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Facundo MémoliPostdocComputational 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 |
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Facundo MémoliPostdocComputational 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 |
shape comparison/matching, object recognition
shape and data analysis
topological data analysis, computational topology
machine learning
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.
Fall 2008: Shape Matching: A metric Geometry Approach
Fall 2007: Math115
"Estimation Of Distance Functions And Geodesics And Its Use For Shape Comparison And Alignment: Theoretical And Computational Results." University of Minnesota, May 2005.
Multiparameter clustering algorithms. G.Carlsson and
F.Memoli. [submitted] Presented at IFCS 2009. [slides]
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] Gromov-Hausdorff
stable signatures for shapes using persistence, with
F. Chazal, D. Cohen-Steiner, S. Oudot and L. Guibas. SGP-2009.
[BibTex]