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CS 205 Mathematical Methods for Robotics and Vision - Chapter 4

Function Optimization Có ba lý do chính lý do tại sao hầu hết các vấn đề về robot, tầm nhìn, và cho là tất cả các khoa học khác hoặc nỗ lực về hình thức của các vấn đề tối ưu hóa. Một là mục tiêu mong muốn có thể không đạt được, và vì vậy chúng tôi cố gắng để có được càng gần càng tốt với nó. Lý do thứ hai là có thể có nhiều cách để đạt được các mục tiêu, và vì vậy chúng tôi có thể chọn một bằng cách gán cho một. | Chapter 4 Function Optimization There are three main reasons why most problems in robotics vision and arguably every other science or endeavor take on the form of optimization problems. One is that the desired goal may not be achievable and so we try to get as close as possible to it. The second reason is that there may be more ways to achieve the goal and so we can choose one by assigning a quality to all the solutions and selecting the best one. The third reason is that we may not know how to solve the system of equations f x 0 so instead we minimize the norm I f x 11 which is a scalar function of the unknown vector x. We have encountered the first two situations when talking about linear systems. The case in which a linear system admits exactly one exact solution is simple but rare. More often the system at hand is either incompatible some say overconstrained or at the opposite end underdetermined. In fact some problems are both in a sense. While these problems admit no exact solution they often admit a multitude of approximate solutions. In addition many problems lead to nonlinear equations. Consider for instance the problem of Structure From Motion SFM in computer vision. Nonlinear equations describe how points in the world project onto the images taken by cameras at given positions in space. Structure from motion goes the other way around and attempts to solve these equations image points are given and one wants to determine where the points in the world and the cameras are. Because image points come from noisy measurements they are not exact and the resulting system is usually incompatible. SFM is then cast as an optimization problem. On the other hand the exact system the one with perfect coefficients is often close to being underdetermined. For instance the images may be insufficient to recover a certain shape under a certain motion. Then an additional criterion must be added to define what a good solution is. In these cases the noisy system admits no .