Group Representation of Global Intrinsic Symmetries · symmetric pair of points for shapes that...

Pacific Graphics 2017J. Barbic, W.-C. Lin, and O. Sorkine-Hornung(Guest Editors)

Volume 36 (2017), Number 7

Group Representation of Global Intrinsic Symmetries

Hui Wang1,2 and Hui Huang†2

1 School of Information Science and Technology, Shijiazhuang Tiedao University, China2 College of Computer Science & Software Engineering, Shenzhen University, China

Figure 1: Global intrinsic rotational (left) and reflectional (right) symmetries that we compute on an Octopus model, where symmetries arerepresented by the colors, comparing them with the original colored shape in the middle.

AbstractGlobal intrinsic symmetry detection of 3D shapes has received considerable attentions in recent years. However, unlike extrinsicsymmetry that can be represented compactly as a combination of an orthogonal matrix and a translation vector, representingthe global intrinsic symmetry itself is still challenging. Most previous works based on point-to-point representations of globalintrinsic symmetries can only find reflectional symmetries, and are inadequate for describing the structure of a global intrinsicsymmetry group. In this paper, we propose a novel group representation of global intrinsic symmetries, which describes eachglobal intrinsic symmetry as a linear transformation of functional space on shapes. If the eigenfunctions of the Laplace-Beltramioperator on shapes are chosen as the basis of functional space, the group representation has a block diagonal structure. We thusprove that the group representation of each symmetry can be uniquely determined from a small number of symmetric pairs ofpoints under certain conditions, where the number of pairs is equal to the maximum multiplicity of eigenvalues of the Laplace-Beltrami operator. Based on solid theoretical analysis, we propose an efficient global intrinsic symmetry detection method,which is the first one able to detect all reflectional and rotational global intrinsic symmetries with a clear group structuredescription. Experimental results demonstrate the effectiveness of our approach.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computer Graphics/ComputationalGeometry and Object Modeling—[Geometric algorithms, languages, and systems]

1. Introduction

Symmetries are universal phenomena, in both natural and man-made shapes, which reflect high-level information about shapestructure. Many applications in geometric modeling and pro-

† Corresponding author: Hui Huang ([email protected])

cessing utilize the symmetry information, such as, segmenta-tion [SKS06], feature correspondence [LKF12], shape match-ing [KFR04, THW∗14], function matching [AVBC13], geometrycompletion [TW05] and meshing [PGR07]. The characterizationand detection of symmetries of 3D shapes, thus receives signifi-cant attention in computational geometry and computer graphics; acomprehensive review is provided in [MPWC13].

c© 2017 The Author(s)Computer Graphics Forum c© 2017 The Eurographics Association and JohnWiley & Sons Ltd. Published by John Wiley & Sons Ltd.

Hui Wang & Hui Huang / Group Representation of Global Intrinsic Symmetries

Figure 2: The composition operation of global intrinsic symme-tries is equal to a product of their 21× 21 representation matri-ces. The middle row presents the symmetries via transferring colorsfrom the top one onto the shape itself. Each matrix at bottom is thefunctional representation of the symmetry, as shown by the above.

The symmetry of a shape can be regarded as a distance pre-serving self-homeomorphism. Based on types of distances, sym-metry can be classified into extrinsic [MGP06, PSG∗06, MSHS06,LJYL16, SAD∗16] and intrinsic [RBBK07, OSG08, XZT∗09,KLC10, XZJ∗12]. While extrinsic symmetry detection finds rigidtransformations that are Euclidean distance preserving, intrin-sic symmetry detection uses geodesic distances and looks forisometric deformations. Symmetry can also be categorized asglobal [MSHS06, OSG08] and partial [XZT∗09, XZJ∗12] by tak-ing consideration all or part of the shape. In this paper, we focus onthe characterization and detection of global intrinsic symmetries on3D shapes, which can be represented as compact manifolds.

While extrinsic symmetry can be compactly represented usingan orthogonal matrix and a translation vector, i.e., just six de-grees of freedom, detecting and representing intrinsic symmetry ismuch more challenging, in particular for rotational intrinsic sym-metries. The most frequently used symmetry representation is thepoint-to-point correspondence, which indicates each point and itssymmetric point as a pair. However, it is generally intractable tosearch intrinsic symmetries directly in this representation, since thenumber of possible point correspondences is exponential in size.Some previous works thus limit the search space to a set of fea-ture points and prune wrong pairs of points using combinatorialmethods [RBBK07, RBBK10], which are still computationally ex-pensive. Furthermore, it is difficult to describe the structure of aglobal intrinsic symmetry group under this representation. Describ-ing the structure of symmetry from point-to-point correspondencesrequires additional information, such as geodesic distances, withO(N2) complexity for a shape of N vertices. Since our method rep-resents the symmetries directly as a matrix of a small dimension,the complexity is significantly reduced.

Based on the eigen-decomposition of the Laplace-Beltrami op-erator, Ovsjanikov et al. [OSG08] transform intrinsic symmetries

Figure 3: Both reflectional (top) and rotational (bottom) global in-trinsic symmetries can be uniquely determined from a small numberof symmetric pairs of points.

on the shape into extrinsic symmetries in a high dimensionalsignature space. The intrinsic symmetry can be uniquely deter-mined from a single pair of points using the proposed Heat KernelMap [OMMG10]. However, the non-repeated eigenvalue restric-tion of the Laplace-Beltrami operator causes these two proposedalgorithms to fail for shapes with rotational intrinsic symmetry or-der larger than two, due to repeated eigenvalues [Ovs11].

Kim et al. [KLC10] use low dimensional Möbius transforma-tions to find intrinsic symmetries. However, their method is limitedon shapes with zero genus and also relies on symmetry represen-tation using point-to-point correspondences. Based on the state-of-the-art functional map [OBCS∗12], Liu et al. [LLL∗15] propose anintrinsic symmetry detection method that still fails to detect rota-tional intrinsic symmetries.

In this paper, inspired by the classical group representation the-ory [TGT07, Gur08] and the functional map in geometry process-ing [OBCS∗12], we represent each global intrinsic symmetry as alinear transformation on the function space defined on the shape(see Figure 1), and have proven that each global intrinsic symmetrycan be uniquely determined from sparse symmetric pairs of pointsunder certain conditions. Based on these analyses, we propose anovel algorithm for global intrinsic symmetry detection.

Our main technical contributions can be summarized as:

• Represent each global intrinsic symmetry of a compact mani-fold in any dimension as a linear transformation on the functionspace defined on the manifold, which can be represented by amatrix under a chosen basis on the function space. This allowsus to study the structure of the global intrinsic symmetry groupthrough the use of simple linear algebra, e.g., the compositionshown in Figure 2.

• Prove that, if the eigenfunctions of the Laplace-Beltrami oper-ator are chosen as the basis for the function space, each globalintrinsic symmetry can be uniquely recovered from some sparsesymmetric pairs of points under mild conditions shown in Fig-ure 3, where the minimal number of pairs is the maximum mul-tiplicity of the eigenvalue of the Laplace-Beltrami operator. Thisprovides a new band-to-band method to compute the functionalmap using much fewer constraints than previous works.

c© 2017 The Author(s)Computer Graphics Forum c© 2017 The Eurographics Association and John Wiley & Sons Ltd.


• Propose a fast symmetry detection algorithm that can not onlyhandle reflectional symmetries but also rotational global intrinsicsymmetries as demonstrated in Figure 1.

2. Related work

The literature on symmetry representation and detection isvast [MPWC13]. Here we mainly concentrate on representation ofglobal intrinsic symmetries of 3D shapes and briefly introduce ex-isting methods of detecting them.

Point-to-point correspondence. Global intrinsic symmetry is anisometric mapping on a shape, which preserves geodesic distancesbetween every two points. Perhaps the most common representa-tion of global intrinsic symmetry is the point-to-point correspon-dence, i.e., representing each point and its symmetric point as apair. It is difficult to investigate the symmetry group structure, suchas computing composition and order of symmetries, using this rep-resentation in practice. In contrast, our proposed representation al-lows much simpler linear algebra computation to delve into thestructure of the symmetry group, such as composition, group table,inverse and order of each symmetry, etc. Furthermore, searchingsymmetries directly based on the point-to-point correspondencesis an NP-hard subclass of the quadratic assignment problem. Thesearch space can be greatly reduced by ruling out bad mappings thatcannot preserve local geometric signatures and global consistentdistances [RBBK07, RBBK10, RBB∗10]. However, these methodsare still quite computationally expensive and it is difficult to handleshapes with multiple symmetries [MPWC13].

Möbius transformation. Based on the observation that the groupof global intrinsic symmetry is a subset of anti-Möbius transforma-tions, which has only six degrees of freedom for genus zero shapes,Kim et al. [KLC10] and Panozzo et al. [PLPZ12] propose novelintrinsic symmetry detection methods. Among candidate Möbiustransformations the one that best maps the shape onto itself is se-lected as the detected symmetry. Those candidate Möbius transfor-mations are generated by enumerating small subsets of a symmetryinvariant point set, which are critical points of average geodesicdistance functions. The candidate maps can be blended to a bet-ter one [KLF11], leading to a higher dimension matrix and thusare computationally costly. These methods are limited to genuszero shapes, whereas our approach is much faster and applicable toshapes of any genus and dimension under light conditions. Further-more, these methods rely on representing symmetries as point-to-point correspondences, which restrict their ability of investigatingthe structure of a symmetry group.

Global point signature. Ovsjanikov et al. [OSG08] prove thatglobal intrinsic symmetries on a compact manifold can be con-verted to Euclidean symmetries on global point signatures via theeigen-decomposition of the Laplace-Beltrami operator [Rus07].Through this theory, they propose a symmetry detection algorithmon a restricted signature space, where each symmetry is representedas a sequence of signs of non-repeating eigenfunctions. However,if a compact manifold has a rotational symmetry g such that g2 6= I,the eigenvalues of the Laplace-Beltrami operator of the manifoldmust contain some repetition [Ovs11]. Therefore, this method fails

to detect rotational global intrinsic symmetries with an order largerthan two.

Heat kernel map. Ovsjanikov et al. [OMMG10] point out thatglobal intrinsic symmetry can be uniquely recovered from only onesymmetric pair of points for shapes that only have non-repeatedeigenvalues of the Laplace-Beltrami operator. This argument isconsistent with our proposed group representation. That is, thegroup representation of a global intrinsic symmetry can be uniquelydetermined from one symmetric pair of points for shapes with-out repeated eigenvalues. However, this restriction on non-repeatedeigenvalues limits the ability of this method to detect rotationalsymmetries, such as the one in [OSG08], where the symmetriesare still represented as point-to-point correspondences.

Functional map. The functional map [OBCS∗12] has been widelyused in geometry processing [OCB∗16], such as shape correspon-dence [PBB∗13], map visualization [OBCCG13], symmetry detec-tion [LLL∗15], measuring shape difference [ROA∗13, CSBC∗17],etc. In particular, the global intrinsic symmetries on one referenceshape can be transferred into another target shape based on decom-posing the functional map into two parts, which act respectivelyon the space of symmetric functions and its orthogonal comple-ment [OMPG13]. However, the symmetries on the reference shapehave to be known in advance.

Note that almost all previous works need to add many con-straints to compute the functional map entirely, e.g., spar-sity [PBB∗13], orthogonality [LLL∗15], commutativity with spe-cific operators [NO17]. Nonetheless, they lack the theoretical anal-ysis to guarantee at least how many constraints are enough touniquely determine the functional map. Our proposed method isbuilt from the functional map. Meanwhile we have proven that thefunctional map of a global intrinsic symmetry can be uniquely re-covered from a small number of symmetric pairs of points undercertain mild conditions. This benefits from when we decompose thefunctional map into the direct sum of several parts, where each partacts respectively on the eigenfunction space of each eigenvalue (re-peated or non-repeated) and thus can be computed separately withmuch fewer constraints.

Group representation. Analyzing a symmetry group through theinduced linear transformation on the function space is a classicalapproach in the Representation Theory [TGT07], where group el-ements can be represented as matrices so that the group operationcan be represented by a matrix product. Many group-theoretic prob-lems can, therefore, be reduced into much simpler problems in lin-ear algebra. The theory about relations of symmetry groups, rep-resentations and Laplacians can also refer to [Gur08]. Inspired bythese analyses, we use the representation theory to analyze and de-tect the global intrinsic symmetries on manifolds.

3. Theoretical analysis

In this section, based on the classical representation theo-ries [TGT07, Gur08], we analyze group representation of globalintrinsic symmetries on a continuous compact Riemannian mani-fold in any dimension and its properties when choosing eigenfunc-tions of the Laplace-Beltrami operator as the basis of the function



space defined on the manifold. For a compact Riemannian man-ifold M with the standard measure induced by the volume form,the space of square-integrable functions on the manifold M is de-noted by L2(M) = { f : M→ R|

∫M f 2ds < ∞} with the inner prod-

uct < f ,g >M=∫

M f gds. All of the bijective linear transformationsL2(M)→ L2(M) on the function space form a group, written asGL(L2(M)), with composition as a group operation.

3.1. Global intrinsic symmetry group

A self-homeomorphism g : M → M on the compact Riemannianmanifold M that preserves geodesic distances is called a global in-trinsic symmetry

d(p,q) = d(g(p),g(q)), ∀p,q ∈M, (1)

where d(p,q) is the geodesic distance of points p and q [RBBK07].

The set of all global intrinsic symmetries of the manifold Mforms a group G(M) with composition as the group operation. Ob-viously, the identity mapping I on M is an element of G(M).

3.2. Functional representation of global intrinsic symmetry

Inspired by the functional map for shape correspondences[OBCS∗12], each global intrinsic symmetry g ∈G(M) on the man-ifold M induces a bijective linear transformation Fg ∈ GL(L2(M))on the space of square-integrable functions L2(M) as

Fg( f )(p) = f (g(p)), (2)

for any function f ∈ L2(M) and point p ∈ M. The above mapFg is usually called the functional representation or functionalmap of the symmetry g. We list the following three propositionsthat are directly taken from the original functional map frame-work [OBCS∗12]. To make the paper self-contained, we providehere the proof of the proposition 3.3.

Proposition 3.1. The original symmetry g can be recovered fromits functional representation Fg.

Proposition 3.2. For each symmetry g ∈ G(M), Fg is a lineartransformation on the function space L2(M).

Proposition 3.3. Given a basis φ1,φ2, . . . ,φi, . . . on the functionspace L2(M), for each symmetry g ∈ G(M), Fg can be representedas an infinite matrix Cg with element Cg

ji =< Fg(φi),φ j > s.t. thecoefficients vector of Fg( f ) is Cga, where f is a function withcoefficients vector a.Proof Suppose that L2(M) is equipped with a basisφ1,φ2, . . . ,φi, . . ., so that f ∈ L2(M) can be represented as alinear combination of the basis f = ∑i aiφi. Then,

Fg( f ) = Fg(∑i

aiφi) = ∑i

aiFg(φi). (3)

Fg(φi) is also a function on the manifold M and can be representedas Fg(φi) = ∑ j Cg

jiφ j, where Cgji =< Fg(φi),φ j >. Therefore,

Fg( f ) = ∑i

ai ∑j

Cgjiφ j = ∑

j∑

iCg

jiaiφ j. (4)

The vectors of coefficients of functions f and Fg( f ) are a =(a1,a2, . . . ,ai, . . .)

T and Cga respectively. �

3.3. Representation of global intrinsic symmetry group

Based on the propositions in the previous section, each global in-trinsic symmetry g ∈ G(M) induces a bijective linear transforma-tion Fg ∈ GL(L2(M)). We can then define a map ρ : G(M) →GL(L2(M)) as follows

ρ(g) = Fg,∀g ∈ G(M). (5)

Proposition 3.4. The map ρ : G(M)→ GL(L2(M)) is a faithfulgroup representation of G(M).Proof ∀g1,g2 ∈ G(M), ∀ f ∈ L2(M) and p ∈ M, we haveρ(g1g2)( f )(p) = Fg1g2( f )(p) = f (g1g2(p)) = f (g1(g2(p))) =Fg1( f (g2(p))) = Fg1 ◦ Fg2( f )(p) = ρ(g1) ◦ ρ(g2)( f )(p), i.e.,ρ(g1g2) = ρ(g1) ◦ ρ(g2). Therefore, the map ρ : G(M) →GL(L2(M)) is a representation of symmetry group G(M).

∀g1,g2 ∈G(M) and g1 6= g2, there must exist one point p∈M s.t.g1(p) 6= g2(p). For the indicator function fq of point q = g1(p), wehave Fg1( fq)(p)= fq(g1(p))= 1 and Fg2( fq)(p)= fq(g2(p))= 0.It means that Fg1 6= Fg2 , i.e., ρ(g1) 6= ρ(g2). As a result, the grouprepresentation ρ : G(M)→ GL(L2(M)) is faithful. �

Proposition 3.5. Given a basis φ1,φ2, . . . ,φi, . . . on L2(M),G′(M) = {Cg,∀g ∈ G(M)} is a matrix group, which is an isomor-phism of the symmetry group G(M).Proof For the basis φ1,φ2, . . . ,φi, . . . on L2(M), Fg ∈ GL(L2(M))can be represented as an infinite matrix Cg defined in Equation (4).Furthermore, since ρ : G(M)→GL(L2(M)) is a faithful group rep-resentation, each global intrinsic symmetry g ∈ G(M) can be rep-resented as a invertible matrix Cg and G(M) is an isomorphism ofthe matrix group G′(M) = {Cg,∀g ∈ G(M)}. �

The above group representation of G(M) allows that manygroup-theoretic problems can be reduced to the much simpler prob-lems in linear algebra. Instances include:

• Composition operation on group G(M) can be represented bythe matrix product on G′(M), that is, ∀g1,g2 ∈G(M), the matrixrepresentation of g1 ·g2 is Cg2 ∗Cg1 shown in Figure 2.

• The matrix representation of identity symmetry I of the manifoldM is the identity matrix I.

• For all symmetry g ∈ G(M), the matrix representation of its in-verse element g−1 is (Cg)−1.

• If symmetry g ∈G(M) has finite order m, then its representationmatrix Cg also has the same order m, and vice versa.

3.4. Basis of the Laplace-Beltrami eigenfunctions

Suppose the Laplace-Beltrami operator ∆M on the compact Rie-mannian manifold M has eigenvalues 0 = λ1 < λ2 < .. .λi < .. .,where λi corresponds to an ik dimensional eigenfunction space Wiwith an orthogonal basis defined as φi1, . . . ,φiik . It is then knownthat φ11, . . . ,φ11k ,φ21, . . . ,φ22k , . . . ,φi1, . . . ,φiik , . . . forms an orthog-onal basis of L2(M). Beside the compactness and stability char-acteristics in the original functional map framework [OBCS∗12],the group representations of global intrinsic symmetries have someother special properties under the above orthogonal basis.



Proposition 3.6. For each symmetry g ∈ G(M), the eigenfunc-tion space Wi is an invariant space under the linear transforma-tion Fg. Furthermore, its representation matrix Cg is orthogonal,i.e., Cg(Cg)T = (Cg)T Cg = I, and has a block diagonal structure,where the i-th block matrix Dg

i is with dimension ik.Proof For the orthogonal basis φi1,φi2, . . . ,φiik of the space Wi,Fg(φi1),Fg(φi2), . . . ,Fg(φiik ) is also an orthogonal basis of Wi[Ros97, OSG08]. Thus, there exists an ik dimensional orthogonalmatrix Dg

i such that

(Fg(φi1),Fg(φi2), . . . ,Fg(φiik )) = (φi1,φi2, . . . ,φiik )Dgi .

Then Cg is also an orthogonal matrix with a block diagonalstructure, whose i-th block matrix is Dg

i . �

The above proposition means L2(M) can be split into the directsum of the eigenfunction spaces L2(M)=⊕iWi, which are invariantunder all linear transformations Fg induced by g ∈ G(M). That is,Fg can be decomposed into the following direct sum

Fg =⊕

iFg

i , (6)

where Fgi is the induced linear transformation of Fg on the func-

tional space Wi, and the representation matrix of Fgi is Dg

i . Iff ∈ L2(M) and its projection on the subspace Wi is ∑

ikj=1 ai jφi j

with a coefficient vector ai = (ai1,ai2, . . . ,aiik )T , then the coeffi-

cient vector bi of function Fgi (∑

ikj=1 ai jφi j) is

bi = Dgi ai. (7)

If we define the coefficient vector of indicator function fp ofpoint p ∈M as its spectral embedding

Φ(p) = (φ11(p), . . . ,φ11k (p) . . . ,φi1(p), . . . ,φiik (p), . . .)T , (8)

then for g ∈ G(M) with its representation matrix Cg, we getΦ(g(p)) = Cg

Φ(p) and

(φi1(g(p)), . . . ,φiik (g(p))T = Dg

i ∗ (φi1(p), . . . ,φiik (p))T . (9)

Each global intrinsic symmetry g ∈ G(M) induces an extrinsicsymmetry on the above spectral embedding, which can be repre-sented as the orthogonal matrix Cg with a block diagonal structure.

3.5. Main theorem

Based on the above, if we choose eigenfunctions of the Laplace-Beltrami operator for the basis of the function space L2(M) on themanifold M, then the functional representation Fg of every globalintrinsic symmetry g ∈ G(M) can be decomposed into the directsum of some linear transformation as in Equation (6). Unlike previ-ous works solving the representation matrix Cg of functional mapFg entirely with many constraints, we compute its block matrixDg

i corresponding to the induced functional map Fgi separately.

This strategy has the advantage that each matrix Dgi can be com-

puted with a much smaller number of constraints. In this section,we prove that the global intrinsic symmetry can be uniquely deter-mined by only a small number of constraints.

Definition 3.1. Functions f j, j = 1,2, . . . ,J are called full rank ifthe rank of their projections on the eigenfunction space Wi of theeigenvalue λi of the Laplace-Beltrami operator is ik, i = 1,2, · · · .

It is obvious that functions f j, j = 1,2, . . . ,J are full rank if andonly if rank(Ei) = ik, the coefficient matrix Ei is defined as

Ei =

a1

i1 a2i1 · · · aJ

i1a1

i2 a2i2 · · · aJ

i2· · · · · · · · · · · ·a1

iik a2iik · · · aJ

iik

. (10)

where a ji1φi1 +a j

i2φi1 + . . .+a jiik φiik is the projection of function f j

on the space Wi, j = 1,2, . . . ,J.

Theorem 3.1. If the eigenvalues of the Laplace-Beltrami operatoron the compact manifold M with a bounded maximum multiplicitym, i.e., m = max{1k,2k, . . . , ik, . . .}, then for each global intrinsicsymmetry g ∈ G(M), m full rank functions f j and their mappingfunctions Fg( f j), j = 1,2, . . . ,m under the action of g can uniquelydetermine the symmetry representation matrix Cg.Proof Suppose a j

i1φi1 + a ji2φi1 + . . .+ a j

iik φiik is the projection offunction f j on the space Wi, and Fg( f j) ∈ Wi is represented asFg( f j) = b j

i1φi1 +b ji2φi1 + . . .+b j

iik φiik , j = 1,2, . . . ,m.

Based on Equation (7), we haveb1

i1 b2i1 · · · bm

i1b1

i2 b2i2 · · · bm

i2· · · · · · · · · · · ·b1

iik b2iik · · · bm

iik

= Dgi

a1

i1 a2i1 · · · am

i1a1

i2 a2i2 · · · am

i2· · · · · · · · · · · ·a1

iik a2iik · · · am

iik

.

(11)Since m ≥ ik and functions f j, j = 1,2, . . . ,m are full rank, thereexists ik linear independent columns from the right matrix Ni. If thecolumn indexes are j1, j2, · · · , jik , the following matrix is invertible

a j1i1 a j2

i1 · · · ajiki1

a j1i2 a j2

i2 · · · ajiki2

· · · · · · · · · · · ·a j1

iik a j2iik · · · a

jikiik

.

Thus, the i-th block diagonal matrix Dgi of Cg can be solved as

Dgi =

b j1

i1 b j2i1 · · · b

jiki1

b j1i2 b j2

i2 · · · bjiki2

· · · · · · · · · · · ·b j1

iik b j2iik · · · b

jikiik

a j1i1 a j2

i1 · · · ajiki1

a j1i2 a j2

i2 · · · ajiki2

· · · · · · · · · · · ·a j1

iik a j2iik · · · a

jikiik

−1

.

(12)

After all of the block diagonal matrices are computed, we canget the representation matrix Cg of the symmetry g. �

The above theorem can also be regarded as a consequence of thecommutativity constraint between the functional representation andthe Laplace-Beltrami operator [OBCS∗12], which means that Cg

must be block diagonal and can be recoverable from ik linearly in-dependent constraints on each block. Theorem 3.1 proposes a newband-by-band method to compute the functional representation us-ing a much smaller number of constraints. In practice, we divide



(a) symmetry pairs (b) point to point (c) function transferring (d) representation matrix

Figure 4: Reflectional global intrinsic symmetry of a C2 shape is recovered from four symmetric pairs of points shown in (a), where (b)-(d)are different symmetry representations, that is, point-to-point correspondence (b), action of transferring colors from left to right (c), and therepresentation matrix (d), respectively.

the eigenfunctions into a few groups. In each group, the differ-ence of their corresponding eigenvalues is small. We can then com-pute the block diagonal matrices for these grouped eigenfunctionsseparately. Although individual eigenfunction suffers from the or-der switching phenomena numerically, the spaces spanned by thegrouped eigenfunctions of the Laplace-Beltrami operator are sta-ble, since the grouped eigenvalues are well separated [OBCS∗12].

However, only using the well-known global intrinsic sym-metry invariant functions, such as, Heat Kernel Signature(HKS) [SOG09], Wave Kernel Signature (WKS) [ASC11] and Av-eraged Geodesic Distance (AGD) [KLC10], in the above theoremcannot recover the symmetries. This is because these functions re-main the same under all global intrinsic symmetries and, therefore,cannot be used to distinguish symmetries. Next, we investigate touse the indicator functions of some points and their symmetric onesto compute the representation matrix of symmetry.

Definition 3.2. Points p j, j = 1,2, . . . ,J are called full rank if theirindicator functions fp j

are full rank.

It is obvious that points p j, j = 1,2, . . . ,J are full rank if and onlyif rank(Fi) = ik, i = 1,2, . . ., where matrix Fi is defined as

Fi =

φi1(p1) φi1(p2) · · · φi1(pJ)φi2(p1) φi2(p2) · · · φi2(pJ)· · · · · · · · · · · ·

φiik (p1) φiik (p2) · · · φiik (pJ)

, (13)

where the j-th column is the spectral embedding of point p j re-stricted on the space Wi. Specifically, when the eigenvalues are allnon-repeated (generic manifold), i.e., m = 1, a point p (genericpoints) is full rank if and only if φi1(p) 6= 0 for every eigenfunc-tion φi1, i = 1,2, · · · [OMMG10]. Some corollaries can be easilyobtained without proofs.

Corollary 3.1. If the eigenvalues of the Laplace-Beltrami operatorhave maximum multiplicity m, then for every global intrinsic sym-metry g ∈ G(M), m full rank points p j and their symmetric pointsg(p j), j = 1,2, . . . ,m can uniquely determine its symmetry repre-sentation matrix Cg.

The condition of symmetry determination in the above corollary

is not strict. Because the probability of choosing m points that arenot full rank in the function space Wi, i.e., their spectral embed-ding vector restricted on the ik (m ≥ ik) dimension space Wi hasrank lower than ik, is very low. This is similar to the probabilityof randomly choosing more than three 3D points which are exactlyon a line is very low. If the m points are not full rank, a small tur-bulence will make them full rank. Furthermore, in addition to theabove indicator functions, we can also add the global intrinsic sym-metry invariant functions, such as Heat Kernel Signature and WaveKernel Signature, to make the constraint functions full rank for thecomputation of symmetry representation matrix in practice.

Corollary 3.2. For a generic compact manifold M, the global in-trinsic symmetry g can be uniquely determined from only one pairof points (p,g(p)) with the condition that p is a generic point. Fur-thermore, each block diagonal matrix Dg

i of Cg is one dimension,i.e., Dg

i = 1 or −1.

The above corollary coincides with a result for the Heat KernelMap [OMMG10] from a different view, i.e., the global intrinsicsymmetry on a generic manifold can be uniquely recovered fromonly one generic point and its symmetric point. It is also proventhat the area of non-generic points is zero, i.e., the generic points arealmost everywhere on the manifold [Che76, OMMG10]. However,their method can only recover the point-to-point correspondencesof the symmetry, while our method can compute the more generalfunctional representation.

4. Algorithm

In this section, based on the continuous theoretical analysis above,we define an algorithm for effectively computing group representa-tions of global intrinsic symmetries of manifolds discretized as tri-angular meshes in R3. If the global intrinsic symmetry group G(M)of shapes only have two elements, i.e., G(M) = {I,g|g2 = I}, wedenote them as C2 shapes. In this paper, symmetric shapes are clas-sified into two categories, C2 shapes and others with more symme-tries. There is no need to classify symmetric pairs of points in thefirst situation, where is only one reflection beside the identity sym-metry, e.g., human and four-legged animals. As far as we know,most previous works can only handle shapes in this class.



input modelsampling

symmetric points set consistent pairs transformations density action of determined symmetrypairing computing clustering clusters

Figure 5: The global intrinsic symmetry extraction pipeline of a screw model, where the maximum multiplicity of eigenvalues is m = 3. Thesampling yields a symmetry-invariant set. Every three consistent pairs of points determine a transformed orthogonal matrix (2D embeddingvia PCA). Each cluster with higher density in the transformation space corresponds to a global intrinsic symmetry.

4.1. C2 shapes

The proposed algorithm consists primarily of the following steps:pairs of points generation, computation of representation matrixand refinement by Iterative Closest Point (ICP) algorithm.

Step 1: pairs of points generation and functional constraints.To compute the only one non-trivial reflectional symmetry g in theC2 shapes, some symmetric pairs of points (p,g(p)) should be pre-pared firstly, where (g(p),p) is also a symmetric pair for g. Basedon Theorem 3.1 and Corollary 3.1, the number of symmetric pairsof points should be larger than the bounded maximum multiplicitym of the eigenvalues of the Laplace-Beltrami operator.

In practice, we find that m is seldom larger than four for thefirst smallest eigenvalues that are used and and generate two mu-tual symmetric pairs coherently, i.e., (pi,g(pi)) (g(pi),pi), i = 1,2,which are selected as the two closest pairs from the local ex-trema of the Heat Kernel Signature (HKS) with a larger time t =4ln10/λ2 defined in [SOG09]. The closeness of HKS is measuredby their Euclidean distances. To improve the stability, we don’t se-lect the closest pairs, i.e., the geodesic distances between pointsp1,g(p1),p2, and g(p2) should be large enough. The selected sym-metric pairs of points are usually at the tips of the hands or legs ofa human or animal as illustrated in Figure 4(a), which are enoughto recover the reflectional symmetry with different representationsshown in Figures 4(b)-(d). In practice, the indicator functions ofsmall geodesic disks of the above four points, p1,p2,g(p1), andg(p2), are used as functional constraints for computing a functionalmap. In addition to the four indicator functions above, one hundredscaled HKSs and WKSs are also added as functional constraints.

Step 2: computation of the representation matrix. Unlike pre-vious works that compute the functional map matrix entirely, wecompute the symmetry representation matrix Cg band-by-band. Inpractice, the coefficient vectors of the constrained functions in thespace Wi are not exactly aligned with the orthogonal matrix Dg

iin Equation (11). The best orthogonal matrix Dg

i is computed viathe Singular Value Decomposition (SVD), i.e., Dg

i = U∗V′ where[U,S,V] = SV D(B∗A′). The first four columns of matrices A andB are coefficient vectors of the four indicator functions and theirsymmetric ones respectively. The other columns of both A and Bare coefficient vectors of 100 HKSs and WKSs with the scaled pa-rameter 0.01 as constraints.

Step 3: ICP refinement. Since the computed matrix Cg can be re-garded as a rigid alignment between the spectral embeddings of allpoints on shapes as defined in Equation (8), we are able to refine itby the iterative closest point algorithm as in Section 6.2 of the origi-nal functional map [OBCS∗12]. During the refinement process, thepoint-to-point correspondence of the global intrinsic symmetry isobtained at the same time.

4.2. Shapes with more symmetries

Because global intrinsic symmetries are represented as matricesand can be recovered from sparse pairs of points, we vote and clus-ter symmetries in the matrix space like extrinsic symmetry extrac-tion [MGP06]. Therefore, our algorithm pipeline consists of pointssampling, pairs of points generation and pruning, and clustering intransformation space, as illustrated in Figure 5.

Step1: symmetric points sampling. Although all pairs of pointson shapes M can be considered, in practice, we use all possi-ble pairs of points of a smaller symmetry-invariant set P = {pi ∈M, i = 1,2, . . . ,n} satisfying g(P) ⊆ P,∀g ∈ G(M) [KLC10]. Theset {g(p),∀g∈G(M)} from a given point p∈M is called a symme-try orbit [LCDF10], a specific type of symmetry-invariant set. Fur-thermore, a symmetry-invariant set can be seen as a union of somesymmetry orbits. Some recent works [KLC10, LCDF10, WSSZ14]have conducted extracting symmetry-invariant sets or orbits.

The number of sampling points n should be larger than the max-imum multiplicity m of the eigenvalues of the Laplace-Beltramioperator. In practice, we find that the number of points in the sym-metry orbit of a point other than the stationary point under all sym-metries, is usually larger than the bounded maximum multiplicitym. In this paper, the symmetry orbit from a single point is usedas symmetric point sampling. We use the method of [WSSZ14] tocompute the symmetry orbit. The single point for symmetry orbitcomputation can be random selected or defined by user.

Step 2: pairs of points generation and matrix computation. Ifthe symmetry-invariant set has n points, then there are C(n,m) ∗C(n,m)∗m! possible m pairs of points (pi,qi), i= 1,2, · · · ,m. Sinceboth n and m are small in practice, it is possible to check all pos-sibilities. However, several pairs can can be pruned out based on

geodesic distance preservation. If one of the ratiosd(pi,p j)

d(qi,q j)(or the

reciprocal) is smaller than a threshold δ for i, j = 1,2, · · · ,m and



Corr rate (%) Mesh rate (%)MT BIM OFM Our MT BIM OFM Our

Cat 66 93.7 90.9 96.5 54.6 90.9 90.9 100Centaur 92 100 96.0 92.0 100 100 100 100David 82 97.4 94.8 92.5 57.1 100 100 100Dog 91 100 93.2 97.4 88.9 100 88.9 100Horse 92 97.1 95.2 99.4 100 100 87.5 100Michael 87 98.9 94.6 91.4 75 100 100 100Victoria 83 98.3 98.7 95.5 63.6 100 100 100Wolf 100 100 100 100 100 100 100 100Gorilla - 98.9 98.9 100 - 100 100 100Average 85 98.0 95.1 94.5 76 98.7 96.2 100

Table 1: Evaluation results on TOSCA data sets based on the man-ually selected ground-truth set used in [KLC10].

MT BIM OFM OurCorr rate (%) 82 84.8 91.7 94.5Mesh rate (%) 71.8 76.1 97.2 98.6

Table 2: Evaluation results on SCAPE data sets based on the man-ually selected ground-truth set used in [KLC10].

i 6= j, then this m pair of points set is pruned. For most cases,δ = 0.9 by default can obtain desirable results. A representationmatrix can be computed for each unpruned m pair of points usingthe SVD decomposition same as the method in Section 4.1.

Step 3: clustering in transformation space. If both the symme-try g on the shape and the sampling pairs of points are perfectlysymmetric, then all m symmetric pairs of points set of a g can re-cover an unique representation matrix Cg. However, due to approx-imate symmetries and inprecise symmetric sampling in practice, weneed to find clusters in the transformed representation matrix spacewith the highest density of distribution. The number of clustersis unknown in advance. Thus, the mean-shift clustering [CM02]is used to detect the symmetries with the highest densities in therepresentation matrix space similar to extrinsic symmetry extrac-tion [MGP06]. In each cluster with the highest densities, the matrixclosest to the center of this cluster is selected as the symmetry rep-resentation matrix, which is also refined by the ICP algorithm.

5. Numerical results and applications

Implementation details. In this paper, we use the classical cotan-gent weight scheme [MDSB02] without area normalization for thediscretization of the Laplace-Beltrami operator, which is less sensi-tive to volume distortion and provides a more compact representa-tion of the functional map as pointed out in [OBCS∗12]. In discretesettings, the dimension of the approximated symmetry representa-tion matrix Cg is about 20 depending on the model, i.e., the sumof multiplicities ∑

si=1 ik of the first used s eigenvalues is closest to

20. One hundred eigenfunctions are used for computing the HKSsand WKSs. We refine the initial computed representation matrixof each symmetry, using the ICP procedure with 20 iterations. Thefast marching method is used to compute the geodesic distances ontriangular meshes [KS98].

The main time-consuming parts are eigen-decomposition of the

Laplace-Beltrami operator and ICP refinement for symmetry detec-tion on C2 shapes. Our MATLAB codes run about 2− 20 secondson a laptop with an Intel Core 2.30 GHz processor on average formodels ranging from 5− 50K points. The computational time forshapes with more symmetries mainly depends on the number ofsampling points n and maximum multiplicity m, where the genera-tion of consistent symmetric pairs of points and mean shift cluster-ing take the most time. It costs 18 seconds for the Table model with13714 vertices shown in Figure 9, where the values of the numberof symmetric sampling points n and maximum multiplicity m are 8and 2, respectively.

Comparisons. For g ∈ G(M), if some points pi ∈ T and theirground-truth symmetric points Gg(pi) are given manually, we com-pare the geodesic distance error between the computed symmetrypoint g(pi) and Gg(pi), where the following two evaluation metricsin [KLC10] are used:

• Correspondence rate: The percentage of points in T for whichthe geodesic error is less than a tolerance error ε .

• Mesh rate: The percentage of shapes for which the correspon-dence rate is above a threshold β .

We set ε =

√area(M)

20πand β = 75% by default, which are the same

as the previous works [KLC10, LLL∗15].

There are benchmarks of reflectional symmetries [KLC10] avail-able on both TOSCA [BBK08] and SCAPE [ASK∗05] data sets,where ground truth symmetric points of T are manually givenfor each shape. We compare the proposed global intrinsic sym-metry detection method for C2 shapes on the two data setswith the state-of-the-art previous works, Möbius Transformations(MT) [KLC10], Blended Intrinsic Maps (BIM) [KLF11], and Or-thonormal Functional Maps (OFM) [LLL∗15], in Table 1 and Ta-ble 2 respectively using the above two metrics. Our method clearlyobtains the best performance on the SCAPE data set and com-parable results on the TOSCA data set with much less computa-tional effort. It is because our band-by-band computation has onlya small number of constraints and is not required to solve com-plex optimization problems. Our MATLAB implementation takes24 minutes to compute the global symmetries for all meshes on theTOSCA data set, while the times of OFM and BIM are more thanone hour and six hours as indicated in [LLL∗15]. Some point-to-point correspondences of our detected symmetries on the SCAPEand TOSCA data sets are illustrated in Figures 6 and 7, respectively.

Evaluation. All of the previous works mainly detect reflectionalsymmetries, so there are no benchmarks available for evaluatingrotational global intrinsic symmetries. To evaluate the performanceof our method on symmetric shapes with symmetry groups otherthan C2, we use the five-pointed star model in Figure 8 for an exam-ple, where the known twenty extrinsic symmetries serve as groundtruth. One hundred points T = {pi, i = 1,2, · · · ,100} obtained byfurthest point sampling are used for evaluation. For each global in-trinsic symmetry g, we compute the mean geodesic distance errorsbetween our computed intrinsic symmetry point g(pi) and its ex-trinsic ground-truth Gg(pi), i = 1,2, · · · ,100, which are recordedunder each symmetry shown in Figure 8. It can be seen that themean errors of the twenty symmetries are far smaller than the tol-erance error ε = 3.92, and all of the mesh rates are 100%.



Figure 6: Some symmetry detection results of our method on the SCAPE data sets [ASK∗05].

Figure 7: Some symmetry detection results of our method on the TOSCA data sets [BBK08].

Figure 8: Mean geodesic distance errors of the computed twenty global intrinsic symmetries (under each symmetry), which are much smallerthan the tolerance error ε = 3.92. The left column is the original color both in front and back view, while the other figures demonstratesymmetries by transferring the original color onto the shape itself in front view.

Rotational Reflectional

g1 g2 g3 g4 g5 g6 g7 g8

Figure 9: The computed eight global intrinsic symmetries on a Table model, where each symmetry is demonstrated by transferring theoriginal color (middle) onto the shape itself. The composition operation table of this global intrinsic symmetry group is show in Table 3.



g1 g2 g3 g4 g5 g6 g7 g8g1 g1 g2 g3 g4 g5 g6 g7 g8g2 g2 g3 g4 g1 g7 g8 g6 g5g3 g3 g4 g1 g2 g6 g5 g8 g7g4 g4 g1 g2 g3 g8 g7 g5 g6g5 g5 g8 g6 g7 g1 g3 g4 g2g6 g6 g7 g5 g8 g3 g1 g2 g4g7 g7 g5 g8 g6 g2 g4 g1 g3g8 g8 g6 g7 g5 g4 g2 g3 g1

Table 3: Group operation table of the global intrinsic symmetrygroup of the Table model shown in Figure 9, where the element inthe i-th row and j-th column is the composition symmetry g j ·gi.

g1 g2 g3 g4 g5 g6 g7 g8Inverse g1 g4 g3 g2 g5 g6 g7 g8Order 1 4 2 4 2 2 2 2

Table 4: The corresponding visuals are shown in Figure 9.

Characterization of symmetry group. Our method can detect allreflectional and rotational global intrinsic symmetries of a 3D shapeand represent them as matrices. Since the composition operation ofsymmetries now becomes a much simpler matrix product operationof their representation matrices, we are able to efficiently apply lin-ear algebra methods to characterize the symmetry group.

When all of the symmetry representation matrices are computed,we can easily obtain the symmetry group table, which describes thestructure of the finite symmetry group by arranging all of the pos-sible composition results of the symmetries. In practice, the infinitesymmetry representation matrix is approximated by a finite dimen-sion one, the multiplication operation of G′(M) is not closed. Thus,the composition symmetry g2 ·g1 of given two symmetries g1 andg2, is the one whose representation matrix has the lowest Euclideandistance with the multiplied product Cg1 ∗Cg2 from G′(M). Thegroup table of the global intrinsic symmetry group with eight ele-ments of the model in Figure 9 is shown in Table 3. Furthermore,the inverse element and order of each symmetry can be easily ob-tained from the group operation table shown in Table 4.

6. Conclusion

In this paper, a novel group representation is introduced for globalintrinsic symmetries on 3D shapes, where each symmetry can berepresented as a matrix. The composition operation of symmetriesbecomes a simple product of their representation matrices undersuch representation, which allows us to fast study the structure ofsymmetry group using linear algebra methods. Furthermore, weprove that each global intrinsic symmetry can be uniquely recov-ered from sparse symmetric pairs of points under mild conditions.

Based on the above solid theories, we have also proposed aneffective algorithm to detect global intrinsic symmetries in prac-tice. For shapes with only one non-trivial reflectional symmetry,our proposed method performs comparably or slightly better thanstate-of-the-arts with much smaller computation cost. Our method

can detect all rotational global intrinsic symmetries and describetheir relations in the symmetry group.

Limitations and future works. Although the proposed group rep-resentation is generally valid for compact manifolds in any dimen-sion and genus as proven in Section 3, the algorithm introducedin Section 4 fails for shapes with continuous global intrinsic sym-metries, such as a sphere and cylinder. This is because the clus-tering method cannot find infinite symmetries on these shapes. Inthe future, we would like to describe the continuous symmetriescombined with the theories of the Lie group and the Lie algebrarepresentation. Secondly, the Euclidean distances of the high di-mensional orthogonal matrix used in the clustering might be re-placed with possibly more robust measures, for example, the log-arithm mapping for describing 3D extrinsic symmetry [SAD∗16].Moreover, if the number of sampling points and the maximum mul-tiplicity are very large, the pairs of points generation and clusteringof the representation matrices could be computationally heavy. Fi-nally, some of the parameters of our method are set manually. Wealso would investigate automatic or data-driven method for optimalparameter selection.

Acknowledgments

We thank the anonymous reviewers for their constructive com-ments. We also thank Maks Ovsjanikov for sharing the Oc-topus models used in Figures 1 and 3, and Alexander Bron-stein et al. for sharing the codes of computing geodesic dis-tances. This work was supported in part by NSFC (61402300,61373160, 61522213, 61379090), 973 Program (2015CB352501),Guangdong Science and Technology Program (2014TX01X033,2015A030312015, 2016A050503036), Shenzhen Innovation Pro-gram (JCYJ20151015151249564, 827-000196) and ExcellentYoung Scholar Fund of Shijiazhuang Tiedao University.

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