High Accuracy Optical Flow Method Based on a Theory for Warping: 3D Extension

This paper describes the implementation and qualitative and quantitative evaluation of a 3D optical flow algorithm, whose derivation is based on the 2D optical flow method published by Brox et al. [ECCV2004]. The optical flowminimizes an energy function built with three assumptions: a brightness constancy assumption, a gradient constancy assumption, and a smoothness assumption. Brox et al. minimize the 2D version of this function using a robust estimator, which make the functional convex and guarantees convergence to a single solution. They propose a numerical solution based on nested fixed points iterations and use this scheme within a coarse-to-fine warping strategy in a 2D hierarchical image pyramid. In our 3D extension, our solution requires the regularization of a 3D function based on 3D extensions of their assumptions in a 3D hierarchical volume pyramid. We solve the corresponding Euler-Lagrange equations iteratively using nested iterations. We present 3D quantitative results on three sets of 3D sinusoidal data (with and without motion discontinuities), where the correct 3D flow is known. We also present a qualitative evaluation on the 3D flow computed using gated MRI beating heart sequence.