Topology-Invariant Similarity of Nonrigid Shapes

Bronstein, Alexander M.; Bronstein, Michael M.; Kimmel, Ron

doi:10.1007/s11263-008-0172-2

Cited by 50 publications

(35 citation statements)

References 69 publications

(79 reference statements)

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“…Several techniques have been proposed for robust symmetry detection -voting procedure in a transformation space [24,28,34,36,48], shape descriptors based on spherical harmonics [23,25,38], and graph matching [2,4]. Change in pose, and non-rigid shapes and transformations make symmetry identification difficult and requires detection of symmetry with respect to non-rigid transformations [8,27,31,37,48].…”

Section: Symmetry In Geometric Shapesmentioning

confidence: 99%

Symmetry in Scalar Field Topology

Thomas

Natarajan

2011

IEEE Trans. Visual. Comput. Graphics

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Abstract-Study of symmetric or repeating patterns in scalar fields is important in scientific data analysis because it gives deep insights into the properties of the underlying phenomenon. Though geometric symmetry has been well studied within areas like shape processing, identifying symmetry in scalar fields has remained largely unexplored due to the high computational cost of the associated algorithms. We propose a computationally efficient algorithm for detecting symmetric patterns in a scalar field distribution by analysing the topology of level sets of the scalar field. Our algorithm computes the contour tree of a given scalar field and identifies subtrees that are similar. We define a robust similarity measure for comparing subtrees of the contour tree and use it to group similar subtrees together. Regions of the domain corresponding to subtrees that belong to a common group are extracted and reported to be symmetric. Identifying symmetry in scalar fields finds applications in visualization, data exploration, and feature detection. We describe two applications in detail: symmetry-aware transfer function design and symmetry-aware isosurface extraction.

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Section: Symmetry In Geometric Shapesmentioning

confidence: 99%

Symmetry in Scalar Field Topology

Thomas

Natarajan

2011

IEEE Trans. Visual. Comput. Graphics

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“…Hence, pose invariance is achieved by embedding the shape into an isometric subspace [21,12,4,29,25,15,16]. This is a very strong assumption for multicamera acquisition systems, as the independently reconstructed shapes can be non-isometric due to presence of topological merges and splits.…”

Section: Related Workmentioning

confidence: 99%

Topologically-robust 3D shape matching based on diffusion geometry and seed growing

et al. 2011

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Abstract3D Shape matching is an important problem in computer vision. One of the major difficulties in finding dense correspondences between 3D shapes is related to the topological discrepancies that often arise due to complex kinematic motions. In this paper we propose a shape matching method that is robust to such changes in topology. The algorithm starts from a sparse set of seed matches and outputs dense matching. We propose to use a shape descriptor based on properties of the heat-kernel and which provides an intrinsic scale-space representation. This descriptor incorporates (i) heat-flow from already matched points and (ii) self diffusion. At small scales the descriptor behaves locally and hence it is robust to global changes in topology. Therefore, it can be used to build a vertex-to-vertex matching score conditioned by an initial correspondence set. This score is then used to iteratively add new correspondences based on a novel seed-growing method that iteratively propagates the seed correspondences to nearby vertices. The matching is farther densified via an EM-like method that explores the congruency between the two shape embeddings. Our method is compared with two recently proposed algorithms and we show that we can deal with substantial topological differences between the two shapes.

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“…For an entirely different class of approaches, based on surface geometry, see, for example, Bronstein et al [12] and their excellent monograph [11] which provides an extensive bibliography. Also, the discussion of methods developed for 2D image registration is beyond scope here.…”

Section: Related Workmentioning

confidence: 99%

Fast nonrigid mesh registration with a data-driven deformation prior

Schneider

Eisert

2009

2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops

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We propose an algorithm for non-rigidly registering a 3D template mesh with a dense point cloud, using a morphable shape model to control the deformation of the template mesh. A cost function involving nonrigid shape as well as rigid pose is proposed. Registration is performed by minimizing a first-order approximation of the cost function in the Iterative Closest Points framework. We show how a complex shape model, consisting of multiple PCA models for individual regions of the template, can be seamlessly integrated in the parameter estimation scheme. An appropriate Tikhonov regularization is introduced to guarantee the smoothness of the full mesh despite the splitting into local models. The proposed algorithm is compared to a recent generic nonrigid registration scheme. We show that the data-driven approach is faster, as the linear systems to be solved in the iterations are significantly smaller when a model is available. Also, we show that simultaneous optimization of pose and shape yields better registration results than shape alone.

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Topology-Invariant Similarity of Nonrigid Shapes

Cited by 50 publications

References 69 publications

Symmetry in Scalar Field Topology

Symmetry in Scalar Field Topology

Topologically-robust 3D shape matching based on diffusion geometry and seed growing

Fast nonrigid mesh registration with a data-driven deformation prior

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