arXiv subjects

Ralf Hielscher

Publications and source records attributed to Ralf Hielscher.

18 recordsLinked to original sources

arXiv2026

A Harmonic Framework for Vector Fields and Differential Operators on SO(3)

We present a comprehensive framework for tangent vector fields and differential operators on the rotation group $\mathrm{SO}(3)$ using harmonic series expansions, which provides a mathematical foundation for their implementation in the crystallographic texture analysis software MTEX. The central idea is to employ the standard left- and right-invariant frames as global orthonormal frames of the tangent bundle, thereby avoiding the numerical instabilities of the classical tangent space basis derived from the Jacobian of the Euler angle parametrization. While the choice of frames is primarily motivated by their geometric and numerical properties, their full potential emerges in the harmonic setting. Representing tangent vector fields through harmonic expansions of their frame components, we derive explicit frequency domain formulas for the gradient, divergence, and curl, allowing these operators to be applied directly to the harmonic coefficients. Moreover, we show that these differential operators preserve harmonic band-limitedness. In addition, the left- and right-invariant representations of tangent vector fields can be transformed into one another directly in the frequency domain, with an increase in harmonic bandwidth of at most one degree.

math.NA↗

arXiv2026

On the Role of the Double Fourier Sphere Method in Fast Algorithms on SO(3)

We analyze the Double Fourier Sphere (DFS) method on the rotation group $\mathcal{SO}(3)$ in the frequency domain and demonstrate its central role in fast algorithms. Fast Fourier algorithms on $\mathcal{SO}(3)$ are commonly formulated as a Wigner transform - mapping harmonic to Fourier coefficients - followed by a Fourier transform. We revisit this formulation and interpret the Wigner transform as an explicit realization of the DFS method, lifting functions from $\mathcal{SO}(3)$ to $\mathbb{T}^3$. In this context, we analyze the Sobolev regularity loss induced by this lifting. Furthermore, we compare different Wigner transform implementations, examine additional symmetry enhancements, and observe that the direct method is often faster and more stable than the fast polynomial transform approaches.

math.NA↗

arXiv2026

A unified framework for grain boundary distributions in textured materials

Grain boundary plane distributions are widely used to infer the mechanisms governing grain boundary formation in polycrystalline materials. We show that such interpretations are inherently ambiguous. Using a unified eight-parameter boundary distribution framework, we derive both the grain boundary character distribution (GBCD) and the grain boundary normal distribution (GBND) and identify two limiting cases of boundary network formation. We show that in macroscopically driven networks, the crystal-frame GBND is given by a convolution of the specimen GBND with the orientation distribution function (ODF), whereas in crystallographically driven networks the specimen GBND is obtained by convolution of the crystal GBND with the ODF. This duality implies that anisotropy in the GBND may arise from macroscopic alignment effects rather than intrinsic crystallographic selection. Conversely, this relationship may be used to identify the dominant formation process in the measured mcirostructures. Evaluation of a wide variety of simulated microstructures confirm the theoretically predicted relationships between texture, GBND and GBCD. In particular, our examples confirm that the GBND or GBCD alone are not sufficient for identifying grain boundary formation mechanisms.

cond-mat.mtrl-sci↗

arXiv2024

An Optimal Ansatz Space for Moving Least Squares Approximation on Spheres

We revisit the moving least squares (MLS) approximation scheme on the sphere $\mathbb S^{d-1} \subset \mathbb R^d$, where $d>1$. It is well known that using the spherical harmonics up to degree $L \in \mathbb N$ as ansatz space yields for functions in $\mathcal C^{L+1}(\mathbb S^{d-1})$ the approximation order $\mathcal O \left( h^{L+1} \right)$, where $h$ denotes the fill distance of the sampling nodes. In this paper we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degree up to $L$, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as $h \to 0$. Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space as ansatz space.

math.NA↗

arXiv2023

Habit plane determination from reconstructed parent phase orientation maps

This study details the development and validation of a new algorithm that determines the dominant habit plane of a transformed child phase from orientation maps of a single planar cross-section. The method describes the habit plane in terms of its five-parameter grain boundary character and couples it to the specific orientation relationship of the identified orientation variant. The symmetry operations associated with the specific orientation relationship of the variants are applied to transform habit plane traces as determined in the specimen-fixed reference frame into the parent or child reference frame, allowing for the fitting of the habit plane. Our algorithm stands out by its robustness, computational efficiency, automation and ability to operate on fully transformed microstructures. Four automated methods for habit plane trace determination are proposed and compared. Detailed sensitivity analysis reveals that the proposed algorithm is exceptionally robust against poor accuracy in the measured traces and distortions in the orientation map, but more sensitive to inaccuracies propagated from parent grain reconstruction. Validation on a synthetic microstructure with a known habit plane and returned consistent results when applied to high and low carbon steels with different prior austenite grain sizes and orientation map resolutions. The habit planes were not significantly affected by the austenite grain sizes. The habit plane of the steel with 0.35 wt.% C was close to (111)γ whereas the habit plane of steel with 0.71 wt.% C was closer to (575)γ, in close agreement with previous work using two-surface stereological analysis and transmission electron microscopy-based trace analysis.

cond-mat.mtrl-sci↗

arXiv2023

Marcinkiewicz--Zygmund inequalities for scattered and random data on the $q$-sphere

The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz--Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the $q$-dimensional sphere $\mathbb{S}^q$, and investigate how well continuous $L_p$-norms of polynomials $f$ of maximum degree $n$ on the sphere $\mathbb{S}^q$ can be discretized by positively weighted $L_p$-sum of finitely many samples, and discuss the relationship between the offset between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points $ξ_1,\ldots,ξ_N$ on $\mathbb{S}^q$, the dimension $q$, and the polynomial degree $n$.

math.NA↗

arXiv2022

Approximating the Derivative of Manifold-valued Functions

We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known that the approximation error for manifold-valued functions is close to the approximation error for vector-valued functions. This is not true anymore if we consider the derivatives of such functions. In our paper we give pre-asymptotic error bounds for the approximation of the derivative of manifold-valued function. In particular, we provide explicit constants that depend on the reach of the embedded manifold.

math.NA↗

arXiv2022

The variant graph approach to improved parent grain reconstruction

The variant graph is a new, hybrid algorithm that combines the strengths of established global grain graph and local neighbor level voting approaches, while alleviating their shortcomings, to reconstruct parent grains from orientation maps of partially or fully phase-transformed microstructures. The variant graph algorithm is versatile and is capable of reconstructing transformation microstructures from any parent-child combination by clustering together child grains based on a common parent orientation variant. The main advantage of the variant graph over the grain graph is its inherent ability to more accurately detect prior austenite grain boundaries. A critical examination of Markovian clustering and neighbor level voting as methods to reconstruct prior austenite orientations is first conducted. Following this, the performance of the variant graph algorithm is showcased by reconstructing the prior austenite grains and boundaries from an example low-carbon lath martensite steel microstructure. Programmatic extensions to the variant graph algorithm for specific morphological conditions and the merging of variants with small mutual disorientation angles are also proposed. The accuracy of the reconstruction and the computational performance of the variant graph algorithm is either on-par or outperforms alternate methods for parent grain reconstruction. The variant graph algorithm is implemented as a new addition to the functionalities for phase transformation analysis in MTEX 5.8 and is freely available for download by the community.

cond-mat.mtrl-sci↗

arXiv2021

Concentration Inequalities for Cross-validation in Scattered Data Approximation

Choosing models from a hypothesis space is a frequent task in approximation theory and inverse problems. Cross-validation is a classical tool in the learner's repertoire to compare the goodness of fit for different reconstruction models. Much work has been dedicated to computing this quantity in a fast manner but tackling its theoretical properties occurs to be difficult. So far, most optimality results are stated in an asymptotic fashion. In this paper we propose a concentration inequality on the difference of cross-validation score and the risk functional with respect to the squared error. This gives a pre-asymptotic bound which holds with high probability. For the assumptions we rely on bounds on the uniform error of the model which allow for a broadly applicable framework. We support our claims by applying this machinery to Shepard's model, where we are able to determine precise constants of the concentration inequality. Numerical experiments in combination with fast algorithms indicate the applicability of our results.

math.NA↗

arXiv2021

An Image Registration Model in Electron Backscatter Diffraction

Recently, variational methods were successfully applied for computing the optical flow in gray and RGB-valued image sequences. A crucial assumption in these models is that pixel-values do not change under transformations. Nowadays, modern image acquisition techniques such as electron backscatter tomography (EBSD), which is used in material sciences, can capture images with values in nonlinear spaces. Here, the image values belong to the quotient space $\text{SO}(3)/ \mathcal S$ of the special orthogonal group modulo the discrete symmetry group of the crystal. For such data, the assumption that pixel-values remain unchanged under transformations appears to be no longer valid. Hence, we propose a variational model for determining the optical flow in $\text{SO}(3)/\mathcal S$-valued image sequences, taking the dependence of pixel-values on the transformation into account. More precisely, the data is transformed according to the rotation part in the polar decomposition of the Jacobian of the transformation. To model non-smooth transformations without obtaining so-called staircasing effects, we propose to use a total generalized variation like prior. Then, we prove existence of a minimizer for our model and explain how it can be discretized and minimized by a primal-dual algorithm. Numerical examples illustrate the performance of our method.

math.OC↗

arXiv2021

The cone-beam transform and spherical convolution operators

The cone-beam transform consists of integrating a function defined on the three-dimensional space along every ray that starts on a certain scanning set. Based on Grangeat's formula, Louis [2016, Inverse Problems 32 115005] states reconstruction formulas based on a new generalized Funk-Radon transform on the sphere. In this article, we give a singular value decomposition of this generalized Funk-Radon transform. We use this result to derive a singular value decomposition of the cone-beam transform with sources on the sphere thus generalizing a result of Kazantsev [2015, J. Inverse Ill-Posed Probl. 23(2):173-185].

math.NA↗

arXiv2021

Parent grain reconstruction from partially or fully transformed microstructures in MTEX

A versatile generic framework for parent grain reconstruction from fully or partially transformed child microstructures was integrated into the open-source crystallographic toolbox MTEX. The framework extends traditional parent grain reconstruction, phase transformation and variant analysis to all parent-child crystal symmetry combinations. The inherent versatility of the universally applicable parent grain reconstruction methods, and the ability to conduct in-depth variant analysis are showcased via example workflows that can be programmatically modified by users to suit their specific applications. This is highlighted by three applications namely, $α$-to-$γ$ reconstruction in a lath martensitic steel, $α$-to-$β$ reconstruction in a Ti alloy, and a two-step reconstruction from $α$-to-$\varepsilon$-to-$γ$ in a twinning and transformation -induced plasticity steel. Advanced orientation relationship discovery and analysis options, including variant analysis, is demonstrated via the add-on function library, ORTools.

cond-mat.mtrl-sci↗

arXiv2020

Locally Isometric Embeddings of Quotients of the Rotation Group Modulo Finite Symmetries

The analysis of manifold valued data using embedding based methods is linked to the problem of finding suitable embeddings. In this paper we are interested in embeddings of quotient manifolds $\mathrm{SO}(3)/\mathcal{S}$ of the rotation group modulo finite symmetry groups. Data on such quotient manifolds naturally occur in crystallography, material science and biochemistry. We provide a generic framework for the construction of such embeddings which generalizes the embeddings constructed in arXiv:1701.01579. The central advantage of our larger class of embeddings is that it comprises isometric embeddings for all crystallographic symmetry groups.

math-ph↗

arXiv2020

Fast Cross-validation in Harmonic Approximation

Finding a good regularization parameter for Tikhonov regularization problems is a though yet often asked question. One approach is to use leave-one-out cross-validation scores to indicate the goodness of fit. This utilizes only the noisy function values but, on the downside, comes with a high computational cost. In this paper we present a general approach to shift the main computations from the function in question to the node distribution and, making use of FFT and FFT-like algorithms, even reduce this cost tremendously to the cost of the Tikhonov regularization problem itself. We apply this technique in different settings on the torus, the unit interval, and the two-dimensional sphere. Given that the sampling points satisfy a quadrature rule our algorithm computes the cross-validations scores in floating-point precision. In the cases of arbitrarily scattered nodes we propose an approximating algorithm with the same complexity. Numerical experiments indicate the applicability of our algorithms.

math.NA↗

arXiv2019

Gazing at crystal balls: Electron backscatter diffraction pattern analysis and cross correlation on the sphere

We present spherical analysis of electron backscatter diffraction (EBSD) patterns with two new algorithms: (1) band localisation and band profile analysis using the spherical Radon transform; (2) orientation determination using spherical cross correlation. These new approaches are formally introduced and their accuracies are determined using dynamically simulated patterns. We demonstrate their utility with an experimental dataset obtained from ferritic iron. Our results indicate that the analysis of EBSD patterns on the surface of the sphere provides an elegant method of revealing information from these rich sources of crystallographic data.

cond-mat.mtrl-sci↗

arXiv2019

Standard-Based EBSD: Fingerprinting of Order and Orientation in Materials

Orientation determination does not necessarily require complete knowledge of the local atomic arrangement in a material. We present a method for microstructural phase discrimination and orientation analysis of phases for which there is only limited information available. In this method, experimental Kikuchi diffraction patterns are utilized to generate self-consistent standards for use in the technique of Electron Backscatter Diffraction (EBSD). As an application example, we map the locally varying orientations in samples of icosahedral quasicrystals observed in a Ti40Zr40Ni20 alloy.

cond-mat.mtrl-sci↗

arXiv2017

An SVD in Spherical Surface Wave Tomography

In spherical surface wave tomography, one measures the integrals of a function defined on the sphere along great circle arcs. This forms a generalization of the Funk--Radon transform, which assigns to a function its integrals along full great circles. We show a singular value decomposition (SVD) for the surface wave tomography provided we have full data. Since the inversion problem is overdetermined, we consider some special cases in which we only know the integrals along certain arcs. For the case of great circle arcs with fixed opening angle, we also obtain an SVD that implies the injectivity, generalizing a previous result for half circles in [Groemer, On a spherical integral transform and sections of star bodies, Monatsh. Math., 126(2):117--124, 1998]. Furthermore, we derive a numerical algorithm based on the SVD and illustrate its merchantability by numerical tests.

math.NA↗

arXiv2015

Restoration of Manifold-Valued Images by Half-Quadratic Minimization

The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete Riemannian manifold. We recall the half-quadratic minimization method using the notation of the c-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group SO(3) and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with SO(3)-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.

math.NA↗