Formulating a deformable mesh fitting problem into an optimization problem

In this section, we are going to talk about how to formulate the mesh fitting problem into an optimization problem. One key observation here is that object surfaces such as pedestrians can always be continuously deformed into a sphere. Thus, the approach we are going to take will start from the surface of a sphere and deform the surface to minimize a cost function.

The cost function should be chosen such that it is a good measurement of how similar the point cloud is to the mesh. Here, we choose the major cost function to be the Chamfer set distance. The Chamfer distance is defined between two sets of points as follows:

The Chamfer distance is symmetric and is a sum of two terms. In the first term, for each point x in the first point cloud, the closest point y in the other point cloud is found. For each such pair x and y, their distance is obtained and the distances for all the...