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
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.
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
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
For those who lack the patience (or reason)
to read the full papers, a "digest" of the results can be found in
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
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,
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 ,
A generalization of the
triangulation method above (and its applications), to metric measure spaces, is given in
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
A Problem Book in Topology
(with Dan Guralnik)
- look for Course Number 104142 (Introduction to Topology).
... and some of my "Applied Math" stuff:
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
Methods in Surface Triangulation, accepted at Mathematics of
Surfaces XIII, Lecture Notes in
Computer Science, 5654, 335-355 .
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
- 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
Some more details can be found here.
An application of metric curvatures to a very different field can be found
Wavelets for Image Processing: Metric Curvature of Wavelets, with
Chen Sagiv and Eli Appleboim,
to appear in Proceedings of SampTA09.
mainly in correlation to the isometric embedding
problem and its applications to
Imaging and related fields are discussed here:
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
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:
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).
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
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
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
Reproducing Kernels for Signal Reconstruction (also with the same
coauthors), to appear in Proceedings
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
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:
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
4647, pp. 275-289, Springer-Verlag,
2007. The slides of the talk can be found here.
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
- 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
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.
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.
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
Annual Conference of the Israel
Mathematical Union, Beer Sheva,
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.
flattening - from local to global, with Eli Appleboim,
Yehoshua Y. Zeevi , given
at the Computer Science Department, SUNY at Sony Brook, September
Curvatures and Applications, (joint work with Eli Appleboim and Yehoshua Y. Zeevi), poster presented at IMA Workshop
"Shape Spaces'', 3-7 April, Minnesota,
A longer presentation can be found here.
(This represents most of the talk I gave
at Freie Universität Berlin, 16.6.2006.)
Shannon's Sampling Theorem for Images, (joint work with Eli Appleboim
and Yehoshua Y. Zeevi), invited talk, presented at
Workshop on Applied Mathematics, Sede-Boker, July
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
There also exists an “alter ego” website, as a member of the Technion Machine Learning Center.
Modified: Tuesday, 8-July-2010.