A Gauss-Newton Method for the Integration of Spatial Normal Fields in Shape Space

作者：Balzer Jonathan^*

来源：Journal of Mathematical Imaging and Vision, 2012, 44(1): 65-79.

DOI：10.1007/s10851-011-0311-1

• Integration
• Normal field
• Normal map
• Gauss-Newton method
• Sobolev flow
• Minimal surface
• Shape space
• Shape Hessian

We address the task of adjusting a surface to a vector field of desired surface normals in space. The described method is entirely geometric in the sense, that it does not depend on a particular parametrization of the surface in question. It amounts to solving a nonlinear least-squares problem in shape space. Previously, the corresponding minimization has been performed by gradient descent, which suffers from slow convergence and susceptibility to local minima. Newton-type methods, although significantly more robust and efficient, have not been attempted as they require second-order Hadamard differentials. These are difficult to compute for the problem of interest and in general fail to be positive-definite symmetric. We propose a novel approximation of the shape Hessian, which is not only rigorously justified but also leads to excellent numerical performance of the actual optimization. Moreover, a remarkable connection to Sobolev flows is exposed. Three other established algorithms from image and geometry processing turn out to be special cases of ours. Our numerical implementation founds on a fast finite-elements formulation on the minimizing sequence of triangulated shapes. A series of examples from a wide range of different applications is discussed to underline flexibility and efficiency of the approach.

• 出版日期2012-9