Facundo Mémoli Postdoc Computational Topology group Math Department Stanford University m e m o l i @ m a t h . s t a n f o r d . e d u Research Publications Teaching Talks

Research Interests

shape comparison/matching, object recognition

shape and data analysis

topological data analysis, computational topology

machine learning, clustering

Research Summary

In the past, I've worked on applications of p-Harmonic Maps to Brain Warping. During my PhD thesis time I proposed looking at the plausible connection between \infty-Harmonic Maps and Lipschitz Extensions as a way of obtaining interpolating maps between two cortical surfaces. Here's a paper with these ideas.

During my masters I became aware of the fast marching method on triangulated surfaces as proposed by Sethian and Kimmel. I had been working with level set representation of surfaces so it seemed natural to look for a direct solution to the problem of estimating geodesic distances without having to triangulate the surface. This led to my master's thesis which is contained in this paper.

Later, my PhD advisor proposed me to try to extend my approach to deal with surfaces represented merely as point clouds. This was a very good idea and led me to having to generalize some ideas of the paper above. In particular, the conditions that guarantee convergence were probabilistic. You can find the paper right here.

I have also been working on Shape comparison. I am interested in applying ideas borrowed from Metric Geometry to this problem. Gromov's ideas about comparison of metric spaces have proved useful for formalizing several preexisting practical ideas in the field. In my PhD thesis I proposed using the Gromov-Hausdorff distance for approaching this problem, see this paper.

In this paper I introduced certain Lp Gromov-Hausdorff distances as an alternative to the standard notion. By doing this, I obtained a much more computationally tractable framework and at the same time, many other seemingly unrelated approaches to the problem of Shape Matching were proved to be lower or upper bounds to this new distance. In a follow paper up I study the problem of partial shape matching using these Lp distances, see below.

With Gunnar Carlsson we have become interested in questions regarding the theoretical properties of clustering algorithms. We have some results characterizing single linkage and establishing metric stability and convergence of hierarchical clustering in this paper and this paper. We have started looking at multiparameter clustering methods see this talk.

In the TMSCSCS group we deal with high dimensional datasets and we try to apply tools inspired in Algebraic Topology to the understanding of these datasets. Here's a paper with our work on applying these ideas to neuroscience data. We have developed an algorithm called mapper that is helpful in analyzing data.

Teaching

Fall 2008: Shape Matching: A metric Geometry Approach

Fall 2007: Math115

Publications

• PhD Thesis

  "Estimation Of Distance Functions And Geodesics And Its Use For Shape Comparison And Alignment: Theoretical And Computational Results." University of Minnesota, May 2005.

• Preprints and manuscripts in preparation

  • L^p Gromov-Hausdorff distances for Partial Shape Matching. F. Memoli. [in preparation].
  • Characterization, stability and convergence of hierarchical clustering algorithms. G.Carlsson and F.Memoli. [submitted]
  • Local Scale Selection for Exploratory Visualization and Analysis of Massive Datasets. G.Carlsson, F.Memoli and G.Singh. [submitted]
  • On a connection between the use of Persistence Topology for shape matching and the Gromov-Haudorff distance}. F. Memoli. [preprint- July 2008].
  • Persistent Clustering and a Theorem of J. Kleinberg. G. Carlsson and F. Memoli. August 2008.

• Journal, Conference, and Workshop publications

  1. Some ideas for formalizing clustering schemes. G.Carlsson and F.Memoli. NIPS 2009 workshop "Clustering: science or art?" . See talks section for the talk materials.
  2. Multiparameter clustering algorithms. G.Carlsson and F.Memoli. [to appear in Proceedings of IFCS 2009]. IFCS 2009.
  3. Spectral Gromov-Wasserstein distances for shape matching NORDIA-ICCV-2009. [BibTex]
  4. Gromov-Hausdorff stable signatures for shapes using persistence, with F. Chazal, D. Cohen-Steiner, S. Oudot and L. Guibas. SGP-2009. [BibTex]
  5. Gromov-Hausdorff distances in Euclidean spaces. F. Memoli. NORDIA-CVPR-2008. [BibTex]
  6. Stability of clustering methods with Gunnar Carlsson. Mathematisches Forschungsinstitut Oberwolfach Report No. 29/2008, "Computational Algebraic Topology". [BibTex]
  7. Topological Analysis of Population Activity in Primary Visual Cortex. G. Singh, F. Memoli, T. Ishkhanov, G. Carlsson, G. Sapiro and D. Ringach. Jounal of Vision, Volume 8, Number 8, Article 11, Pages 1-18. [preprint version is linked here, see the publisher's site for the journal version] [BibTex]
  8. On the Use of Gromov-Hausdorff Distances for Shape Comparison. F. Memoli. Symposium on Point Based Graphics 2007, Prague, September 2007. [BibTex]
  9. Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition. G. Singh, F. Memoli and G. Carlsson. Symposium on Point Based Graphics 2007, Prague, September 2007. [BibTex]
  10. Meshless Geometric Subdivision. Facundo Memoli, Carsten Moenning, Guillermo Sapiro, Nira Dyn and Neil Dodgson. Graphical Models Volume 69, Issues 3-4, May-July 2007, Pages 160-179. [BibTex]
  11. Brain and surface warping via minimizing Lipschitz extensions. Facundo Memoli, Guillermo Sapiro, and Paul Thompson. MFCA-2006 International Workshop on Mathematical Foundations of Computational Anatomy. MICCAI. [BibTex]
  12. A theoretical and computational framework for isometry invariant recognition of point cloud data. Facundo Memoli and Guillermo Sapiro. Found. Comput. Math. 5 (2005), no. 3, 313--347. . [BibTex]
  13. Comparing Point Clouds. Facundo Memoli and Guillermo Sapiro. SGP 2004. . [BibTex]
  14. Implicit Brain Imaging. Facundo Memoli, G. Sapiro and P.M. Thompson. (2004). NeuroImage, Special Issue on Mathematics in Brain Imaging, Thompson, P.M., Miller, M.I., Ratnanather, J.T., Poldrack, R., Nichols, T.E., [eds.]. [BibTex]
  15. Distance functions and geodesics on submanifolds of ${\Bbb R}\sp d$ and point clouds. Facundo Memoli and Guillermo Sapiro. SIAM J. Appl. Math. 65 (2005), no. 4, 1227--1260. [BibTex]
  16. Solving variational problems and partial differential equations mapping into general target manifolds. Facundo Memoli, Guillermo Sapiro and Stanley Osher. J. Comput. Phys. 195 (2004), no. 1, 263--292. [BibTex]
  17. Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces. Facundo Memoli and Guillermo Sapiro. J. Comput. Phys. 173 (2001), no. 2, 730--764. [BibTex]

• Chapters in Books

  • Variational problems and partial differential equations on implicit surfaces: bye bye triangulated surfaces? Memoli, Facundo; Bertalmio, Marcelo; Cheng, Li-Tien; Sapiro, Guillermo; Osher, Stanley Variational problems and partial differential equations on implicit surfaces: bye bye triangulated surfaces? Geometric level set methods in imaging, vision, and graphics, 381--397, Springer, New York, 2003. [BibTex]
  • Computing with Point Cloud Data. Memoli, Facundo; Sapiro, Guillermo. Statistics and Analysis of Shapes (Modeling and Simulation in Science, Engineering and Technology). Birkhauser; 1 edition (May 3, 2006). [BibTex]

• Software

  • Flujos . Software for Curve and Surface evolution.

Talks

• Some ideas for formalizing clustering schemes.. Presented at NIPS 2009 workshop: "Clustering: theory or art?", December 2009.

  [video]

  

• Some recent devolopments in the use of Gromov-Hausdorff and Gromov-Wasserstein metrics.. Presented at IMA-2009 workshop: Research in Imaging Sciences, October 2009.

  [video] Contains an overview of some recent work I've done.

• Multiparameter clustering algorithms. G.Carlsson and F.Memoli. Presented at IFCS 2009.

  

• Gromov-Wasserstein stable signatures for object matching and the role of persistence. TGDA workshop, Paris, July 2009.

  

• Gromov-Hausdorff stable signatures for shape matching using persistence. SGP 2009, Berlin, Germany. July 2009.

  

• Shape Matching using Gromov-Hausdorff distances. Computer Vision group, CS-UC Berkeley-2008.

  

• Gromov-Hausdorff distances in Euclidean spaces. NORDIA-CVPR-2008 (Alaska).

  

• Lp Gromov-Hausdorff distances for Shape Comparison. Hausdorff Center for Mathematics, University of Bonn. Workshop: "Geometry and Statistics of Shapes", June 2008.