Precise characterization of the Distortion-Perception tradeoff
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 signal restoration, where it implies that fidelity to ground truth signals comes at the expense of perceptual quality (e.g. 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 our work, 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.
Our theoretical results shed light on several topics that are subject to much practical activity, such as diverse image restoration and traversing between fidelity and visual quality during test-time, without training different models.
* Dror Freirich is a Ph.D. student in the Electrical Engineering department at the Technion under the supervision of Prof. Ron Meir.