Globally Optimal Solutions to Vision Using Convex and Quasi-convex Optimization

Abstract

Vision Geometry is the are of Computer Vision that deals with computing the geometry of the 3D world from sequences of images. It grew out of Photogrammetry, a field that goes back at least to the start of the 20th century. In the 1990s this field was transformed by the application of methods of Projective Geometry, leading to many new algorithms and deployment of the new methods in a wide variety of applications.

The algorithmic basis for Vision Geometry still ultimately relied on a technique called ”bundle adjustment”, involving iterative refinement of initial solutions by Newton or Gauss-Newton methods. These had the disadvantage of often finding local rather than global minima.

Recent work has focussed on applying different optimization techniques, particularly Convex Optimization techniques to attempt to find guaranteed global solutions to these problems. I will talk about progress in this area, through the use of methods such as Second Order Cone Programming, branch-and-bound fractional programming and semi-definite programming.