Hierarchical Data-Driven Descent for Efficient Optimal Deformation Estimation

Real-world surfaces such as clothing, water and human body deform in complex ways. The image distortions observed are high-dimensional and non-linear, making it hard to estimate these deformations accurately. The recent data-driven descent approach applies Nearest Neighbor estimators iteratively on a particular distribution of training samples to obtain a globally optimal and dense deformation field between a template and a distorted image. In this work, we develop a hierarchical structure for the Nearest Neighbor estimators, each of which can have only a local image support. We demonstrate in both theory and practice that this algorithm has several advantages over the non-hierarchical version: it guarantees global optimality with significantly fewer training samples, is several orders faster, provides a metric to decide whether a given image is ``hard'' (or ``easy'') requiring more (or less) samples, and can handle more complex scenes that include both global motion and local deformation. The proposed algorithm successfully tracks a broad range of non-rigid scenes including water, clothing, and medical images, and compares favorably against several other deformation estimation and tracking approaches that do not provide optimality guarantees.