Nonlinear Evolutionary PDE-Based Refinement of Optical Flow

TL;DR

This paper introduces two nonlinear PDE-based models for refining optical flow estimation, with one model employing a two-phase approach and the other a single functional, both leveraging primal-dual algorithms for efficiency.

Contribution

The paper proposes a novel two-phase variational model for optical flow refinement and compares it with a single functional approach within a unified PDE framework.

Findings

Two-phase model is more efficient than single-phase.

Both models achieve similar accuracy in flow estimation.

Two-phase model has faster convergence rate of O(1/N^2).

Abstract

The goal of this paper is to propose two nonlinear variational models for obtaining a refined motion estimation from an image sequence. Both the proposed models can be considered as a part of a generalized framework for an accurate estimation of physics-based flow fields such as rotational and fluid flow. The first model is novel in the sense that it is divided into two phases: the first phase obtains a crude estimate of the optical flow and then the second phase refines this estimate using additional constraints. The correctness of this model is proved using an evolutionary PDE approach. The second model achieves the same refinement as the first model, but in a standard manner, using a single functional. A special feature of our models is that they permit us to provide efficient numerical implementations through the first-order primal-dual Chambolle-Pock scheme. Both the models are compared in the context of accurate estimation of angle by performing an anisotropic regularization of the divergence and curl of the flow respectively. We observe that, although both the models obtain the same level of accuracy, the two-phase model is more efficient. In fact, we empirically demonstrate that the single-phase and the two-phase models have convergence rates of order $O(1/N^2)$ and $O(1/N)$ respectively.