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A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching
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“In cases with large topological changes, the isometry assumption may fail [38]; handling such issues is a challenging task.”
From Registration of 3D Point Clouds and Meshes: A Survey from Rigid to Nonrigid (2013) · cites this paper as [38]
In the citing paper
“Recent intrinsic techniques [36], [38], [39] use low dimensional embedding to by-pass the high-dimensionality problem by assuming a strong inelastic isometry constraint.”
From Registration of 3D Point Clouds and Meshes: A Survey from Rigid to Nonrigid (2013) · cites this paper as [38]
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Citations
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Cited by
Sorted by citations
Items 201–267 of 267
A Survey on Non-rigid 3D Shape AnalysisPreprint
Laga, H. (2018) · arXiv.org · doi:10.48550/arxiv.1812.10111 · Cited by 26
“Examples of such approaches include Generalized Multidimensional Scaling (GMDS) [86], spectral analysis based on Laplace-Beltrami eigenfunctions [87], [88], and Heat Kernel Signatures (HKS) [89], [90].”
MethodDistance Functions and Their Use in Adaptive Mathematical Morphology
Ćurić, V. (2014) · Cited by 2
Spectral Analysis Driven Sparse Matching of 3D Shapes
Darom, T., & Keller, Y. (2012) · Eurographics Workshop on 3D Object Retrieval · doi:10.2312/3dor/3dor12/059-062 · Cited by 2
“Such schemes may operate in the mesh domain by formulating continuous variational problems [BBK06b], or embed the meshes into a space where they can be aligned [SHM12,BBK ∗ 10].”
BackgroundLocal Signature Quantization by Sparse Coding
Boscaini, D., & Castellani, U. (2013) · Eurographics Workshop on 3D Object Retrieval · doi:10.2312/3dor/3dor13/009-016 · Cited by 2
“The recent improvements of 3D acquisition systems and the large proliferation of 3D models increase the interest in 3D shape retrieval methods [IJL ∗ 05,FKMS05,TV04, LGF ∗ 10, LGB ∗ 11].”
BackgroundMethodReliable Isometric Point Correspondence from Depth
KÜPÇÜ, E., & Yemez, Y. (2017) · IEEE International Conference on Computer Vision Workshops · doi:10.1109/iccvw.2017.152 · Cited by 2
“There exist a mature literature on isometric 3D correspondence methods that can find sparse [6, 45, 47, 51, 63] and/or dense [7, 11, 57, 21, 38, 44, 49, 52] accurate correspondences between complete mesh representations of 3D shapes.”
BackgroundMethodShape correspondence based effective combination of linear and quadratic assignment matrices
Zhang, M., Hassan, M. U., Niu, D., Li, N., Liu, M., Zhou, J., & Zhao, X. (2019) · International Conference on Digital Image Processing · doi:10.1117/12.2539652 · Cited by 2
He, P. L., Emami, P., Ranka, S., & Rangarajan, A. (2023) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2023.3307592 · Cited by 2
“Later, diffusion geometry [62] established invariant metrics based on eigenvalues and eigenvectors of the Laplace-Beltrami operator obtained from shapes, showingsignificantlymorerobustness comparedto the geodesic counterparts [63], [64].”
MethodMarkerless motion capture based on diffusion geometry and silhouettes
Liang, Q., & Miao, Z. (2015) · Journal of Electronic Imaging · doi:10.1117/1.jei.24.5.051003 · Cited by 2
“Diffusion distance is the average length of paths connecting two points on the shape, and can be efficiently computed from the eigenvalues of the Laplace–Beltrami operator.(10) Based on the analysis of spectral properties, the eigenvectors of the Laplace–Beltrami operator can be represented as “shapeDNA”(11) for the c…”
MethodFinding the M-best consistent correspondences between 3D symmetric objects
Sung, M., & Kim, J. (2012) · Computers & Graphics · doi:10.1016/j.cag.2012.11.002 · Cited by 2
Shape Matching for Rigid Objects by Aligning Sequences Based on Boundary Change Points
Arslan, A. N., & Sirakov, N. (2017) · Lecture notes in computer science · doi:10.1007/978-3-319-59108-7_24 · Cited by 2
Principal Curvature Guided Surface Geometry Aware Global Shape Representation
Das, S., & Bhandarkar, S. M. (2018) · IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW) · doi:10.1109/cvprw.2018.00081 · Cited by 2
“Iterative operator-based descriptors capture the geometric changes within a shape by manipulating the entire mesh surface by employing strategies such as smoothing [26], estimation of local diffusion geometry [28] over the mesh surface, or diffusion based variation [29] within the surface mesh.”
BackgroundMethodA unified definition and computation of Laplacian spectral distances
Patanè, G. (2019) · Pattern Recognition · doi:10.1016/j.patcog.2019.04.004 · Cited by 2
Diffusion-based isometric depth correspondence
KÜPÇÜ, E., & Yemez, Y. (2019) · Computer Vision and Image Understanding · doi:10.1016/j.cviu.2019.102808 · Cited by 2
Modeling Shapes with Higher-Order Graphs: Methodology and Applications
Wang, C., Zeng, Y., Samaras, D., & Nikos Paragios, A. (2013) · Advances in computer vision and pattern recognition · doi:10.1007/978-1-4471-5195-1_31 · Cited by 2
Local Diffusion Map Signature for Symmetry-aware Non-rigid Shape Correspondence
Wang, M., & Fang, Y. (2016) · ACM International Conference on Multimedia (ACM MM) · doi:10.1145/2964284.2967277 · Cited by 2
“[15, 3] Although those shape signatures are able to robustly capture the deformation invariant feature, all of them face with the symmetry ambiguity challenge.”
BackgroundMultiscale Representation of 3D Surfaces via Stochastic Mesh Laplacian
Song, R., & Wang, L. (2019) · Computer-Aided Design · doi:10.1016/j.cad.2019.05.006 · Cited by 2
“Among them, the multiscale property reflects the ability of the heat kernel to capture local or global shape information over small or large t respectively, making heat diffusion and the heat kernel a popular tool in a wide range of applications, including mesh fairing [20], data representation and dimensionality redu…”
BackgroundSolving PDEs on manifolds represented as point clouds and applications
Lai, R., & Hongkai Zhao, C. (2019) · Handbook of numerical analysis · doi:10.1016/bs.hna.2019.07.006 · Cited by 2
Partial Gromov-Wasserstein with Applications on Positive-Unlabeled Learning.Preprint
Chapel, L., Alaya, M. Z., & Gasso, G. (2020) · arXiv (Cornell University) · Cited by 2
A cohomology-based Gromov–Hausdorff metric approach for quantifying molecular similarity
Wee, J., Xue, G., Tuschmann, W., & Xia, K. (2025) · Scientific Reports · doi:10.1038/s41598-025-93381-y · Cited by 1
3D shape correspondence under topological noise
Gençtav, A., Sahillioğlu, Y., & Tari, S. Z. (2016) · Signal Processing and Communications Applications Conference (SIU) · doi:10.1109/siu.2016.7495762 · Cited by 1
“Literatürde yer alan ilgili çalışmalarda genel yaklaşım kesel (geodesic) uzaklık yerine difüzyon esaslı uzaklık kullanmak şeklinde olmuştur [4]– [6].”
Efficient Single-View 3D Co-segmentation Using Shape Similarity and Spatial Part Relations
Araslanov, N., Koo, S., Gall, J., & Behnke, S. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-45886-1_24 · Cited by 1
“In future, we plan to experiment with other feature encoding schemes, such as spatial sensitive Bag-of-Words [24] and improve the contextual features using other approximations of diffusion distances [3,18] and spatial relations.”
BackgroundMethodLocal Geometry Inclusive Global Shape Representation
Das, S., & Bhandarkar, S. M. (2017) · IEEE International Conference on Computer Vision Workshops · doi:10.1109/iccvw.2017.151 · Cited by 1
Object Localization Using Non-Euclidean Metrics
Bağcı, U., Udupa, J. K., Chen, X., Torigian, D. A., Medetalibeyoğlu, A., Zhou, D., & Bai, L. (2023) · IEEE International Conference on Internet of Things and Intelligence Systems (IoTaIS) · doi:10.1109/iotais60147.2023.10346049 · Cited by 1
“For instance, shapes and surfaces are better understood using Riemannian geometry over Euclidean space [2].”
BackgroundA Scale-Invariant Diffusion Distance for Non-rigid Shape Analysis
Wang, K., Wu, Z., Jia, T., Ali, S., Zhao, J., Yang, G., & Zhou, M. (2014) · International Conference on Virtual Reality and Visualization · doi:10.1109/icvrv.2014.63 · Cited by 1
“Interpreting the heat kernel probabilistically, it can be associated with the probability of getting to the target point by means of a random walk on manifold of length t starting at the source point[19].”
BackgroundA Descriptor for Voxel Shapes Based on the Skeleton Cut Space
Feng, C., Jalba, A. C., & Telea, A. C. (2016) · Eurographics · doi:10.2312/3dor.20161082 · Cited by 1
Sparse Models for Intrinsic Shape Correspondence
Pokrass, J., Bronstein, A. M., Bronstein, M. M., Sprechmann, P., & Sapiro, G. (2016) · Mathematics and visualization · doi:10.1007/978-3-319-24726-7_10 · Cited by 1
Self-citationSummary of Non-rigid 3D Shape Data Retrieval and Description
Liu, M., & Duan, C. (2018) · Proceedings of the 2018 International Conference on Network, Communication, Computer Engineering (NCCE 2018) · doi:10.2991/ncce-18.2018.11 · Cited by 1
“In addition to the above described shape descriptors, another popular shape-drawing method is to use diffusionbased point signatures [31], [32], [33].”
MethodComputable invariants for curves and surfaces
Halimi, O., Raviv, D., Aflalo, Y., & Kimmel, R. (2019) · Handbook of numerical analysis · doi:10.1016/bs.hna.2019.07.004 · Cited by 1
Self-citationVisualizing Time-Varying Particle Flows with Diffusion GeometryPreprint
Berger, M., & Levine, J. A. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1708.03686 · Cited by 1
“Diffusion distances have proven useful in shape analysis [46], [47] compared to geodesics alone, for instance in handling topological noise due to poor estimation of the local geometry.”
BackgroundMethodGeometric Data Analysis Across Scales via Laplacian Eigenvector CascadingPreprint
Mike, J. L., & Perea, J. A. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1812.02139 · Cited by 1
“Graph Laplacian eigensystems have many applications including dimensionality reduction [47, 3], semi-supervised learning [4], consistent ranking [25], and direct quantification of geometry [7, 40, 12].”
BackgroundA Sketch Based 3D Shape Retrieval Approach Based on Efficient Deep Point-to-Subspace Metric LearningPreprint
Lei, Y., Zhou, Z., Zhang, P., Guo, Y., Ma, Z., & Liu, L. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1903.00117 · Cited by 1
“Since the volume of 3D shape data grows significantly, shape retrieval has been becoming a crucial problem for 3D shape data management [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].”
BackgroundGeodesic Exponential Kernels: When Curvature and Linearity ConflictPreprint
Feragen, A., Lauze, F., & Hauberg, S. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1411.0296 · Cited by 2
Non-Aligned Distribution Distance using Metric Measure Embedding and Optimal TransportPreprint
Alaya, M. Z., Bérar, M., Gasso, G., & Rakotomamonjy, A. (2020) · Cited by 1
Statistical Shape Analysis for Population Studies via Level-Set Based Shape Morphing
Raviv, T. R., Gao, Y., Levitt, J. J., & Sylvain Bouix, A. (2012) · Lecture notes in computer science · doi:10.1007/978-3-642-33863-2_5
“For example in [13] the Gromov-Hausdorff distance was used for calculating the diffusion distance, rather than the geodesic path between points on a surface, allowing the comparison of pairs of non-rigid shapes with different topology.”
MethodNon-rigid Shape Correspondence Using Pointwise Surface Descriptors and Metric Structures
Dubrovina, A., Raviv, D., & Kimmel, R. (2012) · Mathematics and visualization · doi:10.1007/978-3-642-34141-0_15
Self-citationIntraclass Identifiable Region Detection
Lu, M., Zheng, B., Takamatsu, J., & Ikeuchi, K. (2013) · IPSJ Transactions on Computer Vision and Applications · doi:10.2197/ipsjtcva.5.90
“[12] and [2], this method is more appropriate for our problem.”
MethodGBI-SA: GBI Feature with Subtle Adjustment for Robust Non-rigid 3D Shape Retrieval
Kuang, Z., Li, Z., Lv, Q., & Liu, Y. (2013) · International Conference on Computer-Aided Design and Computer Graphics (CAD/Graphics) · doi:10.1109/cadgraphics.2013.29
Pappas, J. L., & Miller, D. J. (2014) · PLoS ONE · doi:10.1371/annotation/fc59dde0-057f-4a59-a5ab-9dcc3b949828
Balanced feature matching in probabilistic framework and its application on object localisation
Chen, Y., & Ai, C. (2013) · International Journal of Computer Applications in Technology · doi:10.1504/ijcat.2013.052291
“, 2011), shape matching (Alexander et al., 2010; Noma and Cesar, 2010), image mosaic(Helen et al., 2011; Guimin, 2011; Sujoy, 2010), 2D and 3D registration (Pascal et al., 2011; Karthik et al., 2010; Murong et al., 2011). Feature matching can be formulated as finding a mapping between two sets of points, from which co…”
BackgroundInfluentialGlobal three‐dimensional‐mesh indexing based on structural analysis and geometrical signatures
Hachani, M., Zaid, A. O., & Puech, W. (2015) · IET Computer Vision · doi:10.1049/iet-cvi.2014.0250
A Novel Canonical Form for the Registration of Non Rigid 3D Shapes
Jribi, M., & Ghorbel, F. (2015) · Lecture notes in computer science · doi:10.1007/978-3-319-23117-4_20
Dubrovina, A., Aflalo, Y., & Kimmel, R. (2016) · Mathematics and visualization · doi:10.1007/978-3-319-24726-7_13
Self-citationAccurate 3D Shape Correspondence by a Local Description Darcyan Principal Curvature Fields
Sboui, I., Jribi, M., & Ghorbel, F. (2017) · Communications in computer and information science · doi:10.1007/978-3-319-60654-5_2
“[3] replaced the geodesic distance in the GromovHausdorff framework by the diffusion distance.”
BackgroundThreshold estimation in self-destructing scheme using regression analysis
Kim, Y. K., & Hong, C. S. (2017) · Asia-Pacific Network Operations and Management Symposium (APNOMS) · doi:10.1109/apnoms.2017.8094192
Gençtav, A., & Tari, S. Z. (2018) · Association for Women in Mathematics series · doi:10.1007/978-3-319-77066-6_6
The Non-rigid 3D Shape Descriptors Analysis
Liu, M., & Duan, C. (2018) · International Conference on Mechanical, Electrical, Electronic Engineering & Science (MEEES) · doi:10.2991/meees-18.2018.24
“Another popular approach to shape retrieval uses the diffusion-based point signatures [9], [10], [11].”
BackgroundZeng, H., Yu, H., Liu, W., & Sun, W. F. (2018) · International Journal of Pattern Recognition and Artificial Intelligence · doi:10.1142/s0218001419550085
Similarity estimation based on sparse spectral correspondence
Han, L., Li, D., Liu, S. N., Liu, Y. N., & Tang, D. (2018) · Multimedia Tools and Applications · doi:10.1007/s11042-018-6623-z
“[6] used diffusion geometry in order to match shapes with topological noise, and Thorstensen and Keriven [40] extended it to handle surfaces with textures.”
Auxiliary Maximum Likelihood Estimation for Noisy Point Cloud Registration
Campton, C., & Sun, X. (2019) · IEEE High Performance Extreme Computing Conference · doi:10.1109/hpec.2019.8916224
“Although ‘topological noise’ is noted [14], no existing work analyzes the relationship between a noise model and the Lp-GH distance for non-rigid registration.”
BackgroundMultiscale Geometric Data Analysis via Laplacian Eigenvector Cascading
Mike, J. L., & Perea, J. A. (2019) · International Conference on Machine Learning and Applications · doi:10.1109/icmla.2019.00183
Shape Matching Based on the Enhancement of Riemannian Structure Information
Cui, L., Zhao, W., Hassan, M. U., & Yaqoob, I. (2020) · International Conference on Vision, Image and Signal Processing (ICVISP) · doi:10.1145/3448823.3448846
“The general method usually formulates the shape matching problem as a linear assignment problem constructed by using pointwise descriptor [5] [6] or a quadratic assignment problem [7] constructed by using pairwise descriptors [8] [9].”
MethodDiscovering Relationships between Object Categories via Universal Canonical MapsPreprint
Neverova, N., Sanakoyeu, A., Labatut, P., Novotný, D., & Vedaldi, A. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2106.09758
“Non-deep learning methods include embeddings of geodesic distance matrices [13, 6] and various kinds of diffusion geometry [11] descriptors — Heat Kernel Signature [45] and its scale-invariant followup [8], Gromov-Hausdorff descriptors [7] and the Wave Kernel Signature [2].”
MethodShape Analysis Methods for 3D Brain and Skull Imaging
Gutman, B. A. (2013) · eScholarship (California Digital Library)
Bronstein, A. M., & Bronstein, M. M. (2015) · Springer eBooks · doi:10.1007/978-1-4939-0790-8_57
Self-citation(2016) · Wiley series in probability and statistics · doi:10.1002/9781119072492.refs
Gromov-Wasserstein Distances and their Lower Bounds
Weitkamp, C. A. (2022) · doi:10.53846/goediss-9587
Grownbb: Gromov–Wasserstein learning of neural best buddies for cross-domain correspondence
Tang, R., Wang, W., Yu, H., & Feng, X. (2024) · The Visual Computer · doi:10.1007/s00371-023-03251-9
“Details are given in 3.1 For graph matching and node embedding, the method in [39] uses the Gromov–Wasserstein discrepancy (GWD) [40, 41] to measure the dissimilarity between two graphs and find their correspondences according to the learned optimal transport.”
InfluentialOn the Calculation of Similarity Measures for Figures by Nonsmooth Global Optimization
Norkin, V. I., & Pflug, G. C. (2025) · Springer optimization and its applications · doi:10.1007/978-3-031-91357-0_13
Tran, T. V., & Nguyen, H. S. (2025) · Lecture notes in computer science · doi:10.1007/978-3-032-06106-5_24
Representing 3D models for alignment and recognition
Aubry, M. (2015) · theses.fr (ABES) · doi:10.70675/642480f0z5b8dz40b5z9d38za2c2c097f4d2 · Cited by 2
Nogneng, D. (2018) · doi:10.70675/33441ae5zd99bz415dz9eefz5ddf2758a863
“Most existing methods formulate the problem using this combinatorial representation of point-to-point correspondences [20,103].”
MethodContributions to optimal transport for Machine Learning : Ground Metric and Generalized Framework
Kerdoncuff, T. (2021) · theses.fr (ABES) · doi:10.70675/67878106zc7daz44a6za35az997426484823
Rigidity of the Subelliptic Heat Kernel on $${\text {SU}}\left( 2 \right) $$
Gordina, M., & Wang, J. (2026) · Potential Analysis · doi:10.1007/s11118-026-10300-0
Metadata
- Authors
- Bronstein, A. M., Bronstein, M. M., Kimmel, R., Mahmoudi, M., & Sapiro, G.
- Affiliations
- Technion – Israel Institute of Technology (IL)
- University of Minnesota (US)
- Corresponding author
- Sapiro, Guillermo
- Venue
- International Journal of Computer Vision
- Published
- 2009-10-08
- Volume
- 89
- Issue
- 2-3
- Pages
- 266-286
- ISSN
- 0920-5691 (ISSN-L)
- 1573-1405
- OpenAlex
- W1982465050
- Cited by
- 267
- References
- 78
- Citations by year
- 2026: 1 · 2025: 5 · 2024: 6 · 2023: 4 · 2022: 5 · 2021: 10 · 2020: 12 · 2019: 16 · 2018: 19 · 2017: 22 · 2016: 17 · 2015: 18 · 2014: 27 · 2013: 30 · 2012: 31
- FWCI
- 25.28
- Type
- journal article
- Language
- en
- Open access
- green
- hdl.handle.net
- Licence
- www.springer.com/tdm (tdm)
- Locations
- International Journal of Computer Vision (publishedVersion)
- University of Minnesota Digital Conservancy (University of Minnesota) (submittedVersion)
- Funders
- National Institutes of Health
- Defense Advanced Research Projects Agency
- Topics
- Physical Sciences › Computer Science › Computer Graphics and Computer-Aided Design › 3D Shape Modeling and Analysis
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Keywords
- non-rigid shape matching, shape recognition, diffusion distance, Gromov-Hausdorff distance, metric geometry, topological robustness, geodesic distance, non-rigid deformation, missing data
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