Emil Saucan

I am now back in Department of Mathematics of the Technion, Israel Institute of Technology, my host being Prof. Shahar Mendelson. Last year I was a Senior Research Fellow in the Vision and Image Sciences Laboratory (VISL), the EE Department and in the Mathematics Department of the Technion, Israel Institute of Technology. In the academic years 2005-2006 and 2006-2007 I was a Viterbi Post-doctoral fellow in the EE Department, the Technion, under the guidance of Prof. Yehoshua Y. Zeevi. I also teach in the Department of Mathematics here, at the Technion.

 

I finished my PhD in "Pure Math"!...

       For those who want to know more about me and for a (approximately) complete list of my publications - click here for a copy of my vita.  For only a list of my publications (in reverse chronological order) click here here.

 

       For those more interested in my Thesis - click here. It is based upon two papers:

  • Note on a Theorem of Munkres , Mediterranean Journal of Mathematics, vol. 2, no. 2(2005), 215-229.  This paper is dedicated to the memory of Robert Brooks. It is concerned with the existence of "fat" triangulations and of quasimeromorphic mappings on manifolds with boundary.
  • The Existence of Quasimeromorphic Mappings  , Annales Academiae Scientiarum Fennicae Mathematica, Vol. 31, 2006, 131-142. It uses the same technical apparatus to completely characterize Kleinian groups that admit non-constant quasimeromorphic automorphic mappings.

       A third paper related to the two above is

         The Existence of Quasimeromorphic Mappings in Dimension 3,  Conform. Geom. Dyn. 10 (2006), 21-40. An elementary, geometric method for fattening triangulations is developed, mainly in dimension 2 and 3.             

      For those who lack the patience (or reason) to read the full papers, a "digest" of the results can be found in

         The Existence of Automorphic Quasimeromorphic  Mappings and a Related Problem, Revue Roumaine de Math. Pures et Appl., vol. 51, no. 5-6, 2006.

         I am currently working on some extensions of the results above - some of them can be found here:  

       Remarks on the The Existence of Quasimeromorphic Mappings, AMS Contemporary Mathematics CONM/455, 2008, 325-331.  A presentation of part of these results, as presented at The Annual Conference of the Israel Mathematical Union, Neve Ilan, May 25-26, 2006, can be found here.

       As can be inferred from the title, an ``elementary’’ proof of the existence of fat triangulations of non-compact manifolds can be found in the following short note (joint work with Meir Katchalski):

       The existence of thick triangulations -- an "elementary" proof, The Open Mathematics Journal, 2, 8-11, 2009.

      Yet another proof is given in

      Intrinsic Differential Geometry and the Existence of Quasimeromorphic Mappings, Revue Roumaine de Math. Pures et Appl., 54(5-6), 565-574 , 2009.

      A generalization of the triangulation method above (and its applications), to metric measure spaces, is given in

     Curvature based triangulation of metric measure spaces, submitted.

     (Applications to Information Geometry, as well as the construction of epsilon-nets are also given in this paper.)

 

  

 

...and here is a paper that appeared as book-chapter in "What is Geometry?":

     For those who know Hebrew and fill inclined to solve a few exercises, you can find some here:

           Exercises and Problems in Complex Functions Theory  - look for Course Number 104215 (Complex Functions).

      and also

           A Problem Book in Topology (with Dan Guralnik) - look for Course Number 104142 (Introduction to Topology).

       

... and some of my "Applied Math" stuff:

  • Surface Triangulation - The Metric Approach  (with Eli Appleboim), preprint. (An earlier version - without numerical experiments an a coauthor - can be found on the ArXiv) The title is rather self- explanatory. This developed, at least as results are concerned into
  • Metric Methods in Surface Triangulation, accepted at Mathematics of Surfaces XIII,  Lecture Notes in Computer Science, 5654, 335-355 . 
  • Curvature Based Clustering for DNA Microarray Data Analysis, with Eli Appleboim, Lecture Notes in Computer Science, IbPRIA 2005, 3523, pp. 405-412, Springer-Verlag, 2005. Another essay in metric curvatures - this one employs a discrete version of Haantjes curvature. More applications of this notion in:
  • A Metric Curvature and Some of its Applications, with Eli Appleboim, in preparation. A very succinct presentation of some of the ideas presented in these papers can be found here. Some more details can be found here. An application of metric curvatures to a very different field can be found in
  • Geometric Wavelets for Image Processing: Metric Curvature of Wavelets, with Chen Sagiv and Eli Appleboim, to appear in Proceedings of SampTA09.

        Curvatures, mainly in correlation to the isometric embedding problem and its applications to Imaging and related fields are discussed here:

  • Sampling and Reconstruction of Manifolds (with Eli Appleboim and Yehoshua Y. Zeevi)., Journal of Mathematical Imaging and Vision, 30(1), 2008, 105-123. It first appeared as Technion CCIT Report #621 (EE Pub. #1578 May 2007). This paper uses some of the basic techniques developed in Note on a Theorem of Munkres and explores their use in the context of the classical Shannon's theorem. You can “get a taste” of the ideas, methods and results presented by following this link. Moreover, some new directions are explored in
  • Geometric Sampling for Signals with Applications to Images, with Eli Appleboim and Yehoshua Y. Zeevi, Proceedings of SampTA 07 - Sampling Theory and Applications. For a different field of application of the methods introduced in this paper see:
  • On the classical -- and not so classical -- Shannon Sampling Theorem, with Eli Appleboim, Dirk A. Lorenz and Yehoshua Y. Zeevi. This is Technion CCIT Report #680 (EE Pub. #1637 January 2008).  A substantially extended (and improved) version is in preparation and will be soon submitted for publication. Meanwhile, this project has grown in some new directions (while others were just “augmented”) – it has become now
  • Geometric Approach to Sampling and Communication, submitted for publication. This represents a slightly extended (and hopefully more clear!) version of Technion CCIT Report #707 (EE Pub. #1664). You can “get a taste” some of the ideas it contains here and also in
  • Geometric Sampling of Images, Vector Quantization and Zador's Theorem, with Eli Appleboim and Yehoshua Y. Zeevi, to appear in Proceedings of SampTA09. Another related direction of study is considered in
  • Geometric Reproducing Kernels for Signal Reconstruction (also with the same coauthors), to appear in Proceedings of SampTA09.
  • Local versus Global in Quasiconformal Mapping for Medical Imaging, with Eli Appleboim, Efrat Barak, Ronen Lev and Yehoshua Y. Zeevi, Journal of Mathematical Imaging and Vision, online first (DOI:10.1007/s10851-008-0101-6)  Also related somewhat to the subject of my PhD, but also introducing, in this context,  the use of quasi-geodesics on triangular meshes. Look here for a presentation (including some new directions that are not included in the article above). A paper combining many of the ideas of the papers above (and exploring them in a few new directions) is the following one:
  • Image Projection and Representation on $S^n$, with Eli Appleboim and Yehoshua Y. Zeevi, Journal of Fourier Analysis and Applications, Special Issue -- ``Analysis on the Sphere II'', 13 (6), 2007, 711-727.  Again, some connections between the diverse ideas above are explored and expanded.
  • Curvature Estimation over Smooth Polygonal Meshes using The Half Tube Formula, with Gershon Elber and Ronen Lev, Lecture Notes in Computer Science, Mathematics of Surfaces: 12th IMA International Conference,  4647, pp. 275-289, Springer-Verlag, 2007. The slides of the talk can be found here.
  • Combinatorial Ricci Curvature for Image Processing, with Eli Appleboim, Gershon Wolanski and  Yehoshua Y. Zeevi, presented at MICCAI 2008 Workshop ``Manifolds in Medical Imaging: Metrics, Learning and Beyond'',  the Midas Journal.  A Combinatorial Ricci curvature and Laplacian operators for grayscale images, based upon theoretical work of R. Forman, are introduced and tested on 2-dimensional medical images. A presentation of the ideas, methods and results presented therein, as well as a few directions of further study can be found here. Further developments, including a 3-dimensional version and flows are currently in preparation.  Some of these can be found in Combinatorial Ricci Curvature and Laplacians for Image Processing, Technion CCIT Report # 722 March 2009 (EE Pub. No. 1679).
  • I am also interested in the Didactical Aspects of Mathematics and Computer Science. Some of this interest is reflected in these papers:
  • Euler's Theorem as the Path Towards Mathematics, Nexus Network Journal -- Special issue dedicated to didactics (Teaching Mathematics to Architects), vol. 7 no. 1 (Spring 2005). A preprint (with colour images!) can be downloaded here. 
  • A Place for Differential Geometry?, Proceedings of The 4th International Colloquium on the Didactics of Mathematics, Vol II, pp. 267-276, 2005.

 

Here you can find some of my recent (and not so recent) talks:

Geometric Approach to Sampling and Communication , with Eli Appleboim and Yehoshua Y. Zeevi, presented at German-Israel Workshop for Vision and Image Sciences 2008, Haifa, November 4-6, 2008.

Combinatorial Ricci Curvature for Image Processing, with Eli Appleboim, Gershon Wolanski and  Yehoshua Y. Zeevi, presented at MICCAI 2008 Workshop “Manifolds in Medical Imaging: Metrics, Learning and Beyond”, September 10, 2008, New York.

Geometric Flow Methods for Image Processing These slides represent the notes for a “mini-mini”-course for graduate students, that I gave in the EE Department, Technion.

 Curvatures, Branched Coverings and Wheeler Foam, invited talk, presented at The Annual Conference of the Israel Mathematical Union, Beer Sheva,

May 18-19, 2007.

 

  Curvature Estimation over Smooth Polygonal Meshes using The Half Tube Formula, (joint work with Gershon Elber and Ronen Lev), presented at 12th IMA International Conference, Sheffield, England, September 4-6, 2007.

  On the existence of quasimeromorphic mappings, presented at The Annual Conference of the Israel Mathematical Union, Neve Ilan, May 25-26, 2006.

 Surface flattening - from local to global, with Eli Appleboim, Yehoshua Y. Zeevi , given at the Computer Science Department, SUNY at Sony Brook, September 11, 2008.

  Metric Curvatures and Applications, (joint work with Eli Appleboim and Yehoshua Y. Zeevi), poster presented at IMA Workshop "Shape Spaces'', 3-7 April, Minnesota, USA. A longer presentation can be found here. (This represents most of the talk I gave at  Freie Universität Berlin, 16.6.2006.)

 Towards a Shannon's Sampling Theorem for Images,  (joint work with Eli Appleboim and Yehoshua Y. Zeevi), invited talk, presented at Sixth Negev Workshop on Applied Mathematics, Sede-Boker, July

1-5, 2007.

 What is Curvature? - Talk given at The Mathematical Club, Department of Mathematics, The Technion, 25.1.2006. A presentation for a general public of many of the ideas in the papers and talks above.

 

An alphabetically ordered list of my collaborators (for published works, including technical reports and arXiv preprints) can be found here.

 

I was recently told that the photo above can hardly be used for identification purposes. I admit this is true, but I must confess I enjoy too much the Mathematics on the blackboard to discard it… However, I am including here a picture of mine that can - I hope! – be better used for my ‘face recognition’ (and it still has some Math in the background!...)

 

Dr. Emil Saucan semil”at”ee.technion.ac.il, semil”at”tx.technion.ac.il     
Electrical Engineering Department 
Technion - Israel Institute of Technology
Haifa 32000, ISRAEL
 
Office: Amado 631
Office Hour: Thursday, 15:00-16:00
 
+972-4-829-2896 (Office)
+972-4-829-4799 (FAX)     
Website http://www.ee.technion.ac.il/people/semil/     
 
There also exists an “alter ego” website, as a member of the Technion Machine Learning Center.
 

Last Modified: Tuesday, 8-July-2010.