setting, $D(P)$ is a piecewise linear function. Breakpoints $P^*_i$ and slopes $2u_i$ are given explicitly by Theorem \ref{['thm::DP:binarysource']} for binary sources.
  • Figure 2: Numerical illustration of Theorem \ref{['thm:piecewiseLinear']} and Theorem \ref{['thm::r2convex']} for $| \mathcal{X} |=3, | \mathcal{Y} |=5$. In the (Middle) pane we present the set $\mathcal{S}_2$ and its convex hull in the $(p_0,p_1)$-plane. The (Right) pane shows the optimal solutions obtained by numerically solving \ref{['eq:DPdiscrete_explicit']} for different values of $P$. We can see that the solutions, corresponding to the linear segments of $D(P)$ ( Left pane), occur at extreme points of $\mathrm{conv}\left(\mathcal{S}_{2}\right)$.

Theorems & Definitions (16)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • Remark 4
  • Lemma 5
  • Theorem 6
  • proof : Proof of Theorem \ref{['thm:piecewiseLinear']}
  • Corollary 7
  • Theorem 8
  • ...and 6 more