arXiv subjects

Lukas Baumgärtner

Publications and source records attributed to Lukas Baumgärtner.

9 recordsLinked to original sources

arXiv2026

Geometry Denoising with Preferred Normal Vectors

We introduce a new paradigm for geometry denoising using prior knowledge about the surface normal vector. This prior knowledge comes in the form of a set of preferred normal vectors, which we refer to as label vectors. A segmentation problem is naturally embedded in the denoising process. The segmentation is based on the similarity of the normal vector to the elements of the set of label vectors. Regularization is achieved by a total variation term. We formulate a split Bregman (ADMM) approach to solve the resulting optimization problem. The vertex update step is based on second-order shape calculus. We present various examples including the denoising of an eroded medieval gravestone inscription.

cs.CV↗

arXiv2026

Two Models for Surface Segmentation using the Total Variation of the Normal Vector

We consider the problem of surface segmentation, where the goal is to partition a surface represented by a triangular mesh. The segmentation is based on the similarity of the normal vector field to a given set of label vectors. We propose a variational approach and compare two different regularizers, both based on a total variation measure. The first regularizer penalizes the total variation of the assignment function directly, while the second regularizer penalizes the total variation in the label space. In order to solve the resulting optimization problems, we use variations of the split Bregman (ADMM) iteration adapted to the problem at hand. While computationally more expensive, the second regularizer yields better results in our experiments. In particular it removes noise more reliably in regions of constant curvature. In order to mitigate the computational cost, we present a manifold Newton scheme for the most expensive subproblem, which is related to the Riemannian center of mass on a sphere. This significantly improves the computational cost.

cs.CV↗

arXiv2025

Total Generalized Variation of the Normal Vector Field and Applications to Mesh Denoising

We propose a novel formulation for the second-order total generalized variation (TGV) of the normal vector on an oriented, triangular mesh embedded in $\R^3$. The normal vector is considered as a manifold-valued function, taking values on the unit sphere. Our formulation extends previous discrete TGV models for piecewise constant scalar data that utilize a Raviart-Thomas function space. To extend this formulation to the manifold setting, a tailor-made tangential Raviart-Thomas type finite element space is constructed in this work. The new regularizer is compared to existing methods in mesh denoising experiments.

cs.CV↗

arXiv2024

How Stringent is the Linear Independence Kink Qualification in Abs-Smooth Optimization?

Abs-smooth functions are given by compositions of smooth functions and the evaluation of the absolute value. The linear independence kink qualification (LIKQ) is a fundamental assumption in optimization problems governed by these abs-smooth functions, generalizing the well-known LICQ from smooth optimization. In particular, provided that LIKQ holds it is possible to derive optimality conditions for abs-smooth optimization problems that can be checked in polynomial time. Utilizing tools from differential topology, namely a version of the jet-transversality theorem, it is shown that assuming LIKQ for all feasible points of an abs-smooth optimization problem is a generic assumption.

math.OC↗

arXiv2024

Computation of Generalized Derivatives for Abs-Smooth Functions by Backward Mode Algorithmic Differentiation and Implications to Deep Learning

Algorithmic differentiation (AD) tools allow to obtain gradient information of a continuously differentiable objective function in a computationally cheap way using the so-called backward mode. It is common practice to use the same tools even in the absence of differentiability, although the resulting vectors may not be generalized gradients in the sense of Clarke. The paper at hand focuses on objectives in which the non-differentiability arises solely from the evaluation of the absolute value function. In that case, an algebraic condition based on the evaluation procedure of the objective is identified, that guarantees that Clarke gradients are correctly computed without requiring any modifications of the AD tool in question. The analysis allows to prove that any standard AD tool is adequate to drive a stochastic generalized gradient descent method for training a dense neural network with ReLU activations. The same is true for generalized batch gradients or the full generalized gradient, provided that the AD tool makes a deterministic and agnostic choice for the derivative information of the absolute value at 0.

math.OC↗

arXiv2024

Medical Image Registration using optimal control of a linear hyperbolic transport equation with a DG discretization

Patient specific brain mesh generation from MRI can be a time consuming task and require manual corrections, e.g., for meshing the ventricular system or defining subdomains. To address this issue, we consider an image registration approach. The idea is to use the registration of an input magnetic resonance image (MRI) to a respective target in order to obtain a new mesh from a template mesh. To obtain the transformation, we solve an optimization problem that is constrained by a linear hyperbolic transport equation. We use a higher-order discontinuous Galerkin finite element method for discretization and motivate the numerical upwind scheme and its limitations from the continuous weak space--time formulation of the transport equation. We present a numerical implementation that builds on the finite element packages FEniCS and dolfin-adjoint. To demonstrate the efficacy of the proposed approach, numerical results for the registration of an input to a target MRI of two distinct individuals are presented. Moreover, it is shown that the registration transforms a manually crafted input mesh into a new mesh for the target subject whilst preserving mesh quality. Challenges of the algorithm are discussed.

math.NA↗

arXiv2024

Mesh Denoising and Inpainting using the Total Variation of the Normal and a Shape Newton Approach

We present a novel approach to denoising and inpainting problems for surface meshes. The purpose of these problems is to remove noise or fill in missing parts while preserving important features such as sharp edges. A discrete variant of the total variation of the unit normal vector field serves as a regularizing functional to achieve these goals. In order to solve the resulting problem, we use a version of the split Bregman (ADMM) iteration adapted to the problem. A new formulation of the total variation regularizer, as well as the use of an inexact Newton method for the shape optimization step, bring significant speed-up compared to earlier methods. Numerical examples are included, demonstrating the performance of our algorithm with some complex 3D geometries.

math.NA↗

arXiv2023

The Proximal Map of the Weighted Mean Absolute Error

We investigate the proximal map for the weighted mean absolute error function. An algorithm for its efficient and vectorized evaluation is presented. As a demonstration, this algorithm is applied as part of a checkerboard algorithm to solve a total-variation image denoising (ROF) problem as well as a non-smooth energy minimization problem.

math.OC↗

arXiv2022

Total Generalized Variation for Piecewise Constant Functions on Triangular Meshes with Applications in Imaging

We propose a novel discrete concept for the total generalized variation (TGV), which has originally been derived to reduce the staircasing effect in classical total variation (TV) regularization, in image denoising problems. We describe discrete, second-order TGV for piecewise constant functions on triangular meshes, thus allowing the TGV functional to be applied to more general data structures than pixel images, and in particular in the context of finite element discretizations. Particular attention is given to the description of the kernel of the TGV functional, which, in the continuous setting, consists of linear polynomials. We discuss how to take advantage of this kernel structure using piecewise constant functions on triangular meshes. Numerical experiments include denoising and inpainting problems for images defined on non-standard grids, including data from a 3D scanner.

math.NA↗