Laplace–Beltrami spectra as ‘Shape-DNA’ of surfaces and solids

Abstract

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.

A patent application based on ideas presented in this paper is pending.

Introduction

The characterization and design of the shape of 3d-objects are central problems in computer graphics and geometric modeling. The development of software and hardware tools to design and visualize the shape of 3d-objects has advanced rapidly during the past twenty years. Nonetheless, fundamental problems pertaining to the characterization of shape are still widely unresolved. It is, for example, a basic question to quickly identify and retrieve a given object stored in a huge database or to find all similarly shaped objects. During the past forty years, a vast number of shape matching and searching techniques have been developed (e.g. using moments, spherical harmonics or Reeb graphs—a recent survey can be found in Iyer et al. [39], see also [30]). It should be pointed out that most approaches dealing with shape matching describe procedures to realign the geometric objects, usually called localization or registration (cf. [53], [60]), and work only on a specific representation (mainly polygonal meshes) of the object. Other techniques describe subdivision or decomposition of an object into smaller features (e.g. [9] or [37]) that are then compared in a second step.

The point-set of a solid 3d-object with smooth boundary may be described in very different ways (cf. [65]), for example in boundary representation (B-Rep) using NURBS surface patches. This may cause difficulties to decide if two objects have the same or different shapes. Even when restricted to NURBS surfaces, it is not easy to decide if the given objects are similar in their shape. A simple comparison of the control points used to represent the boundary surfaces does not help at all, because identical patches can be represented with different control points. Both patches first need to be represented with the same basis functions implying equal knot vectors and equal degrees of the employed NURBS basis functions. The problem becomes even more complicated if we consider the possibility that the solid's boundary surfaces might be represented in various other ways, e.g. by trigonometric functions, by implicitly defined functions or by polygonal meshes (planar polygonal faces), that have to be compared with each other.

In some of these cases, the problem of identifying shapes (for example, to protect the copyright of the designer) has been approached with the help of watermarks. For this purpose, visible or invisible watermark information is embedded into the representation or geometry of an object. Later on, this information can be retrieved and the object can be identified. This is of special interest when dealing with delicate high precision material, e.g. turbine blades, whose design needs major research effort and expensive investments. Even though NURBS patches are very popular today, most watermark techniques deal with polygonal meshes only. Often the watermark data is embedded into these meshes by slightly modifying the vertex location, the connectivity of the mesh or the frequency domain employing mesh-spectral analysis (cf. [6], [49]). For NURBS surfaces watermarking is more difficult and only very few algorithms exist. An algorithm proposed by Ohbuchi et al. [48] does not change the surface, but is not very robust. Generally, watermarks can be destroyed by a representation change or by a reparametrization of the object, if they are not embedded into the geometry. On the other hand, embedding data into the geometry rather than into the representation changes the shape of the object which is unacceptable in many cases. It should be noted that the watermarking technique is limited to the comparison of watermark information, which in general is not related to the shape. Therefore, it cannot be used for shape matching but only for the identification of previously marked objects.

A superior identification method avoiding both problems of watermarking (i.e. geometry changes or representation dependency) is to identify the shape of an object by geometric invariants that we will call fingerprints or signatures. An example for a fingerprint of shape intrinsic information to identify shape via registration/alignment of umbilics can be found in [42] or [43]. However, our approach is different because we use sets of geometric invariants that are sufficiently complete to identify isometric objects so we can avoid realignment procedures as a safeguard for tests of identical shape. Of course, fingerprint techniques cannot distinguish between several copies of the same object, since they only depend on the shape. In such a case, watermarks have to be applied in order to discriminate identical copies. Nevertheless, an advantage of the use of shape related fingerprints is that shape can be compared indirectly through the fingerprints (especially if similar shapes lead to similar fingerprints). In addition to shape identification, the fingerprint technique can therefore be used for shape matching.

Shape intrinsic information does not depend on the given representation of the shape and can be understood as a fingerprint of the shape (if enough information is contained). Many geometric shape invariants (e.g. circumference, surface area, volume, bounding sphere or eigenvalues of the inertia tensor) have strong limitations with respect to the amount of completeness up to which these invariants determine the shape of an object. Therefore, we propose the following properties that optimally should be fulfilled by a shape fingerprint (e.g. a vector of numbers/shape invariants associated with the given object):

• Congruent solids (or isometric surfaces) should have the same fingerprint independently of the object's given representation and location. Therefore, the fingerprint should be an isometry invariant.

• For some applications, it is necessary that the fingerprint is independent of the object's size, therefore the fingerprint should optionally be scaling invariant.

• Similarly, shaped solids should have similar fingerprints. The fingerprint should depend continuously on shape deformations.

• The effort needed to compute those fingerprints should be reasonable.

• Ideally, those fingerprints should give a complete characterization of the shape, thus representing the shape uniquely. One step further it would be desirable that those fingerprints could be used to reconstruct the solid.

• In addition it would also be desirable that the fingerprint data should not be redundant, i.e. a part of it could not be computed from the rest of the data.

• Furthermore, it would be nice if an intuitive geometric or physical interpretation of the meaning of the fingerprints would be available.

Concerning property [ISOMETRY] let us give the following definitions:

Definition 1 Isometry

Two geometric objects are isometric if a homeomorphism from one to the other exists preserving (geodesic) distances, i.e. mapping curves to curves with equal arc length. This homeomorphism is then called an isometry.

Definition 2 Congruency

Two geometric objects are congruent if they can be transformed into each other by rigid motions (translations and rotations) as well as reflections.

It should be noted that isometric planar domains in 2d and isometric solids in 3d Euclidean space are already uniquely determined in their respective space up to rigid motions and reflections. For planar shapes and 3d-solids congruency and isometry are the same. Surfaces on the other hand that are bend or folded without stretching (without changing the metric) stay isometric even though they are not congruent. The property [ISOMETRY] (and of course [SIMILARITY]) is important in situations where near isometric surfaces like hands with different finger positions or faces with different expressions are to be compared and identified. See, e.g. [27] for a method using discrete geodesic distances and multidimensional scaling to generate similar signature surfaces (that still need to be aligned for final comparison), and see [14] for an application to face recognition. An isometry invariant fingerprint is often desired in shape matching, since it depends

• only on the (intrinsic) shape, independent of any representation;

• not on the actual embedding and is therefore independent of the spatial position and isometric deformation of the object.

Very often invariants are used to classify objects. For example in knot theory, a branch of topology, knot invariants (e.g. the Alexander polynomial or the more recent Jones and the Homfly polynomials) are used to distinguish knotted space curves (cf. [1]). For surfaces there exist, for example, topological invariants (cf. [57]) such as homotopy invariants including, e.g. homotopy groups and homology groups or the well-known Euler characteristic. Another important topological invariant of a manifold is its orientability (being a non-homotopic invariant, because the non-orientable Moebius strip and the orientable cylinder have the circle as deformation retract and are therefore homotopic to it). Although all these topological shape invariants are interesting and useful, they cannot distinguish any two homeomorphic objects such as two 3d-solids, e.g. obtained by deforming a topological full 3d-disk.

There exist theoretical invariants determining the isometry type of a surface or solid completely up to isometry. Indeed, the first fundamental tensor (defined independently of a parametrization) is a complete isometry invariant [25]. However, this invariant can generally not be used to check if two given parametrizations represent isometric manifolds. In order to compare this invariant for two objects, they first have to be parametrized on a common parameter space. These parametrizations have to be constructed in a way that they map the same point in parameter space to the two corresponding points on each manifold. This task is as difficult as finding the isometry itself, which generally is a very difficult problem. The first fundamental tensor can be used to check if a diffeomorphism is an isometry. But even this task is difficult, as it requires checking every point.

A manifold can theoretically be determined completely up to translation and rotation by the first and the second fundamental tensor [25]. Like before, those two tensors do not provide an easy tool to check if two distinct parametrizations refer to congruent manifolds. The medial axis transform (MAT) (defined uniquely for a solid body) provides a shape invariant that is a complete shape descriptor (cf. reconstruction theorem [64], [65], [66]). Using the MAT for testing the congruence of two given 3d-solids would also require checking if the respective medial axis sets (usually collections of surface patches) are congruent, a task that again is not easy at all in general. Therefore, all shape invariants listed above cannot be used efficiently to detect if two distinct geometrical object representations refer to congruent or isometric objects. We think that this paper offers a remedy for the aforementioned difficulties occurring in shape comparison problems. This holds because the shape invariants presented in this paper can be used efficiently for shape comparison (once they have been computed).

This paper proposes to use the sequence of eigenvalues (spectrum) of the Laplace operator of a planar domain or 3d-solid or the Laplace–Beltrami operator of a surface or parametrized solid in Euclidean space as a fingerprint. The Laplace operator can be seen as the special case of the Laplace–Beltrami operator with a Euclidean metric. These Laplace operators are linear differential operators defined on a corresponding vector space of differentiable functions, the latter being defined on a domain in Euclidean space or on a Riemannian manifold, respectively. Those differentiable functions are supposed to be zero on the boundary of the surface or of the domain in case the boundary is not empty (Dirichlet boundary condition). The Neumann boundary condition forces their derivatives in the normal direction of the boundary curve to be a fixed function or to be constantly zero. The Laplace operators assign the trace of their Hessian to the latter functions (defined on the domain or on the surface). In the surface case, the Hessian must be defined invariantly of the surface parametrization using only the Riemannian metric of the surface.

This fingerprint (i.e. the eigenvalues) can be calculated for different object representations in different dimensions and can even be calculated for grayscale or color images. We consider a gray scale image as a surface defined by the graph of a height function being the gray scale intensity function of the image. The color image can, e.g. be understood as a surface (two-manifold) in a five-dimensional Euclidean space whose coordinates include the intensity parameters of the red, green, blue values assigned to any (x, y) pixel of the image. It is possible as well to understand other even higher dimensional signals as height functions and therefore as manifolds, whose Laplace–Beltrami spectra can be computed. Another advantage of this method is that it can even be applied to solids containing cavities (solids bounded by several not connected surfaces), for example, an ice-cube containing fully enclosed bubbles. Most techniques only working on boundary representations, not on the solid itself, have difficulties with several boundary components. With our method, one can compare the 2d boundary as well as the 3d volume for two given solids.

The fingerprint presented here fulfills the desired properties above (with the only exception of [COMPLETENESS]). Since the eigenvalues are isometry invariants, this fingerprint is independent of the objects representation (especially its parametrization), its spatial position and, as we will see later, even of the object's size [SCALING] (if desired). This isometry invariance makes registration or localization of the objects completely unnecessary. The isometry invariance is very restrictive compared to the topological invariance. As mentioned before, isometry even determines the congruence of objects in important cases such as planar shapes or 3d-solids. In other words, if limited to these very common solid objects, their shapes are uniquely determined by their isometry class. The fingerprint proposed here consists of a family of non-negative numbers (the eigenvalues) that can be compared easily and fast, permitting this approach to be used in time-critical applications such as database retrieval. Because the spectrum of the Laplace–Beltrami operator contains intrinsic shape information we call it ‘Shape-DNA’. We will show that this Shape-DNA can be used (like DNA-tests) to identify objects in practical applications. As in real life, the DNA does not completely characterize a subject. As we will discuss later identical twins exist with different shape but exactly the same Shape-DNA. Even though these twins are shaped differently they still have quite a few common geometric properties (exactly those properties that are determined by the spectrum). It should be noted that in real life human fingerprints (determined by phenotype) can distinguish identical twins, while DNA-tests (genotype) cannot. Therefore, we think that Shape-DNA is the more appropriate term. Beyond the identification of shapes, the Shape-DNA can even be used to detect similarities.

A special name for the Laplace–Beltrami spectra is even helpful to distinguish it from other spectra. To avoid any misunderstanding, note that the continuous Laplace–Beltrami operator does not operate on any mesh vertices, but rather on the underlying manifold itself. It is therefore different from discrete Laplacians on graphs or meshes. Even though these discrete Laplacians have been used for, e.g. dimensionality reduction [5] or mesh compression [41], the introduction of our computation of the Laplace–Beltrami spectra of the underlying manifolds in the areas of geometric modeling—CAD in particular and in computer graphics in general—is completely new. The only exceptions are our recent proceedings publication [56] outlining briefly some of the ideas and results presented in this paper and [65] containing a sketchy description of some basic ideas and goals. More details and background can be found in [55]. Moreover, the application of the Laplace–Beltrami spectra as Shape-DNA in order to discriminate and search for objects in geometric databases is new (cf. our german patent application [67]). Although a considerable amount of theoretical research has been done in geometry on the Laplace–Beltrami operator, very little work dealing with computational research exists (see e.g. Huntebrinker [38] for a numerical computation of the Laplace–Beltrami spectrum on 3d hyperbolic spaces).

One of the reasons why the spectra of the continuous Laplace and especially the Laplace–Beltrami operator have not yet been considered in the area of geometric modeling and computer graphics is that their computation is not easy at all, with respect to the theoretical effort (employing Riemannian geometry), and somewhat cumbersome with respect to the numerical effort involved. However, with the recent and continuing advancement of hardware development, the computations needed to determine surface spectra (e.g. the first 1000 or more eigenvalues) of the Laplace–Beltrami operator have become conveniently feasible even on a fairly modest personal computer. This shows that the requested [EFFICIENCY] can be achieved as well. Improvements concerning the efficient computation of the spectrum are also foreseeable.

We shall present this paper in a self-contained way such that it should be accessible to a researcher in geometric modeling who is not an expert on the tools from partial differential equations and differential geometry used here. Therefore, we will review some concepts from analysis and elements from differential geometry used for the Laplace–Beltrami operator and its properties (Section 2). We also need some concepts from numerical analysis on finite element methods used to compute solutions for partial differential equations (Section 3) and describe some techniques (like meshing) needed for the actual implementation (Section 4). Then we present a method to numerically extract geometric data from the eigenvalues (Section 5) and show how the Shape-DNA can be used to identify shapes and detect similarities for use in innovative applications (Section 6).