RI: Small: Supervised Descent Method and its Applications to Computer Vision (and Beyond)

本计画发展一种快速的最佳化策略，以解决连续且可能的组合最佳化问题。优化是许多科学学科中的基本问题，从生物学，物理学和统计学到计算机图形学和计算机视觉。该项目的重点是解决计算机视觉中的二维（2D）和三维（3D）图像对齐问题。解决2D和3D图像之间的对应关系是一个开放的研究问题，它是大多数计算机视觉系统的基本组成部分，用于医疗成像，监控，高级驾驶员辅助系统，移动的机器人，增强现实，对象识别和空中视频探索等应用。该项目将研究与教育相结合，并让本科生/研究生参与研究。本研究解决了二阶下降法在优化中的两个主要问题：（1）优化的目标函数可能不是解析可微的，数值逼近是不切实际的，（2）Hessian可能很大，不是正定的。研究小组倡导学习通用血统图的概念（即，平均“下降方向”）。使用通用下降映射，研究小组推导出一种实用的算法，监督下降法（SDM），用于最小化具有连续参数的非线性最小二乘（NLS）问题。在训练过程中，SDM学习一系列适当的映射，使NLS目标最小化。在测试过程中，这些学习的下降映射用于最小化NLS目标，而不需要计算昂贵的雅可比矩阵或海森矩阵。除了NLS之外，研究小组还探索了SDM的使用，以优化组合优化问题，例如在图像和点云中找到2D和3D对应关系。

This project develops a fast optimization strategy for continuous and possibly combinatorial optimization problems. Optimization is a fundamental problem in many scientific disciplines from biology, physics and statistics to computer graphics and computer vision. This project focuses on solving 2 dimensional (2D) and 3 dimensional (3D) image alignment problems in computer vision. Solving the correspondence between 2D and 3D images is an open research problem and it is a fundamental component in most computer vision systems for medical imaging, surveillance, advanced driver assistance systems, mobile robots, augmented reality, object recognition and aerial video exploration, among other applications. The project integrates the research with education, and involves undergraduate/graduate students in the research. This research addresses two main issues with second order descent methods in optimization: (1) the objective function to optimize might not be analytically differentiable and numerical approximations are impractical, and (2) the Hessian might be large and not positive definite. The research team advocates the concept of learning generic descent maps (i.e., average "descent directions") in a supervised manner. Using generic descent maps, the research team derives a practical algorithm, Supervised Descent Method (SDM), for minimizing Nonlinear Least Squares (NLS) problems with continuous parameters. During training, SDM learns a sequence of decent maps that minimize the NLS objective. During testing, these learned descent maps are used to minimize the NLS objective without requiring computation of the expensive Jacobian or Hessian. Beyond NLS, the research team explores the use of SDM to optimize combinatorial optimization problems such as finding 2D and 3D correspondences in images and point-clouds.