Regularizing Optical-Flow Computation Using Tensor Theory and Complex Analysis

Abstract

This paper reports a technique that improves the robustness and accuracy in computing dense optical-flow fields. We propose a global formulation with a regularization term. The regularization expressions are derived based on tensor theory and complex analysis. It is shown that while many regularizers have been proposed (image-driven, flow-driven, homogeneous, inhomogeneous, isotropic, anisotropic), they are all variations of a single base expression nablaunablau <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> + nablavnablav <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> . These regularizers, strictly speaking, are valid for uniform 2D translational motion only, because what they do essentially is to penalize changes in a flow field. However, many flow patterns-such as rotation, zoom, and their combinations, induced by a 3D rigid-body motion.are not constant. The traditional regularizers then incorrectly penalize these legal flow patterns and result in biased estimates. The purpose of this work is then to derive a new suite of regularization expressions that treat all valid flow patterns resulting from a 3D rigid-body motion equally, without unfairly penalizing any of them.