A Calculus for Non-Smooth Shape Optimization with Applications to Geometric Inverse Problems

Start: 2021-06-15
End: 2024-07-31
Principal Investigators: Roland Herzog, Stephan Schmidt
Staff: Manuel Weiß, Lukas Baumgärtner
Funded by: DFG within the Priority Program funding scheme
Part of: Non-Smooth and Complementarity-based Distributed Parameter Systems (SPP 1962)

Project Description

The overall goal of the project is a mathematically rigorous approach to the theory and numerics of non-smooth shape optimization problems. The objective functions are of a geometric nature, i.e., they emphasize certain desired properties of optimal shapes. In the process, we will examine different classes of functionals, which are, as they stand, suitable for surface smoothing or geometry segmentation tasks, respectively. The non-smoothness of these functionals, which ultimately are all based on the normal vector field of the surface, is a crucial feature in each case for their functioning.

The main motivation for the above considerations are so-called geometric inverse problems, in which an unknown geometry is to be reconstructed from data. Numerous examples of applications for this can be found in the field of non-invasive sensing, for example for the detection of inclusions, but also in medical imaging. The novel geometric functionals we are examining allow a detailed control over expected or desired properties of the geometries to be identified. In the first phase of the project, we mainly considered the total surface variation of the normal vector field in this context, which has the property of being edge preserving.

In the second phase, on the other hand, we first look at functionals that can be used for geometry segmentation, that is, the classification of a surface according to certain features. These functionals can, for instance, help to express a preference for specific orientations of the surface segments. This makes it possible to bring in expert knowledge in the field of crystallography, geology and material science, for example. Furthermore, we consider functionals based on the generalized, second-order total variation of the surface normals. Thereby, a preference for certain curvature properties of the surface can be expressed.

As application examples, in each case problems of electrical impedance tomography (EIT) as a classical imaging modality are to be combined with the new geometric functionals into a geometric inverse problem. On an equal level with the investigation of the theoretical properties is always an efficient and robust numerical realization. To achieve this, we will develop an ADMM method, which will have to incorporate tools of differential geometry due to the intrinsic properties of the surface normal.

Associated Publications

Lukas Baumgärtner, Ronny Bergmann, Roland Herzog, Stephan Schmidt, José Vidal-Núñez and Manuel Weiß

Mesh denoising and inpainting using the total variation of the normal and a shape Newton approach

SIAM Journal on Scientific Computing 47(1), p.A300-A324, 2025

doi:10.1137/24m1646121

bibtex

@ARTICLE{BaumgaertnerBergmannHerzogSchmidtVidalNunezWeiss:2025:1,
  AUTHOR = {Baumgärtner, Lukas and Bergmann, Ronny and Herzog, Roland and Schmidt, Stephan and Vidal-Núñez, José and Weiß, Manuel},
  PUBLISHER = {Society for Industrial \& Applied Mathematics (SIAM)},
  DATE = {2025-01},
  DOI = {10.1137/24m1646121},
  JOURNALTITLE = {SIAM Journal on Scientific Computing},
  NUMBER = {1},
  PAGES = {A300--A324},
  TITLE = {Mesh denoising and inpainting using the total variation of the normal and a shape Newton approach},
  VOLUME = {47},
}

Lukas Baumgärtner, Ronny Bergmann, Roland Herzog, Stephan Schmidt and Manuel Weiß

Two models for surface segmentation using the total variation of the normal vector, 2024

arXiv:2412.00445

bibtex

@ONLINE{BaumgaertnerBergmannHerzogSchmidtWeiss:2024:1,
  AUTHOR = {Baumgärtner, Lukas and Bergmann, Ronny and Herzog, Roland and Schmidt, Stephan and Weiß, Manuel},
  DATE = {2024-12},
  EPRINT = {2412.00445},
  EPRINTTYPE = {arXiv},
  TITLE = {Two models for surface segmentation using the total variation of the normal vector},
}

Lukas Baumgärtner, Roland Herzog, Stephan Schmidt and Manuel Weiß

The proximal map of the weighted mean absolute error

Proceedings in Applied Mathematics and Mechanics 23(4), 2023

arXiv:2209.13545 doi:10.1002/pamm.202300042

bibtex

@ARTICLE{BaumgaertnerHerzogSchmidtWeiss:2023:1,
  AUTHOR = {Baumgärtner, Lukas and Herzog, Roland and Schmidt, Stephan and Weiß, Manuel},
  PUBLISHER = {Wiley},
  DATE = {2023-09},
  DOI = {10.1002/pamm.202300042},
  EPRINT = {2209.13545},
  EPRINTTYPE = {arXiv},
  JOURNALTITLE = {Proceedings in Applied Mathematics and Mechanics},
  NUMBER = {4},
  TITLE = {The proximal map of the weighted mean absolute error},
  VOLUME = {23},
}

Lukas Baumgärtner, Ronny Bergmann, Roland Herzog, Stephan Schmidt and José Vidal-Núñez

Total generalized variation for piecewise constant functions on triangular meshes with applications in imaging

SIAM Journal on Imaging Sciences 16(1), p.313-339, 2023

arXiv:2206.12331 doi:10.1137/22m1505281

bibtex

@ARTICLE{BaumgaertnerBergmannHerzogSchmidtVidalNunez:2023:1,
  AUTHOR = {Baumgärtner, Lukas and Bergmann, Ronny and Herzog, Roland and Schmidt, Stephan and Vidal-Núñez, José},
  PUBLISHER = {Society for Industrial \& Applied Mathematics (SIAM)},
  DATE = {2023-02},
  DOI = {10.1137/22m1505281},
  EPRINT = {2206.12331},
  EPRINTTYPE = {arXiv},
  JOURNALTITLE = {SIAM Journal on Imaging Sciences},
  NUMBER = {1},
  PAGES = {313--339},
  TITLE = {Total generalized variation for piecewise constant functions on triangular meshes with applications in imaging},
  VOLUME = {16},
}

Lukas Baumgärtner, Roland Herzog, Stephan Schmidt and Manuel Weiß

The proximal map of the weighted mean absolute error, 2023

doi:10.5281/zenodo.7620815

bibtex

@MISC{BaumgaertnerHerzogSchmidtWeiss:2022:2,
  AUTHOR = {Baumgärtner, Lukas and Herzog, Roland and Schmidt, Stephan and Weiß, Manuel},
  DATE = {2023},
  DOI = {10.5281/zenodo.7620815},
  TITLE = {The proximal map of the weighted mean absolute error},
}