The lower the distortion of an estimator, the more the distribution of its outputs generally deviates from the distribution of the signals it attempts to estimate. This phenomenon, known as the perception-distortion tradeoff, has captured significant attention in image restoration, where it implies that fidelity to ground truth images comes at the expense of perceptual quality (deviation from statistics of natural images). However, despite the increasing popularity of performing comparisons on the perception-distortion plane, there remains an important open question: what is the minimal distortion that can be achieved under a given perception constraint? In this paper, we derive a closed form expression for this distortion-perception (DP) function for the mean squared-error (MSE) distortion and the Wasserstein-2 perception index. We prove that the DP function is always quadratic, regardless of the underlying distribution. This stems from the fact that estimators on the DP curve form a geodesic in Wasserstein space. In the Gaussian setting, we further provide a closed form expression for such estimators. For general distributions, we show how these estimators can be constructed from the estimators at the two extremes of the tradeoff: The global MSE minimizer, and a minimizer of the MSE under a perfect perceptual quality constraint. The latter can be obtained as a stochastic transformation of the former.

本文研究估计器的感知-失真权衡问题，导出了均方误差失真和Wasserstein-2感知指数下的失真-感知（DP)函数的闭式表达式，证明了不论潜在分布如何，DP函数总是二次的。我们还表明，这些估计器可以从感知质量卓越的MSE最小化器和感知质量优良的MSE最小化器两个极端处的估计器构建而来。

Wasserstein空间中失真-感知权衡的理论