Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors

Transcription

1 Eurographis Symposium on Geometry Proessing (003) L. Kobbelt, P. Shröder, H. Hoppe (Editors) Rotation Invariant Spherial Harmoni Representation of 3D Shape Desriptors Mihael Kazhdan, Thomas Funkhouser, and Szymon Rusinkiewiz Department of Computer Siene, Prineton University, Prineton NJ Abstrat One of the hallenges in 3D shape mathing arises from the fat that in many appliations, models should be onsidered to be the same if they differ by a rotation. Consequently, when omparing two models, a similarity metri impliitly provides the measure of similarity at the optimal alignment. Expliitly solving for the optimal alignment is usually impratial. So, two general methods have been proposed for addressing this issue: (1) Every model is represented using rotation invariant desriptors. () Every model is desribed by a rotation dependent desriptor that is aligned into a anonial oordinate system defined by the model. In this paper, we disuss the limitations of anonial alignment and present a new mathematial tool, based on spherial harmonis, for obtaining rotation invariant representations. We desribe the properties of this tool and show how it an be applied to a number of existing, orientation dependent, desriptors to improve their mathing performane. The advantage of this is twofold: First, it improves the mathing performane of many desriptors. Seond, it redues the dimensionality of the desriptor, providing a more ompat representation, whih in turn makes omparing two models more effiient. Categories and Subjet Desriptors (aording to ACM CCS): I.3.6 [Computer Graphis]: Methodology and Tehniques 1. Introdution Over the last deade, tools for aquiring and visualizing 3D models have beome integral omponents of data proessing in a number of disiplines, inluding mediine, hemistry, arhiteture and entertainment. With the proliferation of these tools, we have also witnessed an explosion in the number of available 3D models. As a result, the need for the ability to retrieve models from large databases has gained prominene and a key onern of shape analysis has shifted to the design of effiient and robust mathing algorithms. One of the prinipal hallenges faed in the area of shape mathing is that in many appliations, a model and its image under a similarity transformation are onsidered to be the same. Thus, the hallenge in omparing two shapes is to find the best measure of similarity over the spae of all transformation. The need for effiient retrieval makes it impratial to expliitly query against all the transformations, and two different solutions have been proposed: Normalization: Shapes are plaed into a anonial oordinate frame (normalizing for translation, sale and rotation) and two shapes are assumed to be near-optimally aligned when eah is in its own frame. Thus, the best measure of similarity an be found without expliitly trying all possible transformations. Invariane: Shapes are desribed in a transformation invariant manner, so that any transformation of a shape will be desribed in the same way, and the best measure of similarity is obtained at any transformation. We have found that while traditional methods for translation and sale normalization provide good mathing results, methods for rotation normalization are less robust and hamper the performane of many desriptors. In this paper we present a novel tool, alled the Spherial Harmoni Representation, that transforms rotation dependent shape desriptors into rotation independent ones. This tool ontributes to the hallenges of designing effetive shape retrieval algorithms in three ways. First, it is a general tool that an be applied to many existing shape desriptors. Seond, for most shape desriptors, the spherial har-

2 Kazhdan et al / Spherial Harmoni Representations moni representation provides better mathing results than those obtained by rotation normalization. Finally, the spherial harmoni representation provides a redution in the dimensionality of the shape desriptor, thereby reduing both the spae for storage and the time for omparison key properties for the implementation of interative shape retrieval systems. The rest of this paper is strutured as follows: In Setion we desribe previous work in the area of shape retrieval. The spherial harmoni representation is presented in Setion 3, whih reviews the prinipal properties of spherial harmonis and provides a method for obtaining rotation invariant representations of spherial-based shape desriptors. In Setion 4 we desribe the mathematial properties of the spherial harmoni representation and disuss questions of invertibility. We provide a generalization of our method to voxel grids in Setion 5. In Setion 6 we provide empirial results omparing mathing results of normalized desriptors with their rotation invariant representations. We provide an analysis of these results in Setion 7 and we onlude in Setion 8 by summarizing our results and disussing topis for future work.. Related Work The problem of shape mathing has been well studied in the graphis/vision literature and many methods for evaluating model similarity have been proposed. This paper is motivated by the inreased availability and aessibility of 3D models, and fouses on the problem of shape retrieval from within large databases of models. In this ontext, the hallenge is to provide a robust and effiient method for omputing model similarity. To address this hallenge, many methods have foused on separating the mathing problem into two omponents: An offline step, in whih abstrated distinguishing information is extrated from eah model independently, and an online step, in whih the information between two models is ompared. In order to allow for effiient retrieval, the offline step is usually designed to extrat information whih allows for simple and effiient omparison between models. In partiular, many existing methods desribe a 3D shape with an abstrated shape desriptor that is represented as a funtion defined on a anonial domain. Shapes are then ompared by omputing the differene between their desriptors, so that no expliit establishing of orrespondenes is neessary, and the online proess an be effiient. However, in the ontext of shape retrieval, one of the prinipal diffiulties faed by these approahes is that a model and its image under a similarity transformation are onsidered to be the same. Thus, the hallenge in omparing two models is to find the best measure of similarity over the spae of all transformations. This hallenge has been addressed in two different ways: Normalizing the models by finding a anonial transformation for eah one. Charaterizing models with a transformation invariant desriptor so that all transformations of a model result in the same desriptor. (While expliitly solving for the optimal transformation using either exhaustive searh or methods suh ICP 13 14, the Generalized Hough Transform 15, or Geometri Hashing 16, are also possible, these approahes annot be applied to database retrieval tasks sine the online omparison of models beomes ineffiient.) Many hybrid methods exists and a few representative examples are shown in Table 1, whih desribes how these methods address translation, sale and rotation. Representation Tr S Rot Crease Histograms I N I Shape Distributions 3 I N I Extend Gaussian Images 4 I N N Shape Histograms 5 (Shells) N N I Shape Histograms 5 N N N Spherial Extent Funtions 6 N N N Wavelets 7 N N N Refletive Symmetry Desriptors 8 N N N Higher Order Moments 9 N N N Exponentation EDT 1 N N N Table 1: A summary of a number of shape desriptors, showing if they are (N)ormalized or (I)nvariant to eah of translation, sale and rotation. In general, models are normalized by using the enter of mass for translation, the root of the average square radius for sale, and prinipal axes for rotation. We have found that while the methods for translation and sale normalization are robust for whole objet mathing 10, rotation normalization via PCA-alignment does not provide provide a robust normalization for many mathing appliations. This is due to the fat that PCA-alignment is performed by solving for the eigen-values of the ovariane matrix. This matrix aptures only seond order model information, and the assumption in using PCA is that the alignment of higher frequeny information is strongly orrelated with the alignment of the seond order omponents. (Appendix A provides an analysis of this from a signal proessing framework.) We have found that for many shape desriptors this assumption does not hold, and the use of prinipal axes for alignment hampers the performane of these desriptors. Many of the desriptors that have used PCA-alignment represent a 3D shape as either a spherial funtion or a voxel grid, whih rotates with the model. Examples of suh desriptors have inluded:

3 The Extended Gaussian Image 4, whih desribes the distribution of normals aross the surfae of the model Shape Histograms 5, whih desribe the distribution of points on the model aross all rays from the origin Spherial Extent Funtions 6, whih desribe the maximal extent of a shape aross all rays from the origin Refletive Symmetry Desriptors 8, whih desribe the refletive self-similarity of a shape with respet to refletions about all planes through the origin The voxel desription of Funkhouser et al. 1, whih desribes a model by omputing the negative exponential of its Eulidean Distane Transform For these type of desriptors, we propose a solution to the rotation problem by providing a mathematial tool, based on spherial harmonis, for obtaining a rotation invariant representation of the desriptors. Our approah is a generalization of the Fourier Desriptor 11 method to the sphere, haraterizing spherial funtions by the energies ontained at different frequenies. This idea was intially proposed in 1, and this paper presents a detailed desription of the desriptor, its properties, and empirial results demonstrating its effiay in improving the mathing performane of a number of existing shape desriptors. Kazhdan et al / Spherial Harmoni Representations 3. Spherial Rotation Invariane In this paper, we present a tool for transforming rotation dependent spherial and voxel shape desriptors into rotation invariant ones. The key idea of our approah is to desribe a spherial funtion in terms of the amount of energy it ontains at different frequenies. Sine these values do not hange when the funtion is rotated, the resulting desriptor is rotation invariant. This approah an be viewed as a generalization of the Fourier Desriptor method 11 to the ase of spherial funtions Spherial Harmonis In order to be able to represent a funtion on a sphere in a rotation invariant manner, we utilize the mathematial notion of spherial harmonis to desribe the way that rotations at on a spherial funtion. The theory of spherial harmonis says that any spherial funtion f θ φ an be deomposed as the sum of its harmonis: m l f θ φ a lm Yl m θ φ l 0 m l (This deomposition is visualized in step (1) of Figure 1.) The key property of this deomposition is that if we restrit to some frequeny l, and define the subspae of funtions: then: V l Span Y l l Y l 1 l Y l 1 l Y l l V l is a Representation For the Rotation Group: For any funtion f V l and any rotation R, we have R f V l. Figure 1: We ompute a rotation invariant desriptor of a spherial funtion by (1) deomposing the funtion into its harmonis, () summing the harmonis within eah frequeny, and (3) omputing the norm of eah frequeny omponent. (Spherial funtions are visualized by saling points on the unit sphere in proportion to the value of the funtion at that point, where points with positive value are drawn in light gray and points with negative value are drawn in dark gray.)

4 Kazhdan et al / Spherial Harmoni Representations This an also be expressed in the following manner: if π l is the projetion onto the subspae V l then π l ommutes with rotations: π l R f R π l f V l is Irreduible: V l annot be further deomposed as the diret sum V l V l V l where V l and V l are also (nontrivial) representations of the rotation group. The first property presents a way for deomposing spherial funtions into rotation invariant omponents, while the seond property guarantees that, in a linear sense, this deomposition is optimal. 3.. Rotation Invariant Desriptors Using the properties of spherial harmonis, and the observation that rotating a spherial funtion does not hange its L -norm we represent the energies of a spherial funtion f θ φ as: SH f f 0 θ φ f 1 θ φ where the f l are the frequeny omponents of f : f l θ φ π l f m l a lm Yl m θ φ m l (shown in steps () and (3) of Figure 1.) This representation has the property that it is independent of the orientation of the spherial funtion. To see this we let R be any rotation and we have: SH R f π 0 R f π 1 R f R π 0 f R π 1 f π 0 f π 1 f SH f so that applying a rotation to a spherial funtion f does not hange its energy representation Further Quadrati Invariane We an make our representation still more disriminating by refining the ase of the seond order omponent. Using the results from Appendix A we know that the L -differene between the quadrati omponents of two spherial funtions is minimized when the the two funtions are aligned with their prinipal axes. Thus, instead of desribing the onstant and quadrati omponents by the two salars f 0 and f, we an represent them by the three salars a 1, a, and a 3, where after alignment to prinipal axes: f 0 f a 1 x a y a 3 z However, are must be taken beause as funtions on the unit sphere, x, y, and z are not orthonormal. By fixing an orthonormal basis v 1 v v for the span of x y z we an replae the harmoni representation SH f defined in Setion 3. with the more disriminating representation: SHQ f R 1 a 1 a a 3 f 1 f 3 where R is the matrix whose olumns are the orthonormal vetors v i. 4. Properties of the Spherial Harmoni Representation This setion provides a mathematial analysis of some of the properties and limitations of the spherial harmoni representation. In partiular, we desribe how the similarity of spherial desriptors, defined as the optimum over all rotations, relates to the similarity of their harmoni representations. We also desribe the way in whih information is lost in going from a spherial shape desriptor to its harmoni representation. 1. Similarity: The L -differene between the harmoni representations of two spherial funtions is a lower bound for the minimum of the L -differene between the two funtions, taken over all possible orientations. To see this, we let f θ φ and g θ φ be two spherial funtions, and observe that: SH f SH g f l g l l 0 f l g l f θ φ g θ φ l 0 Similarly, if we onsider the rotation invariant representation desribed in Setion 3.3, we get: SH f SH g SHQ f SHQ g f g But as we have shown, the harmoni representations are invariant to rotation, so we get: SH f SH g SHQ f SHQ g min f R g R SO 3. Information Loss: In general, if a spherial funtion f θ φ is band-limited with bandwidth b, then we an express f as: f θ φ b l l 0 m l a lm Y m l θ φ Thus, the spae of spherial funtions with bandwidth b is of dimension O b. The harmoni representation, however, is of dimension O b so that a full dimension worth of information is lost in going from a spherial funtion to its harmoni representation. This information loss happens in two different ways:

5 First, we treat the different frequeny omponents independently. Thus if we write: Kazhdan et al / Spherial Harmoni Representations f b f l and g l 0 b R l f l l 0 where R l are rotations, then the desriptors of the funtions f and g will be the same. That is, the desriptor is unhanged if we apply different rotations to the different frequeny omponents of a spherial funtion. Figure shows a visualization of this for two spherial funtions. The one on the bottom is obtained from the one on the top by applying a rotation to only one of the frequeny omponents. Though the two funtions differ by more than a single rotation, there spherial harmoni desriptors are the same. (An analagous form of information loss ours with Fourier Desriptors where the phases of different frequenies are disarded independently.) Figure 3: Three spherial funtions of the same amplitude and frequeny are shown. Note that there is no rotation transforming any one of them into the other. 5. Extensions to Voxel Desriptors In Setion 3 we presented a method for obtaining rotation invariant representations of spherial funtions. In this setion we show how this method an be generalized to obtain rotation invariant representations of voxel desriptors Rotation Invariant Representations In order to obtain a rotation invariant representation of a voxel grid we use the obsevation that rotations fix the distane of a point from the origin. Thus, we an restrit the voxel grid to onentri spheres of different radii, and obtain the spherial harmoni representation of eah spherial restrition independently. This proess is demonstrated in Figure 4: First, we restrit the voxel grid to a olletion of onentri spheres. Then, we represent eah spherial restrition in terms of its frequeny deomposition. Finally, we ompute the norm of eah frequeny omponent, at eah radius. The resultant rotation invariant representation is a D grid indexed by radius and frequeny. Figure : The bottom spherial funtion is obtained by rotating one of the frequeny omponents of the top one. Despite the fat that there is no rotation transforming the funtion on the top to the one on the bottom, the desriptors of the two funtions are the same. Seond, for eah frequeny omponent f l, the harmoni representation only stores the energy in that omponent. For l it is not true that if f g then there is a rotation R suh that R f g. Thus knowing only the norm of the l-th frequeny omponent does not provide enough information to reonstrut the omponent up to rotation. (This form of information loss does not our with Fourier Desriptors, as two irular funtions with the same amplitude and frequeny an only differ by phase/rotation.) Figure 3 shows a visualization of this for three spherial funtions. The funtions are all of the same frequeny and have the same amplitude, but there is no rotation that an be applied to transform them into eah other. 5.. Properties In addition to the information loss desribed in Setion 4, the method desribed above also loses information as a result of the fat that the representation is invariant to independent rotations of the different spherial funtions. For example, the plane in Figure 5 (right) is obtained from the one on the left by applying a rotation to the interior part of the model. While the two models are not rotations of eah other, the desriptors obtained are the same. 6. Experimental Results To measure the effiay of the spherial harmoni representation, we omputed a number of spherial shape desriptors, and ompared mathing results when the spherial funtions were aligned by PCA with the results obtained when the spherial harmoni representation was used. In order to evaluate our method we omputed the following spherial desriptors: Extended Gaussian Image 4 : This is a desription of a surfae obtained by binning surfae normals.

6 Kazhdan et al / Spherial Harmoni Representations Figure 5: The model on the right is obtained by applying a rotation to the interior part of the model on the left. While the models differ by more than a single rotation, their rotation invariant representations are the same. Radial Distribution: This is a desription of a surfae that assoiates to every ray through the origin, the average distane and standard deviation of points on the intersetion of the surfae with the ray. Spherial Extent Funtion 6 : This is a desription of a surfae assoiating to eah ray from the origin, the value equal to the distane to the last point of intersetion of the model with the ray. Setors: This is a desription of a surfae assoiating to eah ray from the origin, the amount of surfae area that sits over it. This is a ontinuous implimentation of the shells in Shape Histograms 5, with setors hosen to orrespond to a single ell within the representation of the sphere. Shape Histogram 5 : This a finer resolution of the Setor desriptor that breaks up the bounding sphere of the model into a olletion of shells and omputes the setor desriptor for the intersetion of the model with eah one. Voxel 1 : This is a desription of a shape as a voxel grid, where the value at eah point is given by the negatively exponentiated Eulidean Distane Transform of the surfae. Figure 4: We ompute a rotation invariant desriptor of a voxel grid by interseting the model with onentri spheres, omputing the frequeny deomposition of eah spherial funtion, and omputing the norms of eah frequeny omponent at eah radius. The resultant rotation invariant representation is a D grid indexed by radius and frequeny. We evaluated the performane of eah method by testing how well they lassified models within a test database. The database onsisted of 1890 household" objets provided by Viewpoint 17. The objets were lustered into 85 lasses, based on funtional similarities, largely following the groupings provided by Viewpoint and lasses ranged in size from 5 models to 153 models, with 610 models that did not fit into any meaningful lasses 1. Classifiation performane was measured using preision/reall plots, whih whih gives the perentage of retrieved information that is relevant as a funtion of the perentage of relevant information retrieved. We omputed the spherial representations as grids orresponding to regular sampling along the lines of longitude and lattitude and we used SPharmoniKit.5 18 to obtain the spherial harmoni representation as an array of 33 floating point numbers. Both the spherial desriptors and their spherial harmoni representations were ompared

7 Kazhdan et al / Spherial Harmoni Representations using the L -differene. The results of the lassifiation experiment are show in in Figure 6. general lass of spherial funtions, so that frequeny omponents align independently. However, in ertain shape appliations this may not be the ase and the desriptors obtained may fall into a restritive subset of spherial funtions. In these ases it is possible that the alignment of different frequeny omponents are orrelated and PCAalignment performs well. Suh a ase may our when the spherial funtions are primarily axis aligned, so that, up to rotation, they an be desribed as: a k x k b k y k k z k and the alignments of the different frequeny omponents are strongly orrelated. This is the ase for the Extended Gaussian Image 4 whih desribes a polygonal model by the distribution of normal vetors over the unit sphere. When the database of models is restrited to household objets, the obtained desriptors are primarily axis aligned (see Figure 7) and prinipal axis alignment may provide optimal alignment, (as indiated by the improved performane in Figure 6). Figure 6: Preision vs. Reall plots omparing the performane of aligned spherial desriptors with the performane of their harmoni representations. Note that for most of the representations the harmoni desriptor outperforms the anonially aligned one. Figure 7: Images of models of a vase, a hair, and sissors, with their assoiated Extended Gaussian Images. Note that the EGIs are mainly axial funtions and onsequently are well aligned by PCA. As the results indiate, the appliation of the Spherial Harmoni Representation improves the performane of most of the desriptors. The improvement of the mathing results is partiularly meaningful when we onsider the fat that the Spherial Harmoni Representation redues a D desriptor into a 1D array of energy values. Thus, the representation not only provides better performane, but it does so with fewer bits of information. 7. Disussion In this setion we present a disussion of the results in Setion 6. In partiular, we analyze the ase of the Extended Gaussian Image, and disuss how this reflets on the general limitations of the Spherial Harmoni Representation. We also evaluate the impliations of the Spherial Harmoni Representation for database retrieval Limitations The analysis desribed in Appendix A provides a mathematial interpretation of the failing of PCA-alignment. This analysis makes the assumption that we are looking at the 7.. Impliations for Model Databases Muh of the researh presented in this paper is guided by the inreased proliferation and aessibility of 3D models. These models have been gathered into databases, and one of the hallenges has been to design mathing implementations that are well suited for database retrieval. The spherial harmoni representation presented in this paper addresses this hallenge in two ways: 1. While a spherial funtion of bandwidth b requires O b spae, its spherial harmoni representation is of size O b. Consequently, the spherial harmoni representations provide a more ompat representation of the desriptors, and an be ompared more effiiently. (For eah method ompared in Setion 6, Table shows the spae requirements of the desriptor and its Spherial Harmoni Representation.). Furthermore, the Spherial Harmoni Representations are based on a frequeny deompostion of a spherial funtion. Consequently, the representation is inherently multiresolutional and this property an be used to guide indexing shemes for effiient retrieval.

8 Kazhdan et al / Spherial Harmoni Representations Representation PCA-Aligned Harmoni EGI Spherial Extent Funtion Radial Distribution Setors Shape Histogram Voxel Table : The number of floating point numbers used to desribe eah representation. This table demonstrates that the Spherial Harmoni Representation provides a representation that redues the dimensionality of the spae required for storing the desriptor. 8. Conlusion and Future Work In this paper we have introdued the Spherial Harmoni Representation, a rotation invariant representation of spherial funtions in terms of the energies at different frequenies. We have shown that this representation provides a method for improving the performane of many anonially aligned spherial desriptors in tasks of shape mathing. In addition to providing better mathing performane this rotation invariant representation also redues the dimensionality of the existing desriptors improving both the time and spae requirements of these methods. This work suggests a number of hallenges that we would like to onsider in the future: First, we would like to explore the possibility of generalizing this method to voxel grids using Zernike moments. Seond, we would like to onsider methods for reduing the rotation independene of the different frequeny omponents, and, in the ase of voxel grids, of the different radial omponents. Finally, we would like to explore extending this method to apture more rotation invariant information in the higher frequeny omponents, allowing us to truly reonstrut eah frequeny omponent uniquely up to rotation. Referenes 1. Delingette, H., Hebert, M., Ikeuhi, K.: A spherial representation for the reognition of urved objets. In: Pro. ICCV. (1993) Besl, P.: Triangles as a primary representation. Objet Reognition in Computer Vision LNCS 994 (1995) Osada, R., Funkhouser, T., Chazelle, B., Dobkin, D.: Mathing 3d models with shape distributions. Shape Mathing International (001) B. Horn, B.: Extended gaussian images. PIEEE 7 (1984) Ankerst, M., Kastenmüller, G., Kriegel, H.P., Seidl, T.: 3D shape histograms for similarity searh and lassifiation in spatial databases. In: Pro. SSD. (1999) 6. Vrani, D., Saupe, D.: 3d model retrieval with spherial harmonis and moments. Proeedings of the DAGM (001) Gain, J., Sott, J.: Fast polygon mesh querying by example. SIGGRAPH Tehnial Skethes (1999) Kazhdan, M., Chazelle, B., Dobkin, D., Finkelstein, A., Funkhouser, T.: A refletive symmetry desriptor. European Conferene on Computer Vision (ECCV) (00) Elad, M., Tal, A., Ar, S.: Content based retrieval of vrml objets - an iterative and interative approah. EG Multimedia (001) Horn, B., Hilden, H., Negahdaripour, S.: Closed form solution of absolute orientation using orthonormal matries. Journal of the Optial Soiety (1988) Zahn, C., Roskies, R.: Fourier desriptors for plane losed urves. IEE Transation on Computers 1 (197) Funkhouser, T., Min, P., Kazhdan, M., Chen, J., Halderman, A., Dobkin, D., Jaobs, D.: A searh engine for 3d models. ACM TOG (003) Besl, P., MKay, N.: A method for registration of 3d shapes. IEEE TPAMI (199) Zhang, Z.: Iterative point mathing for registration of freeform urves and surfaes. IJCV (1994) Ballard, D.: Generalized hough transform to detet arbitrary patterns. IEEE PAMI 1 (1981) Lamdan, Y., Wolfson, H.: Geometri hashing: A general and effiient model-based reognition sheme. Proeedings of the nd International Conferene on Computer Vision (1988) Labs, V.D.: (001) 18..5, S.: geelong/sphere/ (1998) Appendix A: A Signal Proessing Framework for PCA This appendix presents a signal proessing framework for analyzing the impliations and limitations of model alignment via PCA. We define a spherial funtion haraterizing the radial variane of a shape along different rays from the origin. In partiular, for a model S and a diretion v we set: RV S v lim α 0 C v α S x π 1 os α where C v α is the one with apex at the origin, angle α and diretion v, and π 1 os α is the area of the intersetion of the one with the unit sphere (see Figure 8). That is, RV S v gives the sum of the square of the distanes of the points lying on the intersetion with S and the ray, from the origin, with diretion v. Figure 9 shows a visualization of the Radial Variane for a ube by saling the radius of eah point on a sphere in proportion to the value of the funtion at that point. Note that the funtion sales the points at the orners of the ube more drastially beause: (1) we integrate dx

9 Kazhdan et al / Spherial Harmoni Representations Figure 8: The value of the Radial Variane in the diretion v is defined by interseting the model with a one, in the diretion v, with small angle α, and integrating the square of the distane over the intersetion of the model with the one. Figure 9: The Radial Variane an be visualized by displaing the radius at a point on the sphere, in proportion to the value of the funtion at that point. the square of the distane to the origin over eah path, and () the angle between the point on the sphere and the surfae normal is large, so that more surfae area projets onto a spherial path. What is valuable about this funtion is that for any surfae S, the funtion has the property that: x i x j dx RV S v x i x S j S That is, the seond (and 0-th) order omponents of the radial variane are preisely the terms of the ovariane matrix of the model. This funtion gives a representation of the initial model in a signal proessing framework that allows us to make two observations: 1. Beause of the orthogonality of the frequeny omponents, prinipal axis registration does not take into aount information at non seond-order frequenies and hene makes no guarantees as to how they align.. Aligning two models using their prinipal axes provides the optimal alignment for their seond order omponents, as will be shown in the following theorem: Theorem: If f and g are two spherial funtions onsisting of only onstant and seond order harmonis, then the L -differene between the two is minimized when eah is aligned to its own prinipal axes. Proof: Beause f and g onsist of only onstant and seond order terms, we an represent the funtions by symmetri matries A and B where f v v t Av and g v v t Bv If we assume that A and B are already aligned to their prinipal axes we get: A a a a 3 and B Thus, if R is any rotation we get: R t f g α β Trae ARBR t β b b b 3 3 a i i b j j 1 where α S x4 dx and β S x y dx define the lengths and angles between the funtions x i on the unit sphere. We would like to show that the dot produt is maximized when R is a permutation matrix so that RAR t is diagonal. Using the fat that the differentials of a rotation R are defined by RS where S is a skew-symmetri matrix, it suffies to solve for: d 0 dt t 0 Trae A R trs B R t tsr t Trae R t AR SB BS But S is a skew-symmetri matrix so that, SB BS is a symmetri matrix with 0 s along the diagonal: SB BS 0 b b 1 S 1 b 3 b 1 S 13 b b 1 S 1 0 b 3 b S 3 b 3 b 1 S 13 b 3 b S 3 0 Thus, if R t AR is a diagonal matrix then the derivative is zero, independent of the hoie of S. Conversely, if the b i are distint and R t AR is not diagonal, we an always hoose values for S 1, S 13, and S 3 suh that the derivative is non-zero, implying that if R t AR is not diagonal it annot maximize the dot produt. (Note that if b1 b b3 then B is a onstant multiple of the identity so that the dot produt is independent of the hoie of rotation. Similarly, if b i b j then rotations in the plane spanned by x i and x j also do not hange the dot produt.) This shows that the L -differene between f and g is at an extremum if and only if A and B are both diagonal matries. The minimum L -differene is then attained when a i b i is maximal. So that if a 1 a a 3 then we must also have b 1 b b 3, and the L -differene between f and g is minimized preisely when f and g are aligned to their prinipal axes.