Graduated optimization

Abstract: Point cloud registration (PCR) is an important task in photogrammetry and remote sensing, whose goal is to seek a seven-parameter similarity transformation to register a pair of point clouds. Traditional iterative closest point (ICP) variants highly rely on the initial parameters, and most of them cannot deal with cross-source (multisource) point clouds with scale changes. In this article, we propose a flexible correspondence-based PCR method, which is initial-guess free, fast, and robust. We first decompose the full seven-parameter registration problem into three subproblems, i.e., scale, rotation, and translation estimations, based on line vectors. Then, we propose a one-point random sample consensus (RANSAC) algorithm to estimate the scale and translation parameters. For the rotation estimation, we introduce a graduated optimization strategy into Tukey’s biweight function and propose a scale-annealing biweight estimator. We evaluate the proposed method on both same-source and cross-source data. Results show that the proposed method is robust against over 99% outliers and is one to two orders of magnitude faster than its competitors. The source code of our method will be made public.

Abstract: The graduated optimization approach, also known as the continuation method, is a popular heuristic to solving non-convex problems that has received renewed interest over the last decade. Despite its popularity, very little is known in terms of theoretical convergence analysis. In this paper we describe a new first-order algorithm based on graduated optimiza- tion and analyze its performance. We characterize a parameterized family of non- convex functions for which this algorithm provably converges to a global optimum. In particular, we prove that the algorithm converges to an {\epsilon}-approximate solution within O(1/\epsilon^2) gradient-based steps. We extend our algorithm and analysis to the setting of stochastic non-convex optimization with noisy gradient feedback, attaining the same convergence rate. Additionally, we discuss the setting of zero-order optimization, and devise a a variant of our algorithm which converges at rate of O(d^2/\epsilon^4).

Abstract: The graduated optimization approach, also known as the continuation method, is a popular heuristic to solving non-convex problems that has received renewed interest over the last decade. Despite being popular, very little is known in terms of its theoretical convergence analysis. In this paper we describe a new first-order algorithm based on graduated optimization and analyze its performance. We characterize a family of non-convex functions for which this algorithm provably converges to a global optimum. In particular, we prove that the algorithm converges to an e-approximate solution within O(1/e2) gradient-based steps. We extend our algorithm and analysis to the setting of stochastic non-convex optimization with noisy gradient feedback, attaining the same convergence rate. Additionally, we discuss the setting of "zero-order optimization", and devise a variant of our algorithm which converges at rate of O(d2/e4).

Abstract: This paper presents a new method for estimating piecewise-smooth optical flow. We propose a global optimization formulation with three-frame matching and local variation and develop an efficient technique to minimize the resultant global energy. This technique takes advantage of local gradient, global gradient, and global matching methods and alleviates their limitations. Experiments on various synthetic and real data show that this method achieves highly competitive accuracy.