Eurographics Symposium on Geometry Processing (2004) R. Scopigno, D. Zorin, (Editors) Symmetry Descriptors and 3D Shape Matching Michael Kazhdan, Thomas Funkhouser and Szymon Rusinkiewicz Department of Computer Science, Princeton University, Princeton NJ Abstract In this paper, we present the Symmetry 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 magnitude 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 sphere 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 symmetries 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 rotation group in three- origin. Thus, for the 2-fold (respectively k-fold) symmetry dimensions 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 perpendicular 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 shapes, 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 squares 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 point correspondences. While this issue could be addressed in the case of 2D curves 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 scaling unit vectors on the sphere in proportion to the measure of rotational symmetry 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 plane 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 area 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 ellipsoid 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 invariant 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 rotations 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 circle, 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 orthogonal group is the average of the with respect to a single axis efficiently and a generalization vector over the different elements in the group.