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Eurographics Symposium on Processing (2004) R. Scopigno, D. Zorin, (Editors)

Symmetry Descriptors and 3D Matching

Michael Kazhdan, Thomas Funkhouser and Szymon Rusinkiewicz

Department of Computer Science, Princeton University, Princeton NJ

Abstract

In this paper, we present the Descriptors of a 3D model. This is a collection of spherical functions that describes the measure of a model’s rotational and reflective symmetry with respect to every axis passing through the center of mass. We show that Symmetry Descriptors can be computed efficiently using fast signal processing techniques, and demonstrate the empirical value of Symmetry Descriptors by showing that they improve matching performance in a variety of shape retrieval experiments.

Categories and Subject Descriptors (according to ACM CCS): I.3.6 [Computer Graphics]: Methodology and Tech- niques

1. Introduction of the projection of the model onto the space of models having that symmetry. Symmetry has long been recognized as playing an inte- gral role in human recognition [Att55, Vet92]. It is char- Figure 1 shows a visualization of the Symmetry Descrip- acteristic of repeating patterns within a model, and can be tors of two models. The descriptors are represented by scal- used to guide reconstruction, compression, and classifica- ing points on the unit in proportion to the measure tion. This awareness has motivated the development of a of symmetry, so that points corresponding to axes of near wide range of techniques for identifying the symmetry are pushed out from the origin and points corre- of a 2D image [Ata85, Wol85, Hig86, Sun95, Mar89]. How- sponding to axes of near anti-symmetry are pulled in to the ever, the increased complexity of the in three- origin. Thus, for the 2-fold (respectively k-fold) symmetry has resulted in little research on symmetry de- descriptors, peaks in the descriptors correspond to axes of tection in 3D. Methods for measuring individual symme- near perfect 2-fold (respectively k-fold) rotational symme- tries have been proposed [Zab94, Zab95], and a general ap- try. Similarly, for the reflective symmetry descriptors, peaks proach for characterizing the measure of all reflective sym- correspond to unit vectors to planes of near metries has been described [Kaz02, Kaz04], but no analo- perfect reflective symmetry. gous method for describing all rotational symmetries exists. In this paper, we present the Symmetry Descriptors of a The contribution of our work is three-fold. First, we define model. This is a generalization of the Reflective Symme- a continuous measure for the reflective and rotational sym- try Descriptor presented in [Kaz02, Kaz04]. It represents a metry of a 3D model. Second, we provide an efficient algo- 3D model as a collection of spherical functions that give rithm for computing the measure for all symmetries about a the measure of a model’s reflective and rotational symme- model’s center of mass. Third, we present experimental re- try, with respect to every axis passing through the center of sults evaluating the empirical value of the symmetry descrip- mass. Thus, it can be used not only to identify axes of per- tors in shape retrieval applications. In these experiments, we fect symmetry, but also to measure the quality of symmetry find that symmetry can be used to augment existing methods with respect to any axis. Specifically, the measure of k-fold for matching 3D , providing enhanced discrimination symmetry of a model around some axis is defined to be the and matching performance without sacrificing efficiency.

c The Eurographics Association 2004. M. Kazhdan T. Funkhouser & S. Rusinkiewicz / Symmetry Descriptors

of work needed to transform a model into a symmetric model, measured as the sum of the of the distances that points would need to be moved. This approach made it possible to evaluate symmetries in the presence of noise, but suffered from the fact that it depended on the establish- ment of correspondences. While this issue could be addressed in the case of 2D with uniform sampling, it made it difficult to generalize the method to 3D where uniformly sampling surfaces is often impossible. The difficulty of establishing point correspondences for matching surfaces in 3D has motivated the development of A visualization of the symmetry descriptors for a stool Figure 1: shape descriptors which represent a 3D model by a func- and an iris. The visualization is obtained by unit vectors on the sphere in proportion to the measure of tion defined on a canonical domain, independent of the ini- about the axis through the center of mass, in the direction of the vec- tial model’s shape or topology. (For a general review of such tor, and the measure of reflective symmetry about the through methods see [Pop94, Tan04].) For these descriptors, match- the center of mass, normal to the vector. ing two models could now be performed without explicitly establishing correspondences, by comparing the values of the corresponding shape descriptors at each point.

The rest of this paper is structured as follows. Section 2 The advantage of the canonical parameterization of shape reviews related work in the of symmetry detection. Sec- descriptors was leveraged in a number of symmetry detec- tion 3 provides a theoretical overview of symmetry, and de- tion algorithms [Oma96, Sun97]. These methods used the fines the Symmetry Descriptors. Section 4 describes an effi- fact that the covariance of a 3D model rotates with cient method for computing the Symmetry Descriptors of the model, so that a model could only have symmetries spherical and voxel representations of a 3D model, while where its covariance ellipsoid had them. Since the only axes Section 5 summarizes some properties of the descriptors. In of symmetry of an ellipsoid have to align with its princi- Section 6, we describe how symmetry information can be pal axes, this provided an efficient way to identify candi- incorporated into existing rotation representations, date axes of symmetry. The actual quality of an axis as an and in Section 7, we evaluate the contribution of symmetry axis of symmetry would then be measured by comparing the augmentation in experiments comparing the retrieval perfor- shape descriptor of the model with the shape descriptors of mance of the original representation with the retrieval per- the and reflections of the model about the candidate formance of the augmented representation. Finally, we con- axis. This method had the advantage of providing a contin- clude in Section 8 by summarizing our work. uous measure of symmetry for candidate axes of symme- try without necessitating the establishment of point corre- spondences. Furthermore, the method was a general one that 2. Related Work could be applied to wide class of shape descriptors. How- ever, the method’s dependence on PCA for the identification Early approaches to symmetry detection focused of candidate axes could only guarantee the correct identifi- on detecting the symmetries of planar point cation of symmetry axes for models with perfect symmetry. sets [Ata85, Wol85, Hig86]. These methods reduced the symmetry detection problem to a detection of symmetry Motivated by the ease of evaluating symmetry using shape in circular strings, and used efficient substring algorithms descriptors, and the efficiency of exhaustive search provided (e.g., [Knu77]) to detect the symmetries by searching for by early substring matching approaches, efficient meth- the appearance of a string within its concatenation with ods for evaluating the symmetries of a 2D model, at ev- itself. While these methods had the theoretical advantage ery symmetry, were developed. The key idea of these ap- of efficiently evaluating all possible symmetries, they were proaches was the generalization of discrete substring match- impractical in empirical settings since they were algorithms ing to continuous correlation with the Fast Fourier Trans- that could only identify the perfect symmetries of a model. form. These methods [Sun95, Mar89] compute the symme- Thus if a symmetric model had even a small amount of tries of a model by using correlation to compare the shape noise, these methods would fail to identify its symmetries. descriptor of a 2D model with all of its rotations and reflec- tions. This approach was a general one that could be applied In order to address this issue, Zabrodsky et to any shape descriptor that represented a model with a func- al. [Zab94, Zab95] defined a continuous measure of tion defined either on a , or in 2D. symmetry which transformed the binary question: “Does a model have a given symmetry?” to the continuous question: The dependence of these methods on the FFT made them “How much of a given symmetry does a model have?” The hard to generalize to shape descriptors that represented a measure of symmetry was defined as the minimum amount 3D model with either a spherical function or a function in

c The Eurographics Association 2004. M. Kazhdan T. Funkhouser & S. Rusinkiewicz / Symmetry Descriptors

3D. In [Kaz02, Kaz04] a method is described for computing In general, computing the projection of v onto the sub- the measure of reflective symmetries for all planes passing space of vectors invariant under the action of G is a difficult through the origin. For a spherical descriptor of size O(N2) task. However, in our case we can use the fact that the ele- (respectively 3D function of size O(N3)) the method com- ments of G are orthogonal transformations. In particular, we putes the measures of reflective symmetry in O(N3 logN) can apply a theorem from representation theory [Ser77] stat- (respectively O(N4 logN)) time. The efficiency of this ap- ing that a projection of a vector onto the subspace invariant proach relies on the use of the FFT to compute correlation under the action of an is the average of the with respect to a single axis efficiently and a generalization vector over the different elements in the group. Thus, in the of this approach to general symmetry detection would result case of a vector v and a group G, we get: 4 in algorithms that have complexity O(N logN) for spherical 2 5 functions and O(N logN) for 3D functions. 2 1 2 1 sdG(v) = v ∑ γ(v) = v ∑ v,γ(v) − G k k − G h i In this work, we show how the analogs of the FFT and in- | | γ∈G | | γ∈G verse FFT on the sphere, namely the Fast Harmonic Trans- giving an expression for the symmetry distance in terms of form and the Fast Inverse Wigner-D Transform, can be used the length of v and the dot-products of v with its image under to compute the measure of all symmetries efficiently. In par- the action of G. ticular, we describe a method for computing the measure of all reflective and rotational symmetries of both spheri- 4 cal functions and 3D functions in O(N ) time, providing a 3.2. Symmetry Descriptors in 3D method for computing all symmetries of a model about its center of mass, in less time than previous methods required In order to evaluate the measure of symmetry of a 3D to compute only the reflective symmetries. model, it is necessary to compare a model with its reflec- tions/rotations. A variety of shape descriptors can be used to compare the model with its transformation, and in this paper 3. Measuring Symmetry we focus on those that represent a model by a spherical, or The first issue we must address is to define a measure of 3D, function that rotates with the model. symmetry for a 3D model with respect to an axis of k-fold rotation or a plane of reflection. To this end, we describe a Notation: For any integer k and any unit vector p we let k method for computing the symmetries of any shape descrip- Gp denote the k-fold rotational with respect k tor that represents a 3D model as a function defined either to p. If k is positive, then Gp is the group generated by the on the sphere or in 3D. We begin by describing a general 2π/k transformation r which is the rotation about the axis p by approach for measuring symmetry, and then present the im- p the angle 2π/k. If k is negative, then Gk is the group gener- plications for measuring the symmetries of a 3D model. p / ated by the transformation r2π k A, where A is the antipodal p · map, sending a point q to the point q. 3.1. Symmetry Detection − 3 For example, G(1,0,0) is the group generated by rotating Definition: Given a vector space V and a group G that acts by 120◦ about the x-axis, consisting of 3 elements, while on V, we say that v V is symmetric with respect to G if G−2 is the group generated by reflecting through the xz- ∈ (0,1,0) γ(v) = v for all γ G. ∈ plane, consisting of two elements.

Definition: We define the symmetry distance of a vector v Definition: Given a shape descriptor f , we define its k-fold with respect to a group G as the L2-distance to the nearest symmetry descriptor as the function on the sphere whose vector that is symmetric with respect to G: value at a point p describes the amount of f that is sym- metric with respect to Gk and the amount of f that is anti- sdG(v) = min v w . p w|G(w)=w k − k symmetric:

⊥ Using the fact that the vectors that are invariant to G de- SDk( f , p) = πGk ( f ) , π k ( f ) k p k k Gp k fine a subspace of V, it follows that the nearest G-invariant   where π k is the projection onto the space of functions that vector w is the projection of v onto the subspace of invariant Gp vectors. That is, if we define πG to be the projection onto the ⊥ are k-fold symmetric about the axis p, and πGk is the pro- subspace invariant under the action of G and we define π⊥ p G jection onto the orthogonal complement. (Note that since to be the projection onto the orthogonal subspace then: 2 2 ⊥ 2 f = πGk ( f ) + πGk ( f ) it suffices to compute one ⊥ k k k p k k p k sdG(v) = v πG(v) = πG (v) ⊥ of πGk ( f ) and π k ( f ) . Despite the redundancy, we k − k k k k p k k Gp k so that the symmetry distance of v with respect to G is the store both values, as they can be used for bounding shape length of the projection of v onto a subspace indexed by G. .)

c The Eurographics Association 2004. M. Kazhdan T. Funkhouser & S. Rusinkiewicz / Symmetry Descriptors

4. Computing the Symmetry Descriptors where p is a unit vector and √4πr2 is the change of variable We will now show how to compute all the k-fold symme- term accounting for the area of the sphere with radius r. try descriptors of a shape descriptor efficiently. We begin by Discretely sampling the different radii, we can express f describing the method for shape descriptors that represent as a collection of spherical functions f ,..., f , with N = { 1 N} a model with a spherical function. Then, we generalize the O(b) where b is the sampling rate. Expressing each of these method to shape descriptors that represent a model with a in terms of its spherical harmonic representation, we get: function defined in 3D. For both cases, the key idea is that in b computing the symmetry descriptors of a shape descriptor f , m fi = ∑ ∑ al,m[i]Yl it is necessary to compare f with its rotations. This amounts l=0 |m|≤l to computing the autocorrelation of f across the group of ro- and for any rotation γ we have tations, f ,γ( f ) , and, while a brute force algorithm would h i b N be prohibitively slow, we show how spherical signal process- m n ing can be used to compute the autocorrelation efficiently. f ,γ( f ) = ∑ ∑ ∑ al,m[i]al,n[i] Yl ,γ(Yl ) . h i l=0 |m|,|n|≤l i=1 !h i

4.1. Efficient Spherical Autocorrelation Computing the autocorrelation of f with all of its rotations 3 4 We begin by describing an efficient method for computing requires: < O(Nb ) = O(b ) time to compute the necessary 3 4 the autocorrelation of a spherical function f across the space spherical harmonics, O(Nb ) = O(b ) time to compute the cross multiplication of harmonic coefficients, and O(b4) of rotations. We express f in terms of its spherical harmonic ≤ decomposition: time to perform the inverse Winger-D transform. Thus, the total complexity of computing the autocorrelation is O(b4). b m f = ∑ ∑ al,mYl Note that increasing the dimensionality of the shape repre- l=0 |m|≤l sentation from a 2D spherical function to a 3D function does where b is the band width of the function f . This decompo- not increase the complexity of computing the symmetry de- sition has the property that for any rotation γ we have: scriptors since the inverse Winger-D transform remains the 0 limiting step. m m 0 Y ,γ(Y 0 ) = 0 l = l . h l l i ∀ 6 This allows us to compute the autocorrelation of f by cross 4.3. Computing the Descriptors multiplying spherical harmonic coefficients within each fre- quency l, ignoring cross-frequency terms: In order to compute the symmetry descriptors of a function f , it suffices to compute the lengths of the projections: b 0 m m 2 f ,γ( f ) = al,mal,m0 Yl ,γ(Yl ) . 1 ∑ ∑0 πGk ( f ) = f ,γ( f ) h i l=0 |m|,|m |≤l h i p k ∑ Gp ∈ k h i | | γ Gp This gives an expression of the function f ,γ( f ) as a 0 h i for all k and all points p on the unit sphere. When k is posi- linear sum of the functions Y m,γ(Y m ) . Since these are k h l l i tive, the elements in Gp are all rotations. Thus, having com- precisely the Wigner-D functions, a fast inverse Wigner-D puted the values the autocorrelation, we can reconstruct the transform [Kos03, Sof03] gives the autocorrelation of f over symmetry descriptors SDk( f , p). However, when k is nega- the space of rotations. tive, some elements of Gk will be of the form γ = rα A – p p · Since the spherical harmonic decomposition can be com- products of a rotation and the antipodal map. In this case, puted in < O(b3) time, the cross multiplication of harmonic we cannot use the computed autocorrelation values directly. 3 coefficients takes O(b ) time, and since the inverse Wigner- In order to be able to compute the symmetry descriptors D transform can be done in O(b4) time, we compute the ≤ SDk( f , p) for negative k, we observe that the antipodal map autocorrelation of f across all rotations in time O(b4). ≤ acts on a function f as follows: 1. If f is an even function, the antipodal map leaves f un- 4.2. Efficient 3D Autocorrelation changed In order to compute the symmetry descriptors of a 3D func- 2. If f is an odd function, the antipodal map sends the func- tion, we decompose the 3D function as a collection of spher- tion f to the function f . ical functions at different radii and use the method described − Thus, we can compute the symmetry descriptors SD ( f , p) if above to compute the symmetry descriptors of the collection k we address the even and odd frequencies of f independently. of spherical functions. In particular, given a function f de- In particular, we express f as the sum of its even and odd fined for all points x 1, we set fr to be the restriction of | | ≤ components, f = f + + f −, with: the function f to the sphere with radius r: + f (p) + f ( p) − f (p) f ( p) 2 f (p) = − f (p) = − − fr(p) = f (rp) 4πr 2 2 p c The Eurographics Association 2004. M. Kazhdan T. Funkhouser & S. Rusinkiewicz / Symmetry Descriptors

Then, instead of computing the autocorrelation of f , we 2 = SDk( f , p) SDk(g, p) compute the autocorrelation of the even and odd parts in- − dependently to get: so that the difference between the symmetry descriptors of

two models, at any point and any type of symmetry, is an Φ+(γ) = f +,γ( f +) Φ−(γ) = f −,γ( f −) f h i f h i explicit bound for the proximity of the two models. This provides a general expression for the value of 2 π k ( f ) for all k = 0 and all axes p as: 5.2. Continuous Symmetry Classification Gp 6 One of the challenges of shape retrieval stems from the fact |2k| 1 + 2 jπ/k j − 2 jπ/k that often 3D models are not a priori aligned, and many Φ f (rp ) + (Sgn(k)) Φ (rp ). 2k ∑ f methods for comparing two models require an initial step | | j=1 ! of pair-wise registration. For these types of applications, the Note that to compute the measure of axial symmetry globality property mentioned above cannot be utilized with- πG∞ ( f ) , it suffices to compute π k ( f ) with k equal to k p k k Gp k out first aligning the models. In this section we show how twice the band width of the function. symmetry information can be used for comparing two mod- els without requiring the initial alignment step. Complexity: For both spherical functions and 3D func- We are motivated in our approach by early work in sym- tions the complexity of computing the autocorrelation over metry detection [Ata85, Wol85, Hig86] where the goal was all rotations is bounded by O(b4). Since computing the k-th to classify models in terms of the types of symmetry that symmetry descriptor requires O(k) summations for each of they have. These methods sought to assign a binary value to O(b2) points on the sphere the overall complexity of com- each integer k, indicating whether or not a model had k-fold puting the O(b) symmetry descriptors is O(b4). symmetry. Since such a representation did not specify the axis of symmetry, it was inherently rotation invariant. 5. Symmetry and Model Similarity Using the symmetry descriptors, we extend these binary Work in symmetry detection has been motivated, in part, by classifications into a continuous framework where for each the recognition that symmetry is a property characterizing k, we store the optimal measure of k-fold symmetry, even global shape information so that storing a small amount of when the model is not k-fold symmetric. In particular, setting symmetry information for each model should provide an ef- sk( f ) to be the maximal value of k-fold symmetry of f : ficient bound for the similarity of two models. In this section, sk( f ) = max πGk ( f ) we formalize this intuition by explicitly describing how the p∈S2 p difference in the symmetries of two models relates to their we define the optimal k-fold symmetry of f as the pair: measure of similarity. Sym ( f ) = s ( f ), f 2 s ( f )2 k k k k − k 5.1. Globality  q  giving a continuous, rotation invariant classification of a A fundamental property of the symmetry descriptors is that model in terms of its symmetries. Furthermore, as a direct they characterize global properties of a model, and hence if corollary of the globality property, it follows that the sym- the symmetry descriptors of two models differ at even one metry classification can be used to bound the proximity of point, we expect this to imply that the models must be dif- two models: ferent. This property can be formulated explicitly by stat- max Symk( f ) Symk(g) f g . ing that the L∞-difference between symmetry descriptors k − ≤ − bounds the L -difference of the models: 2 Thus, symmetry classifications can be used to match models

without requiring an initial step of pair-wise registration. max SDk( f , p) SDk(g, p) f g . k − ∞ ≤ − 2

The explicit proof of this bound deriv es from the fact that the 6. Symmetry Augmentation values of the symmetry descriptors of a function are equal to the lengths of its projections onto two orthogonal subspaces. Motivated by the property described in Section 5, we would Hence, for any k-fold symmetry, and any axis p we have: like to use the continuous symmetry classification for effi- ciently comparing models in a rotation invariant manner. In 2 2 2 ⊥ ⊥ particular, we would like to augment existing shape descrip- f g = πGk ( f ) πGk (g) + πGk ( f ) πGk (g) − p − p p − p tors with symmetry information, but would like to do so in 2 a manner that is not redundant. To this end, we consider the πGk ( f ) πG k (g) + ≥ p − p Spherical Harmonic Representation described in [Kaz03].   2 ⊥ ⊥ + π Gk ( f ) πGk (g) The Spherical Harmonic Representation is a general p − p  

c The Eurographics Association 2004. M. Kazhdan T. Funkhouser & S. Rusinkiewicz / Symmetry Descriptors method for obtaining a rotation invariant representation of spherical (and 3D) shape descriptors that describes the de- scriptors in terms of the distribution of energies across dif- ferent frequencies (and radii). Specifically, given a spherical function f , the Spherical Harmonic Representation decom- poses the function in terms of its frequency components fl:

b m f = ∑ fl, with fl = ∑ al,mYl l=0 |m|≤l where al,m are the coefficients of f with respect to the spheri- m cal harmonic basis Yl . A rotation invariant representation of f is then obtained by storing only the norms of the different frequency components: Figure 2: An example of the improvement gained by augmenting SH( f ) = f0 , f1 ,..., fb . the energy representation with symmetry information. The database {k k k k k k} was queried with the near axially-symmetric table on the left and re- The advantages of this representation are two-fold: First, the sults are shown for retrieval without (top) and with (bottom) symme- representation is rotation invariant by construction, making try augmentation (considering, axial, 2-fold, 3-fold, and reflective it possible to compare models without first aligning them. symmetries). Note that symmetry augmentation improves matching Second, in going from a spherical function to its Spherical performance by introducing a preference for models which have Harmonic Representation, the dimensionality of the repre- near axial symmetry. sentation is reduced, contracting a 2D spherical function to a 1D array of energy values. with the symmetry augmented representations: However, it has been noted that the Spherical Harmonic Representation treats each frequency component indepen- f f SH ( f ) = f ,Sym ( f ) k 1k ,...,Sym ( f ) k bk . dently and does not capture information characterizing the k k 0k k · f k · f  k k k k  alignment between different frequency components. Sym- Then, to compare two descriptors, we find the symmetry metry, by contrast, depends strongly on the manner in which type for which the two models vary most, and compare the the different frequencies align, and therefore captures infor- corresponding symmetry augmented representations: mation that is missing in the Spherical Harmonic Represen- tation. Thus, augmenting the Spherical Harmonic Represen- D( f ,g) = max SHk( f ) SHk(g) . k k − k tation with symmetry information should provide a more discriminating representation, combining the local (in fre- Figure 3 demonstrates the process of symmetry augmen- quency space) information of the Spherical Harmonic Rep- tation. Given a spherical shape descriptor (shown in the top resentation with global symmetry information. left), its Spherical Harmonic Representation is computed by expressing the spherical function in terms of its frequency Figure 2 demonstrates the motivation for this approach. components, f , f ,... , and storing the norm of each com- In this figure, a database is queried with the near-axially { 0 1 } ponent (shown in the top right). The symmetry descriptors symmetric table on the left, and retrieval results are shown are computed, and the continuous, k-fold symmetry of f is without (top) and with (bottom) symmetry augmentation. extracted (shown in the bottom left). Finally, the Spherical Note that the addition of symmetry induces a preference for Harmonic Representation is augmented with symmetry in- models that are near-axially symmetric, and pushes away formation by scaling with the k-fold symmetry of f , to ob- models (such as the table, second model in the non- tain a finer resolution of non-constant frequency information augmented results) that do not have such a symmetry. (shown in bottom right). In order to augment the Spherical Harmonic Represen- tation we make the assumption that symmetry is uniformly 6.1. Comparing the Symmetry Augmented Descriptor distributed across all the non-constant frequencies (constant frequency components are fully symmetric), so that if f is a Despite the fact that the symmetry augmented representa- tion now requires a copy of the Spherical Harmonic Rep- shape descriptor and Symk( f ) is the measure of the k-fold symmetry of f then: resentation for each symmetry type, in theory encumbering both storage and comparison, the symmetry augmented rep- f Sym ( f ) Sym ( f ) k lk resentation is in fact compact and compares efficiently. In k l ≈ k · f k k particular, if we compute the symmetry dot product: where f is the l-th frequency component of f . Thus, we re- Sym ( f ) Sym (g) l SDot( f ,g) = max k , k place the original Spherical Harmonic Representation (SH) k f g  k k k k 

c The Eurographics Association 2004. M. Kazhdan T. Funkhouser & S. Rusinkiewicz / Symmetry Descriptors

the importance of symmetry in the measure of model simi- larity. In particular, we can define the family of metrics: D2 ( f ,g) = f 2 + g 2 α k k k k 2 f g 2SDotα( f ,g) FDot( f ,g). − k 0kk 0k − · indexed by the parameter α. When α = 0 symmetry plays no role in shape comparison and we revert to the Spherical Harmonic Representation. When α = 1 we obtain the sym- metry augmented representation described above. And more generally, as α is increased, symmetry plays a more defining role in evaluation of shape similarity.

7. Experimental Results To measure the efficacy of the symmetry augmented Spher- ical Harmonic Representation in tasks of shape retrieval, we computed a number of spherical shape descriptors, and com- pared matching results when the Spherical Harmonic Repre- sentation was used with the results obtained when the Spher- ical Harmonic Representation was augmented with symme- try information. The descriptors we used in our experiments were: Spherical Extent Function[Vra1]: A description of a • Figure 3: The augmented Spherical Harmonic Representation of associating to each ray from the origin, the value a 3D shape descriptor (top left) is obtained by first computing the equal to the distance to the last point of of the Spherical Harmonic Representation (top right) and the k-fold sym- model with the ray. (If the intersection is empty then the metries of f (bottom left). The k-fold symmetries are then used to associated value is zero.) provide a finer resolution of non-constant frequency information by Radialized Spherical Extent Function[Vra03]: A de- • multiplying each frequency norm by the pair of k-fold symmetry val- scription of a surface which first decomposes 3D space ues (bottom right). into concentric shells and then computes the Spherical Ex- tent Function of the intersection of the model with each shell independently. The resulting descriptor represents a 3D model with a collection of spherical functions. and the frequency dot product: Shape Histogram (Sectors)[Ank99]: A description of a • b surface associating to each ray from the origin, the amount FDot( f ,g) = f g ∑ k lkk lk of surface area that sits over it. l=1 Shape Histogram (Sectors and Shells)[Ank99]: A de- • independently, we can separate the role of symmetry infor- scription of a surface which first decomposes 3D space mation from frequency information in the measure of shape into concentric shells and then computes the Sector repre- similarity: sentation of the intersection of the model with each shell independently. The resulting descriptor represents a 3D D2( f ,g) = f 2 + g 2 k k k k model with a collection of spherical functions. 2 f0 g0 2SDot( f ,g) FDot( f ,g). Voxel: A description of a model as a voxel grid, which − k kk k − · • is obtained by rasterizing the boundary of the model. The Thus, in comparing two descriptors, the symmetry infor- voxel grid is represented as a collection of spherical func- mation is separated from frequency information and only a tions by restricting the grid to concentric and scal- single copy of the Spherical Harmonic Representation needs ing by √4π r2 to account for the change of area. · to be stored. Furthermore, the separation of symmetry infor- Gaussian Euclidean Distance Transform[Fun03]: A de- • mation from frequency information allows for efficient com- scription of a shape as a voxel grid, where the value at parison of two models, since the computations of SDot( f ,g) each point is given by the composition of a Gaussian with and FDot( f ,g) are both efficient computations that can be the Euclidean Distance Transform of the surface. Similar performed independently, and then combined to give the to the Voxel representation, the grid is represented by a measure of similarity. collection of spherical functions. Finally, the separation of symmetry information from fre- We evaluated the performance of each method by measur- quency information provides an easy method for modulating ing how well they classified models within a test database.

c The Eurographics Association 2004. M. Kazhdan T. Funkhouser & S. Rusinkiewicz / Symmetry Descriptors

Figure 4: The improvement in the precision of both the original and augmented Spherical Harmonic Representation over PCA-alignment are shown. Note that symmetry augmentation always improves the matching performance and out-performs the PCA-aligned representation.

The database consisted of 1890 “household" objects pro- ing on the retrieval performance of the original Spherical vided by Viewpoint [Vie01]. The objects were clustered into Harmonic Representation. 85 classes, based on functional similarity, largely following the groupings provided by Viewpoint and classes ranged in Representation Spherical Voxel size from 5 models to 153 models, with 610 models that did not fit into any meaningful classes [Fun03]. Classifica- Size (floats) 256 8192 tion performance was measured using precision/recall plots, PCA Compute (sec) 0.01 0.09 which give the percentage of retrieved information that is Compare (ms) 0.18 5.89 relevant as a function of the percentage of relevant informa- tion retrieved. That is, for each target model in class C and Size (floats) 16 512 any number K of top matches, “recall” represents the ratio of Harmonic Compute (sec) 0.01 0.09 models in class C returned within the top K matches, while Compare (ms) 0.02 0.31 “precision” indicates the ratio of the top K matches that are Size (floats) 28 524 in class C. Thus, plots that appear shifted up generally indi- Symmetry Compute (sec) 0.59 0.72 cate superior retrieval results. Compare (ms) 0.02 0.32

The results of the Precision vs. Recall experiments are Table 1: A table of the sizes and compute and compare times of the shown in Figure 4, where the Spherical Harmonic Represen- spherical and voxel shape descriptors using PCA-alignment, Spher- tation is augmented with k-fold symmetry information, with ical Harmonic Representations, and symmetry augmentation. k = 2,2,3,4,5, corresponding to reflective, 2-fold, 3- − ∞ fold, 4-fold, 5-fold and axial symmetry information. A value Of particular importance is the fact that augmented rep- of α = 2 was used to amplify the importance of symme- resentation always outperforms the PCA-aligned descrip- try in retrieval – this was empirically determined to give the tors (the improved precision plot is always bigger than best results. The plots compare the improvement in precision zero). For a number of descriptors we find that at low re- over PCA-alignment when models are matched by compar- call values the Spherical Harmonic Representation suffers ing the (rotation invariant) Spherical Harmonic Representa- from the information loss inherent in the representation and tions of the descriptors and models are matched by com- performs worse than PCA-aligned descriptors. By contrast, paring the symmetry augmented Spherical Harmonic Rep- the augmented representation always does better than PCA- resentations of the descriptors, (so that PCA-aligned models alignment, despite the fact that the augmented Spherical would give a constant base- of 0% improvement.) Note Harmonic Representation is much smaller than the PCA- that for all of the descriptors, the symmetry augmented rep- aligned representation. The space and time complexity of resentation provides better matching performance, improv- the different representations are compared in Table 1 which

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