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 .