A Theoretical Analysis of a Bilevel Optimization Formulation for Combined Local-Global Optical Flow Models

Abstract

This paper analyzes a bilevel formulation for learning the regularization parameter of combined local-global (CLG) optical flow models. The lower-level problem computes the flow from a Lucas--Kanade data term and a quadratic or total variation (TV) regularizer, and the upper-level problem selects the parameter that minimizes the squared distance to a ground-truth flow. The setting is finite-dimensional, and a full-rank assumption makes the lower-level solution unique. For the quadratic model we derive sufficient conditions for the existence of an optimal parameter, for every optimal parameter to be positive, and for strict local optimality. For TV regularization an optimal parameter always exists, because the lower-level solution vanishes for large parameters, and we give a sufficient condition for positivity that accounts for the coupling of the two flow components through the local structure tensors. Synthetic examples illustrate the conditions, including cases in which they fail.