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82
Genus zero surface conformal mapping and its application to brain surface mapping
 IEEE Transactions on Medical Imaging
, 2004
"... Abstract—We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic oneforms for surfaces with or without boundaries (Gu and Yau, 2002), (Gu and Yau, 2003). For genus zero surfaces, our algorithm can find a unique mapping betwe ..."
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Cited by 152 (68 self)
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Abstract—We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic oneforms for surfaces with or without boundaries (Gu and Yau, 2002), (Gu and Yau, 2003). For genus zero surfaces, our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map. In this paper, we apply the algorithm to the cortical surface matching problem. We use a mesh structure to represent the brain surface. Further constraints are added to ensure that the conformal map is unique. Empirical tests on magnetic resonance imaging (MRI) data show that the mappings preserve angular relationships, are stable in MRIs acquired at different times, and are robust to differences in data triangulation, and resolution. Compared with other brain surface conformal mapping algorithms, our algorithm is more stable and has good extensibility. Index Terms—Brain mapping, conformal map, landmark matching, spherical harmonic transformation. I.
3d model retrieval with spherical harmonics and moments
 DAGM
, 2001
"... Abstract. We consider 3D object retrieval in which a polygonal mesh serves as a query and similar objects are retrieved from a collection of 3D objects. Algorithms proceed first by a normalization step in which models are transformed into canonical coordinates. Second, feature vectors are extracted ..."
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Cited by 74 (6 self)
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Abstract. We consider 3D object retrieval in which a polygonal mesh serves as a query and similar objects are retrieved from a collection of 3D objects. Algorithms proceed first by a normalization step in which models are transformed into canonical coordinates. Second, feature vectors are extracted and compared withthose derived from normalized models in the search space. In the feature vector space nearest neighbors are computed and ranked. Retrieved objects are displayed for inspection, selection, and processing. Our feature vectors are based on rays cast from the center of mass of the object. For each ray the object extent in the ray direction yields a sample of a function on the sphere. We compared two kinds of representations of this function, namely spherical harmonics and moments. Our empirical comparison using precisionrecall diagrams for retrieval results in a data base of 3D models showed that the method using spherical harmonics performed better. 1
Tools for 3Dobject retrieval: KarhunenLoeve Transform and spherical harmonics
 IEEE MMSP 2001
, 2001
"... Abstract We present tools for 3D object retrieval in which a model, a polygonal mesh, serves as a query and similar objects are retrieved from a collection of 3D objects. Algorithms proceed first by a normalization step (pose estimation) in which models are transformed into a canonical coordinate f ..."
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Cited by 69 (12 self)
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Abstract We present tools for 3D object retrieval in which a model, a polygonal mesh, serves as a query and similar objects are retrieved from a collection of 3D objects. Algorithms proceed first by a normalization step (pose estimation) in which models are transformed into a canonical coordinate frame. Second, feature vectors are extracted and compared with those derived from normalized models in the search space. Using a metric in the feature vector space nearest neighbors are computed and ranked. Objects thus retrieved are displayed for inspection, selection, and processing. For the pose estimation we introduce a modified KarhunenLoeve transform that takes into account not only vertices or polygon centroids from the 3D models but all points in the polygons of the objects. Some feature vectors can be regarded as samples of functions on the 2sphere. We use Fourier expansions of these functions as uniform representations allowing embedded multiresolution feature vectors. Our implementation demonstrates and visualizes these tools.
Featurebased similarity search in 3D object databases
 ACM Computing Surveys
, 2005
"... The development of effective contentbased multimedia search systems is an important research issue due to the growing amount of digital audiovisual information. In the case of images and video, the growth of digital data has been observed since the introduction of 2D capture devices. A similar dev ..."
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Cited by 65 (10 self)
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The development of effective contentbased multimedia search systems is an important research issue due to the growing amount of digital audiovisual information. In the case of images and video, the growth of digital data has been observed since the introduction of 2D capture devices. A similar development is expected for 3D data as
Wavelets on the 2sphere: A grouptheoretical approach
 APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
, 1999
"... We present a purely grouptheoretical derivation of the continuous wavelet transform (CWT) on the 2sphere S 2, based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and R + ∗ f ..."
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Cited by 46 (10 self)
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We present a purely grouptheoretical derivation of the continuous wavelet transform (CWT) on the 2sphere S 2, based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and R + ∗ for dilations on S 2,which are embedded into the Lorentz group SO0(3, 1) via the Iwasawa decomposition, so that X � SO 0(3, 1)/N, where N � C. We select an appropriate unitary representation of SO0(3, 1) acting in the space L 2 (S 2,dµ) of finite energy signals on S 2. This representation is square integrable over X; thus it yields immediately the wavelets on S 2 and the associated CWT. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition. Finally, the Euclidean limit of this CWT on S 2 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R →∞. Then the parameter space goes into the similitude group of R 2 and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility.
Ffts on the rotation group
 Santa Fe Institute Working Papers Series Paper
, 2003
"... Earlier work by Driscoll and Healy [4] has produced an efficient O(B 2 log 2 B) algorithm for computing the Fourier transform of bandlimited functions on the 2sphere. In this paper, we discuss an implementation of an O(B 4) algorithm for the numerical computation of Fourier transforms of functions ..."
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Cited by 28 (0 self)
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Earlier work by Driscoll and Healy [4] has produced an efficient O(B 2 log 2 B) algorithm for computing the Fourier transform of bandlimited functions on the 2sphere. In this paper, we discuss an implementation of an O(B 4) algorithm for the numerical computation of Fourier transforms of functions defined on the rotation group, SO(3). This compares with the direct O(B 6) approach. The algorithm we implemented is based on the “Separation of Variables ” technique, e.g. as presented by Maslen and Rockmore [19]. In conjunction with the techniques developed in [4], the SO(3) algorithm we implemented may be made truly fast, O(B 3 log 2 B). Basic results will be presented establishing the algorithm’s numerical stability, and examples of applications will be presented. 1
ExpressionInvariant Representations of Faces
 IEEE TRANS. PAMI
, 2007
"... Addressed here is the problem of constructing and analyzing expressioninvariant representations of human faces. We demonstrate and justify experimentally a simple geometric model that allows to describe facial expressions as isometric deformations of the facial surface. The main step in the constru ..."
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Cited by 24 (6 self)
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Addressed here is the problem of constructing and analyzing expressioninvariant representations of human faces. We demonstrate and justify experimentally a simple geometric model that allows to describe facial expressions as isometric deformations of the facial surface. The main step in the construction of expressioninvariant representation of a face involves embedding of the facial intrinsic geometric structure into some convenient lowdimensional space. We study the influence of the embedding space geometry and dimensionality choice on the representation accuracy and argue that compared to its Euclidean counterpart, spherical embedding leads to notably smaller metric distortions. We experimentally support our claim showing that a smaller embedding error leads to better recognition.
An Experimental Effectiveness Comparison of Methods for 3D Similarity Search
, 2006
"... Methods for contentbased similarity search are fundamental for managing large multimedia repositories, as they make it possible to conduct queries for similar content, and to organize the repositories into classes of similar objects. 3D objects are an important type of multimedia data with many pr ..."
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Cited by 20 (12 self)
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Methods for contentbased similarity search are fundamental for managing large multimedia repositories, as they make it possible to conduct queries for similar content, and to organize the repositories into classes of similar objects. 3D objects are an important type of multimedia data with many promising application possibilities. Defining the aspects that constitute the similarity among 3D objects, and designing algorithms that implement such similarity definitions is a difficult problem. Over the last few years, a strong interest in 3D similarity search has arisen, and a growing number of competing algorithms for the retrieval of 3D objects have been proposed. The contributions of this paper are to survey a body of recently proposed methods for 3D similarity search, to organize them along a descriptor extraction process model, and to present an extensive experimental effectiveness and efficiency evaluation of these methods, using several 3D databases.
Fast algorithms for spherical harmonic expansions
 SIAM J. Sci. Comput
, 2004
"... An algorithm is introduced for the rapid evaluation at appropriately chosen nodes on the twodimensional sphere S 2 in � 3 of functions specified by their spherical harmonic expansions (known as the inverse spherical harmonic transform), and for the evaluation of the coefficients in spherical harmon ..."
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Cited by 18 (3 self)
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An algorithm is introduced for the rapid evaluation at appropriately chosen nodes on the twodimensional sphere S 2 in � 3 of functions specified by their spherical harmonic expansions (known as the inverse spherical harmonic transform), and for the evaluation of the coefficients in spherical harmonic expansions of functions specified by their values at appropriately chosen points on S 2 (known as the forward spherical harmonic transform). The procedure is numerically stable and requires an amount of CPU time proportional to N(log N) log(1/ε), where N is the number of nodes in the discretization of S 2, and ε is the precision of computations. The performance of the algorithm is illustrated via several numerical examples. 1
Largescale modeling of parametric surfaces using spherical harmonics
 In Third International Symposium on 3D Data Processing, Visualization and Transmission (3DPVT
, 2006
"... We present an approach for largescale modeling of parametric surfaces using spherical harmonics (SHs). A standard least square fitting (LSF) method for SH expansion is not scalable and cannot accurately model large 3D surfaces. We propose an iterative residual fitting (IRF) algorithm, and demonstra ..."
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Cited by 15 (10 self)
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We present an approach for largescale modeling of parametric surfaces using spherical harmonics (SHs). A standard least square fitting (LSF) method for SH expansion is not scalable and cannot accurately model large 3D surfaces. We propose an iterative residual fitting (IRF) algorithm, and demonstrate its effectiveness and scalability in creating accurate SH models for large 3D surfaces. These largescale and accurate parametric models can be used in many applications in computer vision, graphics, and biomedical imaging. As a simple extension of LSF, IRF is very easy to implement and requires few machine resources. 1.