“Functional map networks for analyzing and exploring large shape collections” by Huang, Wang and Guibas

  • ©Qixing Huang, Fan Wang, and Leonidas (Leo) J. Guibas




    Functional map networks for analyzing and exploring large shape collections

Session/Category Title:   Shape Collection




    The construction of networks of maps among shapes in a collection enables a variety of applications in data-driven geometry processing. A key task in network construction is to make the maps consistent with each other. This consistency constraint, when properly defined, leads not only to a concise representation of such networks, but more importantly, it serves as a strong regularizer for correcting and improving noisy initial maps computed between pairs of shapes in isolation. Up-to-now, however, the consistency constraint has only been fully formulated for point-based maps or for shape collections that are fully similar.In this paper, we introduce a framework for computing consistent functional maps within heterogeneous shape collections. In such collections not all shapes share the same structure — different types of shared structure may be present within different (but possibly overlapping) sub-collections. Unlike point-based maps, functional maps can encode similarities at multiple levels of detail (points or parts), and thus are particularly suitable for coping with such diversity within a shape collection. We show how to rigorously formulate the consistency constraint in the functional map setting. The formulation leads to a powerful tool for computing consistent functional maps, and also for discovering shared structures, such as meaningful shape parts. We also show how to adapt the procedure for handling very large-scale shape collections. Experimental results on benchmark datasets show that the proposed framework significantly improves upon state-of-the-art data-driven techniques. We demonstrate the usefulness of the framework in shape co-segmentation and various shape exploration tasks.


    1. Bronstein, M. M., Glashoff, K., and Loring, T. A. 2014. Making Laplacians commute: multimodal spectral geometry using closest commuting operators. SIAM Journal on Imaging Sciences (SIIS), submitted.Google Scholar
    2. Candès, E. J., Romberg, J., and Tao, T. 2006. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory 52, 2, 489–509. Google ScholarDigital Library
    3. Candès, E. J., Li, X., Ma, Y., and Wright, J. 2011. Robust principal component analysis? J. ACM 58, 3 (June), 11:1–11:37.Google ScholarDigital Library
    4. Funkhouser, T., Kazhdan, M., Shilane, P., Min, P., Kiefer, W., Tal, A., Rusinkiewicz, S., and Dobkin, D. 2004. Modeling by example. ACM Trans. Graph. 23, 3 (Aug.), 652–663. Google ScholarDigital Library
    5. Giorgi, D., Biasotti, S., and Paraboschi, L., 2007. Shape retrieval contest 2007: Watertight models track.Google Scholar
    6. Grant, M., and Boyd, S., 2011. CVX: Matlab software for disciplined convex programming. http://www.stanford.edu/~boyd/cvx/.Google Scholar
    7. Hu, R., Fan, L., and Liu, L. 2012. Co-Segmentation of 3D shapes via subspace clustering. Computer Graphics Forum 31, 5 (Aug.), 1703–1713. Google ScholarDigital Library
    8. Huang, Q., and Guibas, L. 2013. Consistent shape maps via semidefinite programming. Computer Graphics Forum (SGP) 32, 5, 177–186. Google ScholarDigital Library
    9. Huang, Q., Adams, B., Wicke, M., and Guibas, L. J. 2008. Non-rigid registration under isometric deformations. In Eurogaphics Symposium on Geometry Processing ’08, 1449–1457. Google ScholarDigital Library
    10. Huang, Q., Koltun, V., and Guibas, L. 2011. Joint shape segmentation using linear programming. ACM Trans. Graph. 30, 6 (Dec.), 125:1–125:12. Google ScholarDigital Library
    11. Huang, Q., Zhang, G.-X., Gao, L., Hu, S.-M., Butscher, A., and Guibas, L. 2012. An optimization approach for extracting and encoding consistent maps in a shape collection. ACM Trans. Graph. 31, 6 (Nov.), 167:1–167:11. Google ScholarDigital Library
    12. Huang, Q., Su, H., and Guibas, L. 2013. Fine-grained semi-supervised labeling of large shape collections. ACM Trans. Graph. 32, 6 (Nov.), 190:1–190:10. Google ScholarDigital Library
    13. Kalogerakis, E., Chaudhuri, S., Koller, D., and Koltun, V. 2012. A probabilistic model for component-based shape synthesis. ACM Trans. Graph. 31, 4 (July), 55:1–55:11. Google ScholarDigital Library
    14. Kim, V. G., Lipman, Y., and Funkhouser, T. 2011. Blended intrinsic maps. ACM Trans. Graph. 30, 4 (Aug.), 79:1–79:12. Google ScholarDigital Library
    15. Kim, V. G., Li, W., Mitra, N. J., DiVerdi, S., and Funkhouser, T. 2012. Exploring collections of 3D models using fuzzy correspondences. ACM Trans. Graph. 31, 4 (July), 54:1–54:11. Google ScholarDigital Library
    16. Kim, V. G., Li, W., Mitra, N. J., Chaudhuri, S., DiVerdi, S., and Funkhouser, T. 2013. Learning part-based templates from large collections of 3D shapes. ACM Trans. Graph. 32, 4 (July), 70:1–70:12. Google ScholarDigital Library
    17. Kovnatsky, A., Bronstein, M. M., Bronstein, A. M., Glashoff, K., and Kimmel, R. 2013. Coupled quasi-harmonic bases. In Eurographics’13, 439–448.Google Scholar
    18. Nan, L., Xie, K., and Sharf, A. 2012. A search-classify approach for cluttered indoor scene understanding. ACM Trans. Graph. 31, 6 (Nov.), 137:1–137:10. Google ScholarDigital Library
    19. Nguyen, A., Ben-Chen, M., Welnicka, K., Ye, Y., and Guibas, L. 2011. An optimization approach to improving collections of shape maps. Computer Graphics Forum 30, 5, 1481–1491.Google ScholarCross Ref
    20. Osada, R., Funkhouser, T., Chazelle, B., and Dobkin, D. 2002. Shape distributions. ACM Trans. Graph. 21 (October), 807–832. Google ScholarDigital Library
    21. Ovsjanikov, M., Ben-Chen, M., Solomon, J., Butscher, A., and Guibas, L. 2012. Functional maps: A flexible representation of maps between shapes. ACM Trans. Graph. 31, 4 (July), 30:1–30:11. Google ScholarDigital Library
    22. Rustamov, R. M., Ovsjanikov, M., Azencot, O., Ben-Chen, M., Chazal, F., and Guibas, L. 2013. Map-based exploration of intrinsic shape differences and variability. ACM Trans. Graph. 32, 4 (July), 72:1–72:12. Google ScholarDigital Library
    23. Sidi, O., van Kaick, O., Kleiman, Y., Zhang, H., and Cohen-Or, D. 2011. Unsupervised co-segmentation of a set of shapes via descriptor-space spectral clustering. ACM Trans. Graph. 30, 6 (Dec.), 126:1–126:10. Google ScholarDigital Library
    24. Solomon, J., Nguyen, A., Butscher, A., Ben-Chen, M., and Guibas, L. 2012. Soft maps between surfaces. Computer Graphics Forum 31, 5, 1617–1626. Google ScholarDigital Library
    25. Szeliski, R., Zabih, R., Scharstein, D., Veksler, O., Kolmogorov, V., Agarwala, A., Tappen, M., and Rother, C. 2008. A comparative study of energy minimization methods for Markov random fields with smoothness-based priors. IEEE Trans. Pattern Anal. Mach. Intell. 30, 6 (June), 1068–1080. Google ScholarDigital Library
    26. Wang, L., and Singer, A. 2013. Exact and stable recovery of rotations for robust synchronization. Information and Inference 2, 2, 145–193.Google ScholarCross Ref
    27. Wang, Y., Asafi, S., van Kaick, O., Zhang, H., Cohen-Or, D., and Chen, B. 2012. Active co-analysis of a set of shapes. ACM Trans. Graph. 31, 6 (Nov.), 165:1–165:10. Google ScholarDigital Library
    28. Wang, F., Huang, Q., and Guibas, L. 2013. Image co-segmentation via consistent functional maps. In Proceedings of the 14th International Conference on Computer Vision (ICCV). Google ScholarDigital Library
    29. Wang, F., Huang, Q., Ovsjanikov, M., and Guibas, L. 2014. Unsupervised multi-class joint image segmentation. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR).Google Scholar
    30. Wen, Z., Goldfarb, D., and Yin, W. 2010. Alternating direction augmented Lagrangian methods for semidefinite programming. Math. Prog. Comput. 2, 3–4, 203–230.Google ScholarCross Ref
    31. Yuan, M., and Lin, Y. 2006. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society, Series B 68, 49–67.Google ScholarCross Ref

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