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Nonlinear Evolutionary PDE-Based Refinement of Optical Flow

Hirak Doshi, N. Uday Kiran

TL;DR

This work addresses refining optical flow estimates by embedding physics-based constraints that capture rotational and fluid-like motion. It introduces two nonlinear variational models: a two-phase refinement (M1) and a single-phase refinement (M2), both solved via a first-order primal–dual Chambolle–Pock method, with anisotropic regularization targeting divergence and curl. The authors establish well-posedness through an Evolutionary PDE framework and derive explicit optimality conditions for the CP iterations, including dual projections onto $L^\infty$-balls and 2×2 primal solves. Numerical results on fluid- and rigid-like sequences show the two-phase model can achieve faster practical convergence, while the single-phase model delivers strong accuracy with a $O(1/N^2)$ behavior; modern implementation principles such as pyramidal warping and median filtering further enhance performance. Collectively, the paper provides a principled, efficient approach for physics-informed refinement of motion fields with significant implications for accurate angle estimation in complex flows.

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.

Nonlinear Evolutionary PDE-Based Refinement of Optical Flow

TL;DR

This work addresses refining optical flow estimates by embedding physics-based constraints that capture rotational and fluid-like motion. It introduces two nonlinear variational models: a two-phase refinement (M1) and a single-phase refinement (M2), both solved via a first-order primal–dual Chambolle–Pock method, with anisotropic regularization targeting divergence and curl. The authors establish well-posedness through an Evolutionary PDE framework and derive explicit optimality conditions for the CP iterations, including dual projections onto -balls and 2×2 primal solves. Numerical results on fluid- and rigid-like sequences show the two-phase model can achieve faster practical convergence, while the single-phase model delivers strong accuracy with a behavior; modern implementation principles such as pyramidal warping and median filtering further enhance performance. Collectively, the paper provides a principled, efficient approach for physics-informed refinement of motion fields with significant implications for accurate angle estimation in complex flows.

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 and respectively.

Paper Structure

This paper contains 8 sections, 2 theorems, 55 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Model Description
  3. Well-Posedness
  4. The Primal-Dual Framework
  5. Optimality Condition for Our Model $(M1)$
  6. Optimality Condition for Our Model $(M2)$
  7. Results
  8. Modern Implementation Principles

Key Result

Lemma 1

Figures (5)

  • Figure 1: Oseen Vortex Pair Liu2.
  • Figure 2: Velocity magnitude plot for the Oseen vortex pair with $\alpha=0.1,\beta=0.01, \text{iter}=50$.
  • Figure 3: Estimated flow field for the rubberwhale sequence with $\alpha=10,\beta=1$.
  • Figure 4: Sphere sequence otago
  • Figure 5: Estimated flow fields of the sphere sequence from otago using Algorithm 1$^\dagger$ with $\alpha=1,\beta=0.1$ ($^\dagger$ refers to the algorithm incorporated with the modern implementation principles).

Theorems & Definitions (5)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Lemma 2
  • proof