Eigenvalue spectra of the Laplace-Beltrami operator have successfully been employed as fingerprints for shape and image comparison. Especially notable in this context is the work of Peinecke on Laplace spectrum finger- printing for image data. Recently, new research on greyscale images by Berger et al. introduces the idea of attributing individual eigenfunctions to image parts and describes a mechanism for controlling their localisation. These parts are separated by sufficiently strong variations of grey value, giving the originally global fingerprint a semi-local character. This paper provides an approach to extend this idea to colour images so that not only gradients of brightness but also gradi- ents of hue or chroma lead to localisation of eigenfunctions. This is accomplished by generalising the eigenfunctions to R^2-valued functions and mapping the colours to symmetric 2×2-matrices. The resulting matrix field is then used to mod- ify the Laplacian. Finally, we present a distance function for comparing eigenvalue-based fingerprints that makes use of eigenfunction colocalization information.

This lecture was given by Prof. F.-E. Wolter at Massachusetts Institute of Technology (MIT) on April 8, 2016.

More information:

http://sandlab.mit.edu/index.php/seminars/147-seminar-by-prof-franz-erich-wolter-sept-11-12-00pm-33-117

This talk will start with a brief overview on current research projects pursued at the speaker´s lab, the Welfenlab. Thereafter, the seminar will focus on two ongoing research projects. The first project deals with computations on geodesics in Riemannian manifolds with an emphasis on singular situations caused by singularities of the geodesic exponential map. In this part of the seminar, I will present computations of focal sets, geodesic joins, medial curves and geodesic Voronoi diagrams. The second project employs some insights from the first one; in this part, I will present new differential geometry-based methods for numerical computations, analysis and visualization with respect to special singularities of dynamical systems governed by differential algebraic equations. In this context, we present an apparently new technique showing that an implicitly-defined manifold can be numerically parameterized. 

This lecture was given by Prof. F.-E. Wolter at University of Michigan on June 1, 2016. 

More information:
http://ece.umich.edu/eecs/etc/events/showevent.cgi?3935

This talk will start with a brief overview on current research projects pursued at the speaker´s lab, the Welfenlab. Thereafter, the seminar will focus on two ongoing research projects. The first project deals with computations on geodesics in Riemannian manifolds with an emphasis on singular situations caused by singularities of the geodesic exponential map. In this part of the seminar, I will present computations of focal sets, geodesic joins, medial curves and geodesic Voronoi diagrams. The second project employs some insights from the first one; in this part, I will present new differential geometry-based methods for numerical computations, analysis and visualization with respect to special singularities of dynamical systems governed by differential algebraic equations. In this context, we present an apparently new technique showing that an implicitly-defined manifold can be numerically parameterized.

We present processing methods and visualization techniques for accurately characterizing and interpreting kinematical data of flexion–extension motion of the knee joint based on helical axes. We make use of the Lie group of rigid body motions and particularly its Lie algebra for a natural representation of motion sequences. This allows to analyze and compute the finite helical axis (FHA) and instantaneous helical axis (IHA) in a unified way without redundant degrees of freedom or singularities. A polynomial fitting based on Legendre polynomials within the Lie algebra is applied to provide a smooth description of a given discrete knee motion sequence which is essential for obtaining stable instantaneous helical axes for further analysis. Moreover, this allows for an efficient overall similarity comparison across several motion sequences in order to differentiate among several cases. Our approach combines a specifically designed patient-specific three-dimensional visualization basing on the processed helical axes information and incorporating computed tomography (CT) scans for an intuitive interpretation of the axes and their geometrical relation with respect to the knee joint anatomy. In addition, in the context of the study of diseases affecting the musculoskeletal articulation, we propose to integrate the above tools into a multiscale framework for exploring related data sets distributed across multiple spatial scales. We demonstrate the utility of our methods, exemplarily processing a collection of motion sequences acquired from experimental data involving several surgery techniques. Our approach enables an accurate analysis, visualization and comparison of knee joint articulation, contributing to the evaluation and diagnosis in medical applications.

Link to the paper in the Journal website      Video regarding the paper  

Educational material regarding the earlier course "Computer Graphics II (Graphische Datenverarbeitung II)", currently not being taught. 

Within Riemannian geometry the geodesic exponential map is an essential tool for various distance-related investigations and computations. Several natural questions can be formulated in terms of its preimages, usually leading to quite challenging non-linear problems. In this context we recently proposed an approach for computing multiple geodesics connecting two arbitrary points on two-dimensional surfaces in situations where an ambiguity of these connecting geodesics is indicated by the presence of focal curves. The essence of the approach consists in exploiting the structure of the associated focal curve and using a suitable curve for a homotopy algorithm to collect the geodesic connections. In this follow-up discussion we extend those constructions to overcome a significant limitation inherent in the previous method, i.e. the necessity to construct homotopy curves artificially. We show that considering homotopy curves meet- ing a focal curve tangentially leads to a singularity that we investigate thoroughly. Solving this so-called geodesic bifurcation analytically and dealing with it numerically provides not only theoretical insights, but also allows geodesics to be used as homotopy curves. This yields a stable computational tool in the context of computing distances. This is applicable in common situations where there is a curvature induced non-injectivity of the exponential map. In particular we illustrate how applying geodesic bifurcation approaches the distance problem on compact manifolds with a single closed focal curve. Furthermore, the presented investigations pro- vide natural initial values for computing cut loci using the medial differential equation which directly leads to a discussion on avoiding redundant computations by combining the presented concepts to determine branching points. Link: http://link.springer.com/article/10.1007/s00371-014-1041-3/fulltext.html

In this work we present computational methods for examining dynamical systems. We focus on those systems being characterized by slow–fast vector fields or corresponding differential algebraic equations that commonly occur in physical applications. In the latter ones scientists usually consider a manifold of admissible physical states and a vector field describing the time evolution of the physical system. The manifold is typically implicitly defined within a higher-dimensional space by a system of equations. Certain physical systems, such as relaxation oscillators, perform sudden jumps in their state evolution when they are forced into an unstable state. The main contribution of the present work is to model the dynamical evolution incorporating the jumping behavior from a perspective of computational geometry which not only provides a qualitative analysis but also produces quantitative results. We use geodesic polar coordinates (GPC) to numerically obtain explicit parametrizations of the implicitly defined manifold and of the relevant jump and hit sets. Moreover, to deal with the possibly high co-dimension of the considered implicitly defined manifold we sketch how GPC in combination with the cut locus concept can be used to numerically obtain an essentially injective global parametrization. This allows us to parametrize and visualize the dynamical evolution of the system including the aforementioned jump phenomena. As main tools we use homotopy approaches in conjunction with concepts from differential geometry. We discuss how to numerically realize and how to apply them to several examples from mechanics, electrical engineering and biology. Link: http://link.springer.com/article/10.1007/s00371-014-1036-0/fulltext.html

This publication is a contribution to basic research in image comparison using eigenvalue spectra as features. The differential-geometric approach of eigenvalue spectrum-based descriptors is naturally applicable to shape data, but so far little work has been done to transfer it to the setting of image data painted on a rectangle or general curved surface. We present a new semi-global feature descriptor that also contains information about geometry of shapes visible in the image. This may not only improve the performance of the resulting distance measures, but may even enable us to approach the partial matching problem using eigenvalue spectra, which were previously only considered as global feature descriptors. We introduce some concepts that are useful in designing and understanding the behaviour of similar fingerprinting algorithms for images (and surfaces) and discuss some preliminary results. Link: http://link.springer.com/article/10.1007/s00371-014-1038-y/fulltext.html

Comprehensive descriptions of the motion in articulating joints open new opportunities in biomedical engineering. The helical axis is a established method that describes flexion-extension at joints, which currently lacks an intuitive visualization. In this paper, we present a comprehensive visualization of knee joint motion based on a direct measurement of the helical axis. The proposed approach incorporates the three-dimensional motion of patient-specific bone segments and the representation of helical axes on the bone, facilitating the observation of the flexion-extension motion at the knee joint.

Injuries of the bony orbit that, which contains the eye, not only have aesthetic implications but also impair stereoscopic vision. Different techniques are used for surgical treatment. In this work we will present a tool that allows the reliable segmentation of the reconstructed and the unaffected orbit. For this we present a novel anterior closing that follows the clinical understanding of anatomy. To assist in the post-surgical evaluation, we suggest parameters with respect to clinically relevant regions (orbital floor, medial wall) and demonstrate a method to automatically determine them. We evaluate our methods on clinical cases.

Date: May, 7, 2015

This document summarizes the current lines of research at Welfenlab, including links to the references.

Accepted as a lecture presentation in 29th International Congress and Exhibition of Computer Assisted Radiology and Surgery (CARS 2015). 

Link: http://www.cars-int.org/fileadmin/templates/download/2015/cars2015_program.pdf

Purpose: Objective determination of the orbital volume is important in the diagnostic process and in evaluating the efficacy of medical and/or surgical treatment of orbital diseases. Tools designed to measure orbital volume with computed tomography (CT) often cannot be used with cone beam CT (CBCT) because of inferior tissue representation, although CBCT has the benefit of greater availability and lower patient radiation exposure. Therefore, a model-based segmentation technique is presented as a new method for measuring orbital volume and compared to alternative techniques.

Methods: Both eyes from thirty subjects with no known orbital pathology who had undergone CBCT as a part of routine care were evaluated (n = 60 eyes). Orbital volume was measured with manual, atlas-based, and model-based segmentation methods. Volume measurements, volume determination time, and usability were compared between the three methods. Differences in means were tested for statistical significance using two-tailed Student’s t tests.

Results: Neither atlas-based (26.63±3.15mm3) nor model-based (26.87±2.99mm3) measurements were significantly different from manual volume measurements (26.65 ± 4.0mm3). However, the time required to determine orbital volume was significantly longer for manual measurements (10.24 ± 1.21 min) than for atlas-based (6.96 ± 2.62 min, p < 0.001) or model-based (5.73 ± 1.12 min, p < 0.001) measurements.

Conclusion: All three orbital volume measurement methods examined can accurately measure orbital volume, although atlas-based and model-based methods seem to be more user-friendly and less time-consuming. The new model-based technique achieves fully automated segmentation results, whereas all atlas-based segmentations at least required manipulations to the anterior closing. Additionally, model-based segmentation can provide reliable orbital volume measurements when CT image quality is poor.

The exploration of biomedical data which involves heterogeneous sources coming from different spatial scales and medical domains is a challenging topic in current research. In this work, we combine efforts regarding multi-scale visualiza-tion, multimodal interaction and knowledge formalization for the exploration of multi-scale biomedical data. The knowledge formalization stores and organizes the information sources, the integrated visualization captures all relevant information for the domain expertise of the user and the multimodal interaction provides a natural exploration. We present a concrete example of use of the proposed exploratory system designed for a biologist investigating multi-scale pathologies.

Seminar titled "Man-Machine-Communication. Interaction with the computer", given at the Rotary Club Hannover on October, 16, 2015.

The study and analysis of relationships in a complex and multi-scale data set is a challenge of information and scientific visualization. This work proposes an integrated visualization to capture all the important aspects of multi-scale data into the same view by leveraging the multi-scale biomedical knowledge encoded into an underly- ing ontology. Ontology supports visualization by providing semantic means to identify relevant items that must be presented to the user. The study and analysis of relationships across the scales are presented as results of queries to the multi-scale biomedical knowledge space. We demonstrate the prototype of the graphical interface of an integrated visualization framework and the knowledge formalization support in an example scenario related to the musculoskeletal diseases.

Pathologies which simultaneously involve different spatial scales are often difficult to understand. Biomedical data from different modalities and spatio-temporal scales needs to be combined to obtain an understandable representation for its examination. Despite of requests to improve the exploration of multi-scale biomedical data, no major progress has been made in terms of a common strategy combining the state of art in visualization and interaction. This work presents a multimodal approach for a natural biomedical multi-scale exploration. The synergy of a multi-layered visualization environment based on spatial scales with hand gestures and haptic interfaces opens new perspectives for a natural data manipulation.

Lecture given at Massachusetts Institute of Technology (september 11, 2014) by Prof. F.-E. Wolter. More information: 

http://sandlab.mit.edu/index.php/seminars/95-seminar-by-prof-franz-erich-wolter-sept-11-12-00pm-33-116

We shall describe how concepts from differential geometry have been providing powerful tools creating major advances in geometric modeling, geometry processing and image analysis dealing with the topics presented in the title of this address. This talk includes a retrospective compiling contributions of the author's works showing how concepts from local and global differential geometry have introduced new methods into geometric modeling and shape interrogation and classification finally ending with modern state of the art research on geometry processing and image processing. A major part of this seminar starting with works in the late nineties at the author's laboratory is dedicated to discussing how "efficient finger prints" useful for indexing and clustering digital data collections can be derived from spectra of Laplace operators being naturally associated with geometric objects such as surfaces and solids as well as (colored) images including medical 2d- and 3d-images. Recently the latter works obtained particular attention in the area of medical imaging. Geometric aspects of the Laplacian operator lead to generalizations of the Laplacian operating on line bundles allowing to compute eigen-functions whose iso-surfaces yield smooth non-orientable Seifert surfaces spanning knot complements in three-dimensional manifolds. Next we focus on cut loci, the medial axis and its inverse in Euclidean and Riemannian worlds.

This work starts with basic medial axis results presented by the the author in the early nineties. Those results state: The Medial Axis Transform can be used to reconstruct, modify and design a given shape ("Shape Reconstruction Theorem"). Under some weak assumptions the medial axis contains the essence of the topological shape of the geometric object as it is a deformation retract of the given shape ("Topological Shape Theorem"). Therefore the medial axis contains the homotopy type of the given shape. We present recent results showing how geodesic Voronoi diagrams, geodesic medial axis and its inverse can be computed in 3d- or higher-dimensional Riemannian spaces. The "medial axis inverse" allows to construct a medial modeler providing efficient features for shape optimization with respect to shape dependent mechanical properties.

Both future air traffic management programs SESAR and NextGen foresee trajectory based operations as one of the major enablers for more efficient handling of air traffic. Furthermore, both initiatives claim to support user preferred trajectories. Using a common airspace it is unlikely to get a conflict-free scenario from independently optimized preferred trajectories. This paper investigates if the concept of user preferred trajectories is feasible, and what constraints are necessary to reach a conflict-free global scenario. Restricting aircraft’s trajectories a priori by forcing them to fly on pre-defined en-route segments, respect semi-circular routes, or apply given departure and arrival times reduces the achievable efficiency, but helps to avoid conflicts and keep the concept simple. On the other hand, trying to combine independently optimized trajectories to a full traffic scenario needs a lot of adaptations to achieve conflict-free operations. Even though the good answer to a problem lies often in-between the extreme solutions, this paper investigates the second approach. Based on a real day of actual traffic over Europe, an optimized sample is generated applying user’s preferred trajectory for each flight. Based on an individually optimized traffic scenario yielding nearly 29,000 conflicts, strategic resolution is performed generating least possible penalties for involved aircraft. The main technique of solving conflicts applied is shifting whole flights in time by a few minutes. Direct and recursive algorithms are presented; a focus is put on solving airport-related conflicts first. The efficiency of the original optimized scenario stays untouched, as lateral, vertical, and speed profiles remain identical; flights are only rescheduled to a slightly earlier or later departure (and arrival) slot in order to minimize conflicts. More than 94% of all conflicts can be solved by shifting whole flights in time respecting a maximum offset of 10 minutes. In a subsequent step, lateral and vertical maneuvers are facilitated; resulting in a resolution for more than 97% of all conflicts.

In the present work, we extend the theoretical and numerical discussion of the well-known Laplace–Beltrami operator by equipping the underlying manifolds with additional structure provided by vector bundles. Focusing on the particular class of flat complex line bundles, we examine a whole family of Laplacians including the Laplace–Beltrami operator as a special case. To demonstrate that our proposed approach is numerically feasible, we describe a robust and efficient finite-element discretization, supplementing the theoretical discussion with first numerical spectral decompositions of those Laplacians.

Our method is based on the concept of introducing complex phase discontinuities into the finite element basis functions across a set of homology generators of the given manifold. More precisely, given an m-dimensional manifold M and a set of n generators that span the relative homology group H_m-1(M, ∂ M), we have the freedom to choose n phase shifts, one for each generator, resulting in a n-dimensional family of Laplacians with associated spectra and eigenfunctions. The spectra and absolute magnitudes of the eigenfunctions are not influenced by the exact location of the paths, depending only on their corresponding homology classes.

Employing our discretization technique, we provide and discuss several interesting computational examples highlighting special properties of the resulting spectral decompositions. We examine the spectrum, the eigenfunctions and their zero sets which depend continuously on the choice of phase shifts.

The well-known Laplace-Beltrami operator, established as a basic tool in shape processing, builds on a long history of mathematical investigations that have induced several numerical models for computational purposes. However, the Laplace-Beltrami operator is only one special case of many possible generalizations that have been researched theoretically. Thereby it is natural to supplement some of those extensions with concrete computational frameworks. In this work we study a particularly interesting class of extended Laplacians acting on sections of flat line bundles over compact Riemannian manifolds. Numerical computations for these operators have recently been accomplished on two-dimensional surfaces. Using the notions of line bundles and differential forms, we follow up on that work giving a more general theoretical and computational account of the underlying ideas and their relationships. Building on this we describe how the modified Laplacians and the corresponding computations can be extended to three-dimensional Riemannian manifolds, yielding a method that is able to deal robustly with volumetric objects of intricate shape and topology. We investigate and visualize the two-dimensional zero sets of the first eigenfunctions of the modified Laplacians, yielding an approach for constructing characteristic well-behaving, particularly robust homology generators invariant under isometric deformation. The latter include nicely embedded Seifert surfaces and their non-orientable counterparts for knot complements.

In this paper, we present a novel method for computing multiple geodesic connections between two arbitrary points on a smooth surface. Our method is based on a homotopy approach that is able to capture the ambiguity of geodesic connections in the presence of positive Gaussian curvature that generates focal curves. Contrary to previous approaches, we exploit focal curves to gain theoretical insights on the number of connecting geodesics and a practical algorithm for collecting these. We consider our method as a contribution to the contemporary debate regarding the calculation of distances in general situations, applying continuous concepts of classical differential geometry which are not immediately transferable in purely discrete settings.

Computing the spectral decomposition of the Laplace–Beltrami operator on a manifold M has proven useful for applications such as shape retrieval and geometry processing. The standard operator acts on scalar functions which can be identified with sections of the trivial line bundle M ×R. In this work we propose to extend the discussion to Laplacians on nontrivial real line bundles. These line bundles are in one-to-one correspondence with elements of the first cohomology group of the manifold with Z₂ coefficients. While we focus on the case of two-dimensional closed surfaces, we show that our method also applies to surfaces with boundaries.
Denoting by β the rank of the first cohomology group, there are 2^β different line bundles to consider and each of these has a naturally associated Laplacian that possesses a spectral decomposition. Using our new method it is possible for the first time to compute the spectra of these Laplacians by a simple modification of the finite element basis functions used in the standard trivial bundle case. Our method is robust and efficient. We illustrate some properties of the modified spectra and eigenfunctions and indicate possible applications for shape processing. As an example, using our method, we are able to create spectral shape descriptors with increased sensitivity in the eigenvalues with respect to geometric deformations and to compute cycles aligned to object symmetries in a chosen homology class.

Die Bestimmung der Vorform- und Zwischenformgeometrien nimmt bei der Auslegung der Stadienfolge für das Gesenkschmieden eine zentrale Stellung ein. Sie erfolgt bislang iterativ und zum größten Teil wissensbasiert. In diesem Beitrag wird ein Ansatz vorgestellt, mit dem das Ziel verfolgt wird, durch die zeitlich rückwärtsschreitende Betrachtung des Schmiedeprozesses die Einlegegeometrie der jeweiligen Vorstufe zu berechnen. Es wird hierfür ein Materialflussmodell angewendet, welches auf der Ermittlung des geometrischen Widerstandes im Schmiedeteil basiert und schnelle Rechenzeiten ermöglicht.

We shall describe how concepts from differential geometry, especially from Riemannian geometry have been providing powerful tools creating major advances in geometric modeling, geometry processing and image analysis dealing with the topics presented in the title of this address. This talk includes a retrospective compiling contributions of the author's works showing how concepts from local and global differential geometry have introduced new methods into geometric modeling and shape interrogation and classification finally ending with modern state of the art research on geometry processing and image processing.

A major part of this seminar is dedicated to discussing how "efficient finger prints" useful for indexing and clustering digital data collections can be derived from spectra of Laplace operators. Laplace operators can be naturally associated with geometric objects such as surfaces and solids as well as (colored) images - including 2d- and 3d-image data. Recently the latter works obtained particular attention in the area of medical imaging.

Next we focus on cut loci, the medial axis and its inverse in Euclidean and Riemannian worlds. This work starts with basic medial axis results presented by the author in the early nineties. Those results state: The Medial Axis Transform can be used to reconstruct, modify and design a given shape ("Shape Reconstruction Theorem"). Under some weak assumptions, the medial axis contains the essence of the topological shape of the geometric object as it is a deformation retract of the given shape ("Topological Shape Theorem"). Therefore the medial axis contains the homotopy type of the given shape. We present recent results showing how geodesic Voronoi diagrams, geodesic medial axis and its inverse can be computed in 3d- or higher dimensional Riemannian spaces. The "medial axis inverse" allows to construct a medial modeler providing efficient features for shape optimization with respect to shape dependent mechanical properties.

This talk includes a retrospective compiling contributions of the author`s works showing how concepts from local and global differential geometry have introduced new methods into CAGD. The seminar is essentially covering two areas surface contact and applications and distance computations and applications.

We start sketching early contributions on curvature analysis in the context of surface contact. Here we are developing criteria for second order surface contact in two different situations, i.e., in case both surfaces have tangential contact along a curve and in case they have tangential contact in a single point. The second result can be used for computing curvature entities of surfaces with a degenerate parametrization (Wolter, 1992). The first result has been used in the context of the constructing curvature continuous blend surfaces. The results of both contact cases have been generalized by the speaker to surface contact of arbitrary order.

The second part of the talk dealing with distance computations is looking into the problems of computing points nearest to a given surface as well as tracing on a surface a curve of foot point being nearest to a point moving on a given space curve. The latter problems offer a natural setting for discussing in the given context:

1. Local computational methods such as tensorial differential equations for computing the orthogonal projection curve (J. Pegna, F.-E. Wolter (1989).
2. (Semi-) global methods, e.g., Vectorfield Index - Method, (Kriezis, Patrikalakis, Wolter, 1991).
3. Global differential geometric methods naturally yielding the concept of Cut Locus / Medial Axis, (e.g. Wolter, 1985,1992).

The aforementioned distance geometric concepts yield applications useful for finding small loops in surface intersections and have been used also in the context of quality control of manufacturing, e.g., submarine propulsors and turbine blades.

The human peripheral auditory system is divided into the outer, middle and inner ear. The acoustic signal innervates the eardrum. The ossicles carry the signal forward to the inner ear. The system of the three ossicles transforms the acoustic signal. Another function of the middle ear is to protect the inner ear against loud noise. In the inner ear, the structure being responsible for hearing is the cochlea. It incorporates the organ of Corti containing the sensory cells that perform the transduction from mechanical excitation into auditory nerve signals. The cochlear can be described physically by three sections filled with fluid and separated by membranes. One of these is the basilar membrane. Acoustic stimulation by external sound is transformed into traveling waves with amplitude peaks at different places on the basilar membrane depending on the frequency of stimulation. An active mechanism including the outer hair cell somatic electro motility can add energy to the basilar membrane by providing positive feedback to the membrane vibrations. This phenomenon of cochlear amplification results in high sensitivity and increased frequency selectivity of the ear.

In the area of image retrieval from data bases and for copyright protection of large image collections there is a growing demand for unique but easily computable fingerprints for images. These fingerprints can be used to quickly identify every image within a larger set of possibly similar images. This paper introduces a novel method to automatically obtain such fingerprints from an image. It is based on a re-interpretation of an image as a Riemannian manifold. This representation is feasible for gray value images and color images. We discuss the use of the spectrum of eigenvalues of different variants of the Laplace operator as a fingerprint and show the usability of this approach in several use cases. Contrary to existing works in this area we do not only use the discrete Laplacian, but also with a particular emphasis the underlying continuous operator. This allows better results in comparing the resulting spectra and deeper insights in the problems arising. We show how the well known discrete Laplacian is related to the continuous Laplace-Beltrami operator. Furthermore we introduce the new concept of solid height functions to overcome some potential limitations of the method.

This paper introduces a method to extract "Shape-DNA", a numerical fingerprint or signature, of any 2d or 3d manifold (surface or solid) by taking the eigenvalues (i.e. the spectrum) of its Laplace-Beltrami operator. Employing the Laplace-Beltrami spectra (not the spectra of the mesh Laplacian) as fingerprints of surfaces and solids is a novel approach. Since the spectrum is an isometry invariant it is independent of the object's representation including parametrization and spatial position. Additionally the eigenvalues can be normalized, so that uniform scaling factors for the geometric objects can be obtained easily. Therefore, checking if two objects are isometric needs no prior alignment (registration/localization) of the objects but only a comparison of their spectra. In this paper we describe the computation of the spectra and their comparison for objects represented by NURBS or other parametrized surfaces (possibly glued to each other), polygonal meshes as well as solid polyhedra. Exploiting the isometry invariance of the Laplace-Beltrami operator, we succeed in computing eigenvalues for smoothly bounded objects without discretization errors caused by approximation of the boundary. Furthermore, we present two non-isometric but isospectral solids that cannot be distinguished by the spectra of their bodies and present evidence that the spectra of their boundary shells can tell them apart. Moreover, we show the rapid convergence of the heat trace series and demonstrate that it is computationally feasible to extract geometrical data such as the volume, the boundary length and even the Euler characteristic from the numerically calculated eigenvalues. This fact not only confirms the accuracy of our computed eigenvalues, but also underlines the geometrical importance of the spectrum. With the help of this "Shape-DNA" it is possible to support copyright protection, database retrieval and quality assessment of digital data representing surfaces and solids.

Seminar at the Brown University on Tuesday the 30th of September 2003. More on our mirror page

The author explains how stable umbilics, the spectra of the Laplace – and of the Laplace-Beltrame-operator and also how the Medial Axis can be used to classify the shape of surfaces and solids. The approach suggesting to distinguish shapes by using the Spectra of the Laplace – and of the Laplace-Beltrami-operator appears to be completely new in the areas of geometric modeling, CAD and Computer Graphics in general. In the second part of the talk it is shown that the Medial Axis transform provides geometric concepts that might be employed to develop intuitive user interfaces useful to mould shape via haptic man/machine interaction.

Seminar at the Brown University on Friday the 11th of February 2000. More on our mirror page.

A novel technique for designing curves on surfaces is presented. The design specifications for this technique derive from other works on curvature continuous surface fairing. Briefly stated, the technique must provide a computationally efficient method for the design of surface curves that is applicable to a very general class of surface formulations. It must also provide means to define a smooth natural map relating two or more surface curves. The resulting technique is formulated as a geometric construction that maps a space curve onto a surface curve. It is designed to be coordinate independent and provides isoparametric maps for multiple surface curves. Generality ofthe formulation is attained by solving a tensorial differential equation formulated in terms of local differential properties of the surfaces. For an implicit surface, the differential equation is solved in three-space. For a parametric surface the tensorial differential equation is solved in the parametric space associated with the surface representation. This technique has been tested on a broad class of examples including polynomials, splines, transcendental parametric and implicit surface representations.

The medial axis transform is a representation of an object which has been shown to be useful in design, interrogation, animation, finite element mesh generation, performance analysis, manufacturing simulation, path planning, and tolerance specification. In this paper, the theory of the medial axis transform for 3-D objects is developed. For objects with piecewise C² boundaries, relationships between the curvature of the boundary and the position of the medial axis are developed. For n -dimensional submanifolds of nwith boundaries which are piecewise C² and completely G¹, a deformation retract is set up between each object and its medial axis, which demonstrates that if the object is path connected, then so is its medial axis. Finally, it is proven that path connected polyhedral solids without cavities have path connected medial axes.

This paper presents a method to compute curvatures of a surface patch S obtained from a degenerate representation defined over a rectangular domain. To compute curvatures at a point q in S where the surface representation is degenerate one has to assume that the point set S has a tangent plane and well defined curvatures at the point q where the curvatures are to be computed. Our computation method employs a local height function representation of S in a neighborhood of the point q where the height function h is defined over the tangent plane of S at the point q . In our method the second order partial derivatives of the height function h at q are computed by using second order derivatives of any three surfaces curves on S which end up in q with pairwise linearly independent tangent directions. The curvature entities of S at q are then computed by using the second order partial derivatives of the height function h at q . We also show how the method is extended to compute partial derivatives of any order of the function h at q . For this purpose we show (in Theorem 1) how the partial derivatives up to n th order of a surface at a point q can be computed using derivatives up to order n of n + 1 surface curves emanating from q . We also present a definition of a concept of generalized surface curvatures under weaker assumptions appropriate for degenerate surfaces. Under those weaker assumptions one does not require a locally C ²-smooth height function representation of the surface over the tangent plane at the degenerate point q , nor a locally well defined (single-valued) height function representation of the surface in a neighborhood of the point q . One only needs the existence of a unique local second order approximation of the degenerate surface with a quadratic function defined over the tangent plane at q . The surface curvatures of the corresponding approximating quadratic surface then define the curvatures of the approximated degenerate surface in the contact point q .