complex line bundle laplacians

Vis ComputDOI 10.1007/s00371-012-0737-5

O R I G I NA L A RT I C L E

Complex line bundle Laplacians

Alexander Vais · Benjamin Berger · Franz-Erich Wolter

© Springer-Verlag 2012

Abstract In the present work, we extend the theoretical andnumerical discussion of the well-known Laplace–Beltramioperator by equipping the underlying manifolds with addi-tional structure provided by vector bundles. Focusing on theparticular class of flat complex line bundles, we examine awhole family of Laplacians including the Laplace–Beltramioperator as a special case.

To demonstrate that our proposed approach is numer-ically feasible, we describe a robust and efficient finite-element discretization, supplementing the theoretical dis-cussion with first numerical spectral decompositions ofthose Laplacians.

Our method is based on the concept of introducing com-plex phase discontinuities into the finite element basis func-tions across a set of homology generators of the given mani-fold. More precisely, given an m-dimensional manifold M

and a set of n generators that span the relative homol-ogy group Hm−1(M,∂M), we have the freedom to choosen phase shifts, one for each generator, resulting in a n-dimensional family of Laplacians with associated spectraand eigenfunctions. The spectra and absolute magnitudes ofthe eigenfunctions are not influenced by the exact location ofthe paths, depending only on their corresponding homologyclasses.

Employing our discretization technique, we provide anddiscuss several interesting computational examples high-

A. Vais (�) · B. Berger · F.-E. WolterWelfenlab, Division of Computer Graphics, Leibniz Universityof Hannover, Hannover, Germanye-mail: [email protected]

B. Bergere-mail: [email protected]

F.-E. Woltere-mail: [email protected]

lighting special properties of the resulting spectral decom-positions. We examine the spectrum, the eigenfunctions andtheir zero sets which depend continuously on the choice ofphase shifts.

Keywords Laplace operator · Complex vector bundles ·Spectral decomposition · Finite elements

1 Introduction

The Laplace–Beltrami operator was first introduced andconsidered in purely theoretical settings in order to adaptthe well-known Euclidean Laplacian to the realm of smoothmanifolds. Typically, it is defined to act on real-valued scalarfunctions f by Δf = d∗df where d is the differential andd∗ its adjoint.

Since manifolds provide a general setting for describinggeometrical shapes, the feasibility of computable discretiza-tions of the Laplace–Beltrami operator, see, e.g. [1, 5, 7, 24,37, 42], has led to numerous applications in geometry pro-cessing algorithms, exploiting the close connection betweenthe geometry of M and properties of the spectral decompo-sition of this operator. Computing the spectral decomposi-tion of Δ amounts to solving the equation Δf = λf whichresults in an infinite sequence of real non-negative eigen-values λ1, λ2, . . . , called spectrum of Δ and a sequence ofeigenfunctions f1, f2, . . . corresponding to the eigenvalues.Important features are its invariance under isometric defor-mations being a consequence of its purely intrinsic defini-tion and its ability to capture geometric similarity in a com-pressed manner.

Among the known applications are shape and image re-trieval using a prefix of the spectrum as a fingerprint or“Shape-DNA” [21, 22, 26, 27]. For the apparently earliest

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A. Vais et al.

reference to the latter application, see [38], with details in[39]. A corresponding patent application was presented in[40]. Further applications are geometric signal processingoperations [16, 33, 35], surface remeshing [6], creating de-scriptors for shape matching [2, 13, 18, 29, 30, 32], shapesegmentation and registration [14, 23], statistical shape anal-ysis for medical studies [19, 25, 28], and symmetry detec-tion [20], just to mention a few. A survey of some applica-tions can be found in [43].

Along with the familiar concept of manifolds, the con-cept of vector bundles is a very natural construction in differ-ential geometry and algebraic topology. Common geometri-cally motivated examples include the tangent and cotangentbundle. Bundles also provide a natural setting for study-ing differential operators and partial differential equations.However, while there is a great variety of other bundles thatcan be constructed, few concrete computational exampleshave been given so far.

Most of the above mentioned works focus on the La-place–Beltrami operator which can be interpreted as aLaplacian on the trivial vector bundle M ×R. Recently, [34]considered an approach for obtaining spectral decomposi-tions for a class of Laplacian operators on non-trivial realline bundles over two-dimensional surfaces, introducing aglobal topological “twist” into the bundle M × R.

The contribution of the present work is to extend theabove discussion by considering complex line bundleswhich offer significantly more flexibility for shapes withnon-trivial topology. While the simplest complex line bun-dle M × C does not give any new information compared toM × R, it can be modified in order to introduce a globaltwist analogously to the real-valued case.

The twisting in the real-valued case is determined by achoice of signs, yielding a finite number of distinct bun-dles with associated Laplacians. However, our newly pro-posed complex-valued approach allows for a continuum ofphase parameters and accordingly many bundle Laplacians.The real-valued case is completely covered by choosing thephase parameters as integer multiples of π . However, thereis a rich variety of Laplacians corresponding to other param-eter choices, leading to spectral properties which cannot beobtained from a real-valued approach. All considered Lapla-cians only depend on intrinsic quantities and are thereforeinvariant under isometric deformations.

Introducing the concept of a connection and its associ-ated curvature, it is possible to speak of flat bundles andit turns out that even the special case of flat complex linebundles considered here already offers a rich structure. Ourcurrent contribution can also be seen as a first step toward anumerical approach for considering Laplacians also on non-flat complex line bundles.

To substantiate the theoretical discussion with con-crete computational examples, we focus first on the two-dimensional case and propose a discretization method. We

Fig. 1 (a) Graph of a smooth real-valued function (red) over the circle(black) interpreted as a section of the trivial bundle. (b) Introduction ofa sign-flip discontinuity. (c) By going to the twisted bundle, the graphbecomes continuous again

also include first three-dimensional examples indicating ourapproach to be generalizable to higher dimensions.

2 Geometric background

Roughly spoken, our approach is based on forcing any func-tion on the surface to have a complex phase shift across aspecific set of paths lying on the surface. This is accom-plished by modifying the standard finite element basis func-tions accordingly.

At first, one might think that an artificial introduction ofdiscontinuities would produce results that depend on the pre-cise choice of paths, or in the worst case, to produce no re-sults at all as the finite element method for the Laplacianeigenvalue problem is typically formulated in terms of con-tinuous basis functions. However, it turns out that the resultsare invariant with respect to the choice of paths up to ho-mology and that the theory of vector bundles provides theappropriate context for interpretation.

To give an intuitive picture of the main idea, consider asmooth function f : S1 → R over the circle S1. We can drawits graph as a smooth curve on a cylinder as shown in Fig. 1.Introducing a sign flip at a certain place c ∈ S1 makes thegraph discontinuous. However, this discontinuity disappearsif we cut up the cylinder at the vertical line centered at c

and glue it back together after a twist. The result obtained isthe Möbius band. On this object, the graph of the functionf has become a smooth curve again and it seems reasonableto be able to differentiate it. The Laplace–Beltrami operator

on the circle is given locally by Δ = − d2

dt2 with t ∈ [0,2π]being the arc-length parameter. Assuming that the cuttingpoint c corresponds to t = 0 (mod 2π), computing the spec-tral decomposition amounts to solving the boundary valueproblem −f ′′(t) = λf (t) with antiperiodic boundary condi-tions. Note that this procedure does not depend essentiallyon where c is located.


Following this line of thought leads to the notion of vec-tor bundles, an important concept in topology and differen-tial geometry. Using this theoretical framework, the discon-tinuities explicitly present in the functions disappear whenthe functions are replaced by sections of an appropriatelytwisted vector bundle. In the following section, we willbriefly review this framework. We will also need some ba-sics of algebraic topology in order to classify and constructdifferent vector bundles, as briefly collected in Appendix.A comprehensive treatment can be found in textbooks ondifferential geometry and topology such as [11, 17].

2.1 Vector bundles

Intuitively, a real (or complex) rank k vector bundle E overa manifold M is obtained by assigning to each point p ∈M a real (or complex) k-dimensional vector space Ep in acontinuous way. The vector space Ep is called fiber over p.Vector bundles of rank one are called line bundles. For aformal definition of vector bundles, see, e.g. [9].

The cylinder and the Möbius band in Fig. 1 are examplesfor real line bundles over the circle M = S1. In the first case,a copy of R is assigned to each point, resulting in a bun-dle E = M × R. The second case, while locally similar, istopologically different because of the global twist. A bundlehaving the structure of a global Cartesian product M ×V forsome vector space V is called trivial. However, not all vectorbundles are trivial, as the Möbius band shows. Instead, theysatisfy only a local Cartesian product condition: For eachpoint, there is a neighborhood U such that the bundle overU is homeomorphic to U × R

k (or U × Ck). This is called

a trivialization of the bundle over U . If V is another neigh-borhood, then one has a transition function hU �→ hV , thatrelates both trivializations via a linear isomorphism on therespective fibers; see Fig. 2.

An important real rank m vector bundle associated to anm-dimensional manifold is its tangent bundle T M which isthe collection of all tangent spaces. The cotangent bundleT ∗M is the dual bundle of T M , consisting of the collectionof all vector spaces being dual to the tangent spaces. Fortwo-dimensional manifolds, these bundles can be viewed ei-ther as real bundles of rank two or as complex line bundles.

A map s : M → E with the property that s(p) ∈ Ep iscalled a section of E. The space of all smooth sections isdenoted by Γ (E). For example, a section of the trivial linebundle M × C is just a function f : M → C. A section ofthe tangent bundle T M is a vector field. A section of thecotangent bundle T ∗M is a differential one-form.

Often, vector bundles are equipped with additional struc-ture, such as a metric or a connection: An Euclidean vec-tor bundle is a real vector bundle which is equipped with asmoothly varying symmetric positive definite bilinear form〈·, ·〉 : Ep × Ep �→ R on its fibers, called fiber metric.

Fig. 2 Local Cartesian product structure of vector bundles

The space of sections Γ (E) of an Euclidean vector bundle(E, 〈·, ·〉) becomes a Hilbert space by introducing the scalarproduct

(φ,ψ) :=∫

M

〈φ,ψ〉dM.

These constructions carry over to complex vector bun-dles using a Hermitean inner products instead of an Eu-clidean inner product. In this case, one speaks of Hermiteanvector bundles. Note that a Hermitean inner product is notsymmetric. Instead, we have 〈u,v〉 = 〈v,u〉 and, therefore,(u, v) = (v,u) where z denotes complex conjugation ofz ∈ C.

A differentiable manifold whose tangent bundle T M isequipped with a fiber metric is called Riemannian manifold.Generally, a metric on a bundle E induces a metric on thedual bundle E∗. Metrics on two bundles E1,E2 induce ametric in the product bundle E1 ⊗ E2.

A connection ∇ on a vector bundle E is a differential op-erator ∇ : Γ (E) → Γ (E ⊗ T ∗M) that satisfies the Leibnizrule

∇(f s) = s ⊗ df + f ∇s f ∈ C∞(M), s ∈ Γ (E).

Here, d denotes the exterior derivative or differential whichmaps functions (zero-forms) to one-forms. A connection canbe considered as an extension of this concept, mapping fromthe space of E-sections (E-valued zero-forms) to the spaceof E-valued one-forms. Those spaces are equipped with in-ner products induced by those on E and on T M .

A connection is called flat if its associated curvature, see[17] vanishes. Bundles equipped with a flat connection arecalled flat. Note that the exterior derivative on functions canbe considered as a flat connection on the trivial line bundleover M .

2.2 Connection Laplacians

Given an Euclidean or Hermitean vector bundle E witha connection ∇ , the adjoint of ∇ is denoted by the map∇∗ : Γ (E ⊗ T ∗M) → Γ (E). Using the connection and itsadjoint, the connection Laplacian associated to ∇ is definedby

Δ := ∇∗∇ : Γ (E) → Γ (E).

A. Vais et al.

The well-known Laplace–Beltrami operator Δ = d∗darises as a special case of this general construction. We willexploit this point of view in order to construct Laplacian op-erators on flat complex line bundles and to study their prop-erties numerically.

2.3 Classification of line bundles

Considering the variety of possible line bundles, the ques-tion of classification arises naturally. Let E1,E2 be twovector bundles over M . A vector bundle isomorphism is ahomeomorphism φ : E1 → E2 that maps the fiber E1|p tothe fiber E2|p via a linear isomorphism.

Up to vector bundle isomorphisms, real line bundles overa manifold are classified by elements of the first cohomologygroup H 1(M,Z2), while complex line bundles are classifiedby elements of the second cohomology group H 2(M,Z).

Note that for the circle M = S1 this implies that there areexactly two real line bundles up to isomorphism, the non-trivial one being given by the Möbius band example. How-ever, any complex line bundle over the circle is trivial, sincethe second cohomology group of the circle vanishes.

For a closed orientable surface M of genus g, the first co-homology group has rank 2g so there are 22g different realline bundles up to isomorphism. Embedding these bundlesin the complex setting, they become topologically equiva-lent as they all correspond to the zero cohomology class inH 2(M,Z). However, there is a whole continuum of topo-logically equivalent bundles containing those 22g bundles,each giving rise to its own bundle Laplacian.

For surfaces with boundary H 2 vanishes, but for closedsurfaces there is a countable number of topologically distinctcomplex line bundles, since H 2(M,Z) ∼= Z. In this work,we focus on the zero class.

3 Description of our approach

In the following, we assume M to be a two-dimensional sur-face. Later on, we will briefly indicate that it is possible toextend the construction to higher dimensions. Let c1, . . . , cn

be a set of loops which generate the first fundamental groupπ1(M) of M . Cutting the surface along the loops, ci yieldsa tile of the universal cover M . Note that the surface can berecovered by a corresponding inverse gluing operation, i.e.by identifying two points p, q in the universal cover if theydiffer by the action of an element of π1(M) in the sense ofFig. 12.

Let E1 = M × C be the trivial complex line bundle overM . It can be constructed from the trivial complex line bundleM × C over M by identifying (p, v) with (q,w) if p and q

differ by the action of an element of π1(M) and if v = w.Given a set of phase angles β1, . . . , βn, one βk for each

generator ck , we construct a second complex line bundle E2

over M by identifying two points (p, v) and (q,w) in M ×C

if the points differ by the action of an element ck1 · · · ckr ofπ1(M) and if v and w differ by a complex phase of βk1 +· · · + βkr .

The natural connection ∇ = d on E1 lifts to a connectionon the bundle M ×C and induces a connection on E2, whichwe will also denote by ∇ . Our goal is to compute the spectraldecomposition of the connection Laplacian ∇∗∇ acting onsections of E2.

Both E1 and E2 are topologically equivalent. If all phaseangles βk are integer multiples of π , the underlying real bun-dles obtained by the above construction using R instead ofC are in general non-trivial. The main point is, that for thenon-zero phase angles the twisting introduced by the phaseshifts gives rise to different Laplacians and different spectralinformation.

The construction above implies that we have an assign-ment of phase shifts to the loops ck . By interpreting theseloops as elements of the homology group H1(M), this as-signment yields cohomology elements in H 1(M). In orderto localize the transitions described by those cohomology el-ements, we employ their duals γk with respect to Poincaré orLefschetz duality. Roughly spoken, instead of accumulatinga phase shift of βk as we go around a loop ck , we induce asudden transition across the cycle γk ∈ H1(M,∂M).

In order to obtain the spectral information, our algorithmproceeds in four steps:

1. Determine a basis γ1, . . . , γn of the first relative homol-ogy group H1(M,∂M).

2. Discretize the eigenvalue problem d∗df = λf using fi-nite element basis functions that have phase discontinu-ities βk across the generators γk determined in the firststep.

3. Solve the resulting complex generalized Hermitean eigen-value problem with a numerical sparse solver.

4. Visualize the resulting spectra and eigenfunctions.

3.1 Computing homology generators

We assume that our manifold is furnished with a triangu-lation (V ,E,F ) where V is the set of vertices, E ⊂ V 2 isthe set of edges and F ⊂ V 3 is the set of faces. The set ofvertices together with the edges form a graph (V ,E). Thedual of this graph is obtained as the graph (F,E∗) whereE∗ ⊂ F 2 is given by all pairs of faces sharing an edge in E.In the following, we identify E and E∗ with each other.The problem of computing cycles on surfaces has been re-searched before, see, e.g. [3, 4, 8, 36], and the referencestherein. For our purposes, we apply the algorithm of Erick-son and Whittlesey [8]:

1. Compute a spanning tree T ⊂ E of the graph (V ,E).2. Compute a spanning tree T ∗ ⊂ E∗ of the dual graph

(F,E∗) using only edges not occurring in T .


3. Compute the set L of all edges not occurring in T or T ∗.Each edge e ∈ L induces a cycle in T .

This algorithm yields a set of 2g cycles for a closed mani-fold of genus g. For manifolds with k boundary loops, the al-gorithm is modified by treating each boundary loop as an ad-ditional face and augmenting the resulting cycles with k − 1independent paths connecting distinct boundary loops.

3.2 Finite element formulation

The general outline for applying a finite element discretiza-tion to the Laplacian eigenvalue problem d∗df = λf is ob-tained in two steps: First, taking the inner product with anarbitrary test function ϕ, we obtain the equation:

(d∗df,ϕ

) = (df, dϕ) = λ(f,ϕ) ∀ϕ.

This weak variational formulation is discretized by writ-ing the unknown function f as a linear combination f =f 1ϕ1 + · · · + f NϕN of a collection (ϕk) of suitable basisfunctions and solving the discrete generalized eigenvalueproblem

Af = λBf,

where A and B are N ×N matrices and f = (f k) is a vectorof dimension N . The entries of the matrices are computed byevaluating the inner products

Aij =∫

M

〈dϕi, dϕj 〉dM, Bij =∫

M

ϕiϕj dM.

Notice that a complex conjugation of the second factor ispresent in the evaluation of these integrals in order to ac-commodate for the potentially complex valued basis func-tions ϕi .

According to standard finite element constructions, see,e.g. [31], the basis functions are constructed by piecing to-gether polynomials over the individual triangles in a triangu-lation of M to yield functions with local support that satisfythe interpolation conditions ϕi(qj ) = δij on a set of N nodes(qk) spaced regularly at the vertices, on the edges and in theinterior of the triangles. This establishes a one-to-one cor-respondence between every node qk and the basis functionϕk evaluating to one precisely at that node and to zero at allother nodes. Depending on where qk is located, the corre-sponding basis function is called a vertex, edge, or bubblefunction, respectively.

For the classical Laplace–Beltrami eigenvalue problem,the basis functions are real-valued and continuous. In ourmethod, we modify the basis functions to be complex-valuedand to have a phase discontinuity across the set of homol-ogy generators computed in the previous section. Of course,these discontinuities reflect the twisting of the bundle. More

Fig. 3 Modified quadratic finite element basis functions in the real-valued case of phase shifts corresponding to sign flips

precisely, any basis function whose support is crossed byone or more generators is affected. We will denote the gen-erators as γ1, . . . , γn and the corresponding phase anglesby β1, . . . , βn. Depending on the type of basis function, wehave three cases to consider:

– Bubble functions are not affected since their support isconstrained to one triangle and they vanish on the edgesof the triangle.

– An edge basis function has a support consisting of two tri-angles. Assuming that some generator γk passes throughthe common edge and denoting by TR the triangle on theright-hand side of γk , we multiply the values of the stan-dard basis function over TR with the factor eiβk .

– A vertex basis function has a support consisting of all tri-angles incident to a vertex v. Assuming that a single gen-erator γk passes through v, it divides the triangle fan intotwo sets TL and TR corresponding to the triangles to theleft and to the right of γk . We apply a phase shift by mul-tiplying the values of the standard basis function over TR

with the factor eiβk . If more than one generator passesthrough v, the modifications are accumulated by multipli-cation as shown in Fig. 4.

In case of a phase of βk = π corresponding to a sign flip,Fig. 3 illustrates the modified quadratic edge and basis func-tions, showing only the real component as the imaginarycomponent is zero. For phases βk which are not integer mul-tiples of π , the modified basis function becomes complex-valued with a non-zero imaginary component. In this case,

A. Vais et al.

Fig. 4 Accumulation of phase shifts in the presence of multiple gen-erators

the matrix entries Aij and Bij have to be evaluated using theHermitean inner products which amounts to using complexarithmetic and taking care of the conjugation of the secondfactor.

Note that our computations work with linear finite ele-ments requiring only vertex basis functions as well as withhigher order elements that require edge and bubble func-tions. The latter can be employed if high accuracy is desired.

3.3 Solving the discrete eigenvalue problem

The resulting matrices A and B define a generalized Her-mitean eigenvalue problem, in which A and B are Her-mitean and B is positive definite. There are two possibilitiesto deal with the complex case: Either use a solver with com-plex arithmetic or reduce the problem to a real-valued sym-metric eigenvalue problem with double dimension. The re-duction is performed by assigning to a complex valued n×n

matrix A a real valued 2n× 2n matrix A, replacing each en-try z of A by a 2 × 2 block in A: z = x + iy �→ ( x −y

y x

).

Note that if A is Hermitean (AT = A), then A is symmetric(AT = A). In our implementation, we used the SLEPc li-brary [12] with real arithmetic for solving the matrix eigen-value problems.

4 Computations and discussion

After having laid out the theoretical basics for the construc-tion of a continuous family of flat complex bundle Lapla-cians, we continue with a discussion of several of their no-table features in comparison to the Laplace–Beltrami opera-tor. Therefore, we employ the proposed finite element baseddiscretization method in order to obtain numerical spectraldecompositions for various geometric example shapes.

After focusing on the invariance properties of the con-struction, we describe how our approach provides a differen-tiable family of Laplacians containing the ordinary Laplace–Beltrami operator and compare the resulting spectra. Usingan example, we demonstrate how the eigenvalues of dif-ferent bundle Laplacians adapt to geometric deformationsin a predictable manner, showing that our method provides

additional flexibility to supplement the data obtained fromthe Laplace–Beltrami operator. Afterward, we focus on theresulting eigenfunctions and especially their characteristiczero sets. Finally, we indicate that our construction can begeneralized to a three-dimensional setting by giving firstcomputational examples.

The spectral computations in our experiments are basedon the following input data:

– A triangle mesh describing our manifold with or withoutboundary. We assume our mesh to be topologically cor-rect.

– A set of paths γ1, . . . , γn generating the first relative ho-mology group. The paths can be obtained by the methodin Sect. 3.1.

– A set of phase angles βk , one for each generator γk .

To visualize the resulting eigenfunctions, we map thecomplex phase information to a cyclic color palette whilean impression of the complex modulus is conveyed by trac-ing iso-lines. Note that for any eigenfunction f , the productcf for a constant c ∈ C is also an eigenfunction. By conven-tion, the eigenfunctions are normalized to be orthonormalwith respect to the Hermitean inner product which restrictsc to have absolute value one. However, there remains an in-herent phase ambiguity, which can be irritating especiallywhen comparing results in a series of computations. To re-solve this ambiguity, we fix a common reference vertex androtate all resulting eigenfunctions to become real-valued atthat vertex.

4.1 Invariance

As an example shape with non-trivial topology, we considerfirst a model of genus one with two boundary loops as shownin Fig. 5. A set of paths generating the first relative homol-ogy group H1(M,∂M) is given by the green handle loopγ1, the red tunnel loop γ2, and the blue path γ3 connectingthe two boundaries. For the spectral decomposition, we ap-plied Neumann boundary conditions on the two boundariesand phase shifts of βk = π

2 , k = 1,2,3 across all three paths.The plot shows the second eigenfunction.

We want to place emphasis on the fact that the resultsdo not essentially depend on the specific choice of genera-tors: Replacing a generator γk by a homologous generatorγ ′k does not change the eigenvalue spectrum and also does

not affect the absolute value (and, therefore, the correspond-ing contour lines) of the resulting eigenfunctions as shownin the bottom part of Fig. 5. This is explained by the con-struction involving transitions of the form eiβ which leavethe modulus invariant.

4.2 Spectrum

If all phase angles are chosen as zero modulo 2π , we obtainthe spectrum of the classical Laplace–Beltrami operator. For


Fig. 5 Eigenfunction resulting from a phase shift of π2 across all gen-

erators. The color palette indicates the complex phase while the con-tour lines show the complex amplitude which is a smooth functioneven across the generators. Changing the homology generator repre-sentatives only affects the phase information

a closed object, the spectrum starts with the eigenvalue zero,which corresponds to the constant function. However, intro-ducing a non-zero phase shift makes the zero eigenvalue dis-appear since it is impossible for a locally constant functionto satisfy the required phase jump.

To illustrate this behavior, consider the closed object ofgenus two shown in Fig. 6. The red handle loop is one of thefour generators that span the first homology group. We grad-ually increase the phase angle β across this loop from zero toπ and plot the resulting first eigenfunctions while the evolu-tion of the first few eigenvalues is shown in Fig. 7. The con-stant eigenfunction gradually changes until it becomes realagain for β = π . The corresponding eigenvalue zero rises toa positive eigenvalue.

4.3 Modified Shape-DNA and HKS

As an application of the modified Laplacians for shape re-trieval we suggest the extension of spectral-based conceptssuch as Shape-DNA [26, 27], in which a prefix of the spec-trum is used as a feature vector characterizing the shape. Byvarying our phase shift parameters βk , we are able to obtaindifferent spectra that are more or less sensitive in responseto specific shape deformations as indicated below.

As an example, we created a smooth deformation of thegenus one kitten model by thickening its tail over 30 timeframes as shown in the top of Fig. 8. The dimension of thehomology group is two, spanned by a handle and a tunnelloop. Inducing a phase shift β across the handle loop createsa one-parameter family of bundles EH (β) and analogouslywe obtain a family ET (β) for phase shifts across the tunnelloop. We performed spectral decompositions of the corre-sponding Laplacians for each of the thirty kitten models anda sample of phases given by β = k π

16 with k = 0, . . . ,16.Figure 8 plots the distance between the resulting spectra

and the classical Laplace–Beltrami spectrum obtained forthe initial object. We have used the lowest 20 eigenvaluesand the L2 distance to perform the comparison of the spec-

Fig. 6 Effect of continuously varying a phase between zero and π on the first eigenfunction. Each snapshot corresponds to a phase shift of kπ8 for

k = 1, . . . ,8 across the handle loop shown in red

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Fig. 7 Effect of continuously varying a phase between zero and π onthe first eigenvalues for the double torus in Fig. 6

Fig. 8 Effect of a shape deformation on the spectra as measured by theL2-distance from the Laplace–Beltrami spectrum of the un-deformedobject

tra. Note that the deformation has almost no influence onthe spectra obtained from EH (β), whereas it is clearly visi-ble in ET (β). Moreover, as β increases from zero to π , theeffect is amplified. Therefore, the spectra of the tunnel loopLaplacians are more sensitive to the deformation.

This example indicates how geometric deformations arereflected differently in spectra of the modified bundle Lapla-

Fig. 9 Isolated zeros of a complex eigenfunction

cians. Obviously, the modifications also have impact on allshape descriptors that are derived from spectral information.Choosing different phase shift parameters offers versatiledata containing the information obtained by the spectral de-composition of the Laplace–Beltrami operator and, there-fore, our approach allows to extend previous approacheswhich depend on spectral data, such as Shape-DNA [27] orthe Heat Kernel Signature [20].

4.4 Zero sets of complex eigenfunctions

We now shift our focus to the zero sets of the eigenfunc-tions. In contrast to the zero set of a regular real-valued func-tion which is one-dimensional, the zero sets of our complex-valued eigenfunctions are in general zero-dimensional, con-sisting of isolated points. We remark that the invariance ofthe contour lines with respect to the choice of homologygenerator representatives implies that the location of the iso-lated zeroes of any eigenfunction are characteristic for theintrinsic geometry of M and the concrete choice of phaseshifts.

To each isolated zero p of f , the complex phase can beused to define the index of f at p, denoted by ind(f,p). Thisinteger value intuitively captures the signed number of timesthat the phase rotates around that point. The index of a vectorfield v ∈ Γ (T M) at a point p with v(p) = 0 is essentiallythe same concept.

According to a theorem of Chern, see, e.g. [9], the sumof the indices of any section f of a complex line bundle E

with a connection ∇ is a constant that can be computed byintegrating the curvature of ∇ and which turns out to de-pend only on the topology of the bundle. Since we are deal-ing with flat line bundles this integral is zero. In Fig. 9, weplotted an eigenfunction of the Laplacian corresponding to aphase shift of π

4 across the red loop around the tail of a kit-ten model. The two views show that there are four isolatedzeros, two with index +1 and two with index −1. Their sumis zero, in accordance with the theory.


Fig. 10 Homology generators calculated from the zero sets of the firsteigenfunctions

4.5 Zero sets of real eigenfunctions

Returning to the real-valued case, notice that the one-dimensional zero set of the first eigenfunction correspondingto the phase β = π in Fig. 6 is a smooth curve that alignswell with a symmetry of the object and which lies in thesame homology class as the handle loop that was used toinduce the phase shift.

We performed several experiments with the zero setsof the first eigenfunctions and obtained results similar to[4] who compute cycles aligned with the principal curva-ture direction fields on the surface or [3] who computeshort cycles that wrap around the handles and tunnels ofthe surface. In our approach, we compute the zero sets ofall 2n first eigenfunctions corresponding to the phase shiftsβ1, . . . , βn ∈ {0,π} and sort these by total length. Then wepick out a set of n cycles that are linearly independent. Fig-ure 10 shows the output of this method for objects of genusone to four. Note that these computations can also be ob-tained by using only real arithmetic; see [34].

The obtained zero sets form cycles that are characteristicfor their respective homology classes and depend on the in-trinsic properties of the surface. Therefore, they can be usedin the context of isometry invariant shape analysis for ob-jects with non-trivial topology. An example application em-

phasizing the use of characteristic cycles in medical shapeanalysis can be found in [41].

4.6 Extension to three-dimensional solids

Up to now we focused our discussion on two-dimensionalsurfaces. The underlying theoretical framework discussedhere allows the manifold in question to have any dimen-sion, while the rank of the bundle remains one. In threedimensions, we are dealing with volumetric objects andthe sections of the considered complex line bundles canbe modeled as functions that have phase discontinuitiesacross some of the two-dimensional homology generatorsin H2(M,∂M). Elements of this group can be pictured asequivalence classes of surfaces whose boundaries are con-tained in the boundary of M . The correspondence estab-lished above remains valid, i.e. the generator surface ishomologous to any zero surface of a non-degenerate sec-tion of the constructed bundle; especially, it is homolo-gous to the zero surface of the first eigenfunction, yield-ing smooth characteristic homology generator representa-tives.

Two computational examples are shown in Fig. 11. InFig. 11(a), we generated a tetrahedral mesh for the two-dimensional double torus to yield a solid double torus. Itssecond relative cohomology group is two-dimensional anda generator has been computed using algebraic techniques.Inducing a phase shift of π yields the zero set of the firsteigenfunction shown in red.

In Fig. 11(b), a solid deformed torus has been carved outof a solid box. The second relative homology group is one-dimensional in this case. The representative obtained fromthe first eigenfunction is the red membrane clamped on theboundary of the torus.

We note that this three-dimensional extension of our ap-proach could be useful in computing well-behaving surfacesgenerating the second relative homology group. These sur-faces may serve as cuts for magnetic scalar potentials as con-sidered, for example in the works of Kotiuga [10, 15].

5 Conclusion and outlook

In this paper, we studied the numerical eigenvalue problemfor flat complex line bundle Laplacians over a surface M . Inthis framework, the well-known Laplace–Beltrami operatorΔ = d∗d acting on real-valued functions is identified withthe bundle Laplacian acting on sections of the trivial real linebundle M ×R. The underlying connection is the differentiald that maps functions to differential one-forms.

This point of view leads naturally to a generalizationin which we can study the problem of spectral compu-tations for other connection Laplacians in the context of

A. Vais et al.

Fig. 11 Three-dimensional example

non-trivial real line bundles as well as complex line bun-dles.

The essence of the presented algorithm consists in dis-cretizing the continuous eigenvalue problem using finite el-ement basis functions that are modified by applying phaseshifts across a set of paths that span the first relative homol-ogy group of the surface. Our method therefore is limited tosurfaces with non-trivial topology as measured by the pres-ence of handles or multiple boundaries. A surface is rep-resented by a piecewise linear or smooth triangulation thatcarries the construction of the finite element basis functions.The homology generators we choose are paths along theedges of the triangulation. Following the standard Galerkindiscretization leads to a sparse generalized Hermitean eigen-value problem that can be solved using available softwarepackages.

An important special case of our approach arises if thephase shifts are chosen as integer multiples of π , corre-sponding to sign flips across the generators. This yields thesame results as previously introduced in [34] which featuresreal-valued eigenfunctions that can be interpreted as sec-

tions of generally non-trivial real line bundles. However, bycontinuously varying the phase from zero to π , we can cre-ate a smooth transition from the classical Laplace–Beltramioperator on the trivial real line bundle to the aforementionedLaplacian operators acting on sections of non-trivial real linebundles.

While in this work we have focused on computationalaspects, exploring the full range of geometry processing ap-plications remains a topic for further research. It should beemphasized in this context that the spectra of the consid-ered Laplacians and certain features that can be derived fromtheir eigenfunctions, such as the zero sets, depend on the in-trinsic geometry and are invariant with respect to the choiceof homology generator representatives. The precise locationof the homology generators on the surface is immaterial aspredicted by the theory and confirmed by the numerical re-sults. First examples indicating the applicability of our ap-proach include the computation of intrinsic characteristiccycles and the extension of classical shape descriptors suchas Shape-DNA and the heat kernel signature, consideringthe phase shifts as a newly available tool giving these appli-cations more flexibility according to topological considera-tions.

Another topic for further research is the extension of thepresented concepts to non-flat and non-trivial complex linebundles, noting that the tangent bundle and cotangent bundleprovide natural examples.

Based on the first examples, we expect our methodto adapt also to topologically more challenging three-dimensional situations.

Acknowledgements The authors would like to thank H. Thiel-helm for valuable comments and suggestions. We also thank theAIM@SHAPE Repository for making accessible the three-dimensionalmodels used in this paper.

Appendix: Algebraic topology

Let M be a path-connected topological space. A continuousmap γ : [0,1] → M with γ (0) = γ (1) = p is called a loopwith base point p. Two loops are called homotopic if one canbe gradually deformed into the other. The set of equivalenceclasses of loops with a fixed base point is called the firstfundamental group of M , denoted by π1(M,p).

A covering of M is a space M with a surjective lo-cal homeomorphism ρ : M → M . If M is simply con-nected, then ρ is called a universal covering of M . A decktransformation is a homeomorphism h : M → M such thatρ ◦ h = ρ. Using these notions, another geometric interpre-tation of the first fundamental group is given by the fact thatit is isomorphic to the deck transformation group of the uni-versal cover of M .


Fig. 12 Fundamental group of the torus acting on its universal cover-ing space by deck transformations

To visualize this construction, consider a topologicaltorus M = S1 × S1 as shown in Fig. 12. Its fundamentalgroup is generated by two loops, denoted by a and b. Cut-ting the torus along these loops creates a rectangle, a so-called fundamental domain. The universal covering is theEuclidean plane M = R

2, which is tiled with copies ofthe fundamental domain. The fundamental group acts onthe points q ∈ M by translation, sending q to the cor-responding point in another copy of the fundamental do-main.

Now, assume M is a manifold represented by a singularsimplicial complex and let R be an arbitrary ring. A k-chainis a formal linear combination of oriented k-simplices withcoefficients in the ring R. The set of k-chains form a groupCk under addition. The boundary operator ∂k : Ck → Ck−1

is a linear operator that maps any oriented simplex to thechain consisting of its appropriately signed oriented bound-ary simplices. A chain α ∈ Ck is called closed, or cycle, if∂α = 0 and it is called exact if it can be written as α = ∂γ

for some γ ∈ Ck+1. Any exact chain is closed as a conse-quence of the fact that the boundary of a boundary is empty,i.e. ∂2 = 0. Therefore, the sequence of chain groups Ck

with the boundary operators in between form a chain com-plex.

Now let Zk be the group of closed k-chains and let Bk

be the group of exact k-cycles. Two cycles α,β ∈ Zk arecalled homologous, if α−β ∈ Bk . The kth homology groupsare the quotient groups Hk(M,R) := Zk/Bk induced by thisequivalence relation.

The set of homomorphisms from Hk to R form the k-thcohomology group Hk(M,R).

For manifolds M with non-empty boundary ∂M , wewill also need the so called relative homology groupsHk(M,∂M) which are obtained by modifying the homol-ogy equivalence relation to treat any chain on the boundaryas zero. In this relative homology, two relative k-cycles a,β

are called homologous if α −β = ∂γ + δ for a (k +1)-chainγ and some k-chain δ contained in the boundary ∂M .

The well-known Lefschetz duality theorem guaranteesthe existence of an isomorphism between the cohomo-

logy group Hk(M) and the relative homology groupHm−k(M,∂M) where m denotes the dimension of the man-ifold.

For the annulus above, the cycle α is a generatorof H1(M). It is also a generator of the first fundamentalgroup π1(M). The blue path β is not a cycle, since its bound-ary consists of two points. However, it is a relative cyclesince these points lie on the boundary ∂M . In fact, β is agenerator of H1(M,∂M).

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Alexander Vais gained a B.Sc. andM.Sc. degree in Computer Sciencefrom the Leibniz University of Han-nover (LUH) in 2007 and 2009, re-spectively. He is currently a Ph.D.student at LUH Division of Com-puter Graphics. His main interestsare computational differential ge-ometry and geometry processing al-gorithms.

Benjamin Berger obtained his mas-ter’s degree in 2011 at the Welfelabof the Leibniz University, Hanover,investigating modifications of theLaplace operator. He is currently aPh.D. candidate at the Welfenlab,studying these modified operatorsin the context of image recognition,partial image matching, and imageprocessing.

http://doi.ieeecomputersociety.org/10.1109/TVCG.2011.267

http://doi.ieeecomputersociety.org/10.1109/TVCG.2011.267


Franz-Erich Wolter has beenChaired Full Professor of Com-puter Science at Leibniz Univer-sity Hanover since 1994 where hedirects the Division of ComputerGraphics called Welfenlab. He heldfaculty positions at University ofHamburg (1994), MIT (1989–1994)and Purdue University USA (1987–1989). He was software and devel-opment engineer with AEG (Ger-many) (1986–1987). Dr. Wolter ob-tained his Ph.D. (1985) mathemat-ics, TU Berlin, Germany, in the areaof Riemannian manifolds, Diploma

(1980), FU Berlin, mathematics and theoretical physics. At MIT he co-developed the geometric modeling system Praxiteles for the US Navyand published papers that broke new ground applying concepts fromdifferential geometry and topology on problems in geometric model-ing. Dr. Wolter is a research affiliate of MIT.

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