# Second-Order Shape Optimization for Geometric Inverse Problems in Vision

@article{Balzer2014SecondOrderSO, title={Second-Order Shape Optimization for Geometric Inverse Problems in Vision}, author={Jonathan Balzer and Stefano Soatto}, journal={2014 IEEE Conference on Computer Vision and Pattern Recognition}, year={2014}, pages={3850-3857} }

We develop a method for optimization in shape spaces, i.e., sets of surfaces modulo re-parametrization. Unlike previously proposed gradient flows, we achieve superlinear convergence rates through an approximation of the shape Hessian, which is generally hard to compute and suffers from a series of degeneracies. Our analysis highlights the role of mean curvature motion in comparison with first-order schemes: instead of surface area, our approach penalizes deformation, either by its Dirichlet…

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## References

SHOWING 1-10 OF 37 REFERENCES

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

- Computer ScienceJournal of Mathematical Imaging and Vision
- 2011

A novel approximation of the shape Hessian is proposed, which is not only rigorously justified but also leads to excellent numerical performance of the actual optimization.

An Integral Solution to Surface Evolution PDEs Via Geo-cuts

- Computer ScienceECCV
- 2006

This work formulate an optimization problem directly based on an integral characterization of gradient flow as an infinitesimal move of the (whole) surface giving the largest energy decrease among all moves of equal size and shows that this problem can be efficiently solved using recent advances in algorithms for global hypersurface optimization.

Weighted Minimal Hypersurface Reconstruction

- MathematicsIEEE Transactions on Pattern Analysis and Machine Intelligence
- 2007

This work derives the Euler-Lagrange equation in arbitrary dimensional space without the need for any surface parameterization, generalizing existing proofs and opening up the possibility of solving problems involving minimal hypersurfaces in a dimension higher than three, which were previously impossible to solve in practice.

Weighted Minimal Surface Reconstruction

- Mathematics
- 2005

Many problems in computer vision can be formulated as a minimization problem for an energy functional. If this functional is given as an integral of a scalar-valued weight function over an unknown…

ℓ1-Sparse reconstruction of sharp point set surfaces

- Computer ScienceTOGS
- 2010

An ℓ1-sparse method for the reconstruction of a piecewise smooth point set surface that consists mainly of smooth modes, with the residual of the objective function strongly concentrated near sharp features.

Continuous Global Optimization in Multiview 3D Reconstruction

- Computer ScienceInternational Journal of Computer Vision
- 2009

A new global optimization method to the field of multiview 3D reconstruction is introduced to cast the problem of 3D shape reconstruction as one of minimizing a spatially continuous convex functional.

Discrete Willmore flow

- Computer ScienceSGP '05
- 2005

The relevant gradient expressions including a linearization (approximation of the Hessian) are derived which are required for non-linear numerical solvers and demonstrate the utility of this approach for surface restoration, n-sided hole filling, and non-shrinking surface smoothing.

A Geometric Formulation of Gradient Descent for Variational Problems with Moving Surfaces

- MathematicsScale-Space
- 2005

The manifold of admissible surfaces and a scalar product on its tangent spaces is introduced, which makes it possible to properly define gradients and gradient descent procedures for variational problems involving m-surfaces.

Multiview normal field integration using level set methods

- Computer Science2007 IEEE Conference on Computer Vision and Pattern Recognition
- 2007

This paper forms this multiview normal integration problem by an energy minimization framework and finds an optimal solution in a least square sense using a variational technique and shows that the resultant flow is composed of the well known mean curvature and flux maximizing flows.

Gradient Flows for Optimizing Triangular Mesh-based Surfaces: Applications to 3D Reconstruction Problems Dealing with Visibility

- Computer ScienceInternational Journal of Computer Vision
- 2010

This article shows how to rigorously account for visibility in the surface optimization process, and presents different applications including 3D reconstruction from multiple views for which the visibility is fundamental.