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

该项目为连续和可能的组合优化问题开发了一种快速优化策略。从生物学、物理学和统计学到计算机图形学和计算机视觉，最优化问题是许多科学学科中的一个基本问题。本项目致力于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，用于医学成像、监视、先进的驾驶辅助系统、移动机器人、增强现实、对象识别和航空视频探索等应用。该项目将研究与教育相结合，并让本科生/研究生参与研究。本研究主要解决二阶下降法在最优化问题中的两个主要问题：(1)优化的目标函数可能不是解析可微的，数值近似是不切实际的；(2)黑森函数可能很大且不是正定的。研究小组提倡以有监督的方式学习通用下降图(即平均“下降方向”)的概念。利用通用下降图，研究小组得出了一种实用的算法，即监督下降法(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学习一系列使NLS目标最小化的像样的地图。在测试过程中，这些学习的下降映射被用来最小化NLS目标，而不需要计算昂贵的雅可比或Hessian。除了NLS之外，研究团队还探索使用SDM来优化组合优化问题，例如在图像和点云中找到2D和3D对应。

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.