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

Abstract

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.

Normal Integration: A Survey

“…A more general approach to overcome the possible nonintegrability of the gradient field g = [p, q] is to first define a set I of integrable vector fields i.e., of vector fields of the form ∇z, and then compute the projection ∇z of g on I i.e., the vector field ∇z of I the closest to g, according to some norm. Afterwards, the (approximate) solution of Equation (20) is easily obtained using (5), (9) or (18), since ∇z is integrable.…”

“…The way to cope with a possible non-integrable normal field was seen as a property of primary importance in the first papers on normal integration. The most obvious way to solve the problem amounts to use different paths in the integrals of (5), (9) or (18), and to average the different values. Apart from this approach, which has given rise to several heuristics [12,26,59], we propose to separate the main existing normal integration methods into two classes, depending on whether they care about discontinuities or not.…”

“…The integration method proposed by Balzer in [9] is based on second order shape derivatives, allowing for the use of a fast Gauss-Newton algorithm. Hence P Fast is satisfied.…”

Cavlectometry: Towards Holistic Reconstruction of Large Mirror Objects

“…7(c). (2) Observe that there are degenerate cases in which the specular surface itself affords a homography, e.g., if it is locally planar or reflects into a single point (i.e., the light map is constant). We can, however, safely assume that the background occupies a significant amount of the image.…”

Robust surface reconstruction from gradients via adaptive dictionary regularization

“…Additional approaches include line-integral based methods [12,13] and reconstructions based on the calculus of variations [14,15,16]. A range of other methods have also been proposed with mixed results [17,18,19,20,21,22].…”