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Shape Analysis Using the Auto Diffusion Function
 Comp. Graph. Forum
"... Scalar functions defined on manifold triangle meshes is a starting point for many geometry processing algorithms such as mesh parametrization, skeletonization, and segmentation. In this paper, we propose the Auto Diffusion Function (ADF) which is a linear combination of the eigenfunctions of the Lap ..."
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Cited by 41 (0 self)
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Scalar functions defined on manifold triangle meshes is a starting point for many geometry processing algorithms such as mesh parametrization, skeletonization, and segmentation. In this paper, we propose the Auto Diffusion Function (ADF) which is a linear combination of the eigenfunctions of the LaplaceBeltrami operator in a way that has a simple physical interpretation. The ADF of a given 3D object has a number of further desirable properties: Its extrema are generally at the tips of features of a given object, its gradients and level sets follow or encircle features, respectively, it is controlled by a single parameter which can be interpreted as feature scale, and, finally, the ADF is invariant to rigid and isometric deformations. We describe the ADF and its properties in detail and compare it to other choices of scalar functions on manifolds. As an example of an application, we present a pose invariant, hierarchical skeletonization and segmentation algorithm which makes direct use of the ADF.
LaplaceBeltrami Eigenvalues and Topological Features of Eigenfunctions for Statistical Shape Analysis
, 2009
"... This paper proposes the use of the surfacebased LaplaceBeltrami and the volumetric Laplace eigenvalues and eigenfunctions as shape descriptors for the comparison and analysis of shapes. These spectral measures are isometry invariant and therefore allow for shape comparisons with minimal shape pre ..."
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Cited by 22 (3 self)
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This paper proposes the use of the surfacebased LaplaceBeltrami and the volumetric Laplace eigenvalues and eigenfunctions as shape descriptors for the comparison and analysis of shapes. These spectral measures are isometry invariant and therefore allow for shape comparisons with minimal shape preprocessing. In particular, no registration, mapping, or remeshing is necessary. The discriminatory power of the 2D surface and 3D solid methods is demonstrated on a population of female caudate nuclei (a subcortical gray matter structure of the brain, involved in memory function, emotion processing, and learning) of normal control subjects and of subjects with schizotypal personality disorder. The behavior and properties of the LaplaceBeltrami eigenvalues and eigenfunctions are discussed extensively for both the Dirichlet and Neumann boundary condition showing advantages of the Neumann vs. the Dirichlet spectra in 3D. Furthermore, topological analyses employing the MorseSmale complex (on the surfaces) and the Reeb graph (in the solids) are performed on selected eigenfunctions, yielding shape descriptors, that are capable of localizing geometric properties and detecting shape differences by indirectly registering topological features such as critical points, level sets and integral lines of the gradient field across subjects. The use of these topological features of the LaplaceBeltrami eigenfunctions in 2D and 3D for statistical shape analysis is novel.
A Randomized O(m log m) Time Algorithm for Computing Reeb Graphs of Arbitrary Simplicial Complexes
"... Given a continuous scalar field f: X → IR where X is a topological space, a level set of f is a set {x ∈ X: f(x) = α} for some value α ∈ IR. The level sets of f can be subdivided into connected components. As α changes continuously, the connected components in the level sets appear, disappear, spli ..."
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Given a continuous scalar field f: X → IR where X is a topological space, a level set of f is a set {x ∈ X: f(x) = α} for some value α ∈ IR. The level sets of f can be subdivided into connected components. As α changes continuously, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of f encodes these changes in connected components of level sets. It provides a simple yet meaningful abstraction of the input domain. As such, it has been used in a range of applications in fields such as graphics and scientific visualization. In this paper, we present the first subquadratic algorithm to compute the Reeb graph for a function on an arbitrary simplicial complex K. Our algorithm is randomized with an expected running time O(m log n), where m is the size of the 2skeleton of K (i.e, total number of vertices, edges and triangles), and n is the number of vertices. This presents a significant improvement over the previous Θ(mn) time complexity for arbitrary complex, matches (although in expectation only) the best known result for the special case of 2manifolds, and is faster than current algorithms for any other special cases (e.g, 3manifolds). Our algorithm is also very simple to implement. Preliminary experimental results show that it performs well in practice.
Hot Spots Conjecture and Its Application to Modeling Tubular Structures
"... Abstract. The second eigenfunction of the LaplaceBeltrami operator follows the pattern of the overall shape of an object. This geometric property is well known and used for various applications including mesh processing, feature extraction, manifold learning, data embedding and the minimum linear a ..."
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Abstract. The second eigenfunction of the LaplaceBeltrami operator follows the pattern of the overall shape of an object. This geometric property is well known and used for various applications including mesh processing, feature extraction, manifold learning, data embedding and the minimum linear arrangement problem. Surprisingly, this geometric property has not been mathematically formulated yet. This problem is directly related to the somewhat obscure hot spots conjecture in differential geometry. The aim of the paper is to raise the awareness of this nontrivial issue and formulate the problem more concretely. As an application, we show how the second eigenfunction alone can be used for complex shape modeling of tubular structures such as the human mandible. 1
Robust Surface Reconstruction via LaplaceBeltrami EigenProjection and Boundary Deformation
"... Abstract—In medical shape analysis, a critical problem is reconstructing a smooth surface of correct topology from a binary mask that typically has spurious features due to segmentation artifacts. The challenge is the robust removal of these outliers without affecting the accuracy of other parts of ..."
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Abstract—In medical shape analysis, a critical problem is reconstructing a smooth surface of correct topology from a binary mask that typically has spurious features due to segmentation artifacts. The challenge is the robust removal of these outliers without affecting the accuracy of other parts of the boundary. In this paper, we propose a novel approach for this problem based on the Laplace–Beltrami (LB) eigenprojection and properly designed boundary deformations. Using the metric distortion during the LB eigenprojection, our method automatically detects the location of outliers and feeds this information to a wellcomposed and topologypreserving deformation. By iterating between these two steps of outlier detection and boundary deformation, we can robustly filter out the outliers without moving the smooth part of the boundary. The final surface is the eigenprojection of the filtered mask boundary that has the correct topology, desired accuracy and smoothness. In our experiments, we illustrate the robustness of our method on different input masks of the same structure, and compare with the popular SPHARM tool and the topology preserving level set method to show that our method can reconstruct accurate surface representations without introducing artificial oscillations. We also successfully validate our method on a large data set of more than 900 hippocampal masks and demonstrate that the reconstructed surfaces retain volume information accurately. Index Terms—Deformation, eigenprojection, LaplaceBeltrami eigenfunction, mask, outlier, surface reconstruction, topology.
FAST AND ROBUST ALGORITHMS FOR HARMONIC ENERGY MINIMIZATION ON GENUS0 SURFACES
"... Abstract. Surface harmonic map between genus0 surfaces plays an important role in applied mathematics and engineering, with applications in medical imaging and computer graphics. Previous work [1] introduces a variational approach for computing surface harmonic maps. It obtains global conformal par ..."
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Abstract. Surface harmonic map between genus0 surfaces plays an important role in applied mathematics and engineering, with applications in medical imaging and computer graphics. Previous work [1] introduces a variational approach for computing surface harmonic maps. It obtains global conformal parameterizations of genus0 surfaces through minimizing the harmonic energy. Two weaknesses of this approach are addressed in this paper: the slow speed of its gradient descent iterations and the presence of undesired parameterization foldings when the underlying surface has long sharp features. This paper proposes an algorithm that significantly accelerates the harmonic map computation and a method that helps this algorithm produce global conformal parameterizations without foldings. These are achieved by applying recent results of optimization on manifolds and taking advantages of the weighted LaplaceBeltrami eigenprojection. Experimental results show that the proposed approaches compute genus0 surface harmonic maps much faster than the existing algorithm in [1] and the results contain no foldings.
IEEE INTERNATIONAL CONFERENCE ON SHAPE MODELING AND APPLICATIONS (SMI) 2010 1 Fiedler Trees for Multiscale Surface Analysis
"... Abstract—In this work we introduce a new hierarchical surface decomposition method for multiscale analysis of surface meshes. In contrast to other multiresolution methods, our approach relies on spectral properties of the surface to build a binary hierarchical decomposition. Namely, we utilize the f ..."
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Abstract—In this work we introduce a new hierarchical surface decomposition method for multiscale analysis of surface meshes. In contrast to other multiresolution methods, our approach relies on spectral properties of the surface to build a binary hierarchical decomposition. Namely, we utilize the first nontrivial eigenfunction of the LaplaceBeltrami operator to recursively decompose the surface. For this reason we coin our surface decomposition the Fiedler tree. Using the Fiedler tree ensures a number of attractive properties, including: meshindependent decomposition, wellformed and nearly equiareal surface patches, and noise robustness. We show how the evenly distributed patches can be exploited for generating multiresolution high quality uniform meshes. Additionally, our decomposition permits a natural means for carrying out wavelet methods, resulting in an intuitive method for producing featuresensitive meshes at multiple scales. 1.
Contents lists available at SciVerse ScienceDirect International Journal of Developmental Neuroscience
"... j ourna l ho me page: www.elsev ier.com / locate / i jdevneu Developmental changes in hippocampal shape among ..."
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j ourna l ho me page: www.elsev ier.com / locate / i jdevneu Developmental changes in hippocampal shape among