Geometric Hashing
In the late 80's, I shifted my research focus from the problem of apparent motion to that of object recognition. Since I was doing my Ph.D. at the birthplace of Geometric Hashing, it was not long before I found myself studying the method from a theoretical standpoint.
At the core of the geometric hashing method is the issue of an effective representation that remains invariant under the set of allowed transformations. One such transformation, the affine, proved to be a source of frustration (initially) and of several interesting results (eventually).
In the context of my Ph.D. thesis, I had the opportunity to study the Geometric Hashing method with Bob Hummel and Haim Wolfson (now a Professor at the University of Tel Aviv). I extended the geometric hashing method by augmenting it with a noise model and embedding it in a Bayesian framework. I implemented the resulting algorithm on Thinking Machines' Connection Machine and my message-passing-based prototype system was able to locate and recognize the identity of automobiles and flying military aircraft from real-world photographs.
I continued my work on geometric hashing even after my graduation, focusing especially on the problem of well-behaved affine invariants for 2 and 3 dimensions. The work resulted in several interesting results with direct application to the problem of ligand-receptor docking.
In 1997, I was the Guest Editor together with Haim Wolson for a special issue of IEEE Computational Science and Engineering (now IEEE Computing in Science and Engineering) on Shape Matching With Geometric Hashing.. Pointers to relevant publications can be found under Publications.
Invariants
In the mid-90's, I developed a set of affine invariants for both the 2-dimensional and 3-dimensional space. These invariant guarantee uniform distribution of the generated invariant-tuples over the unit square (resp. unit cube), and the invariant-generation scheme can be tuned to take into account the shape of the objects' feature domain. Pointers to relevant publications can be found under Publications.
Applied Mathematics
Some of my earlier work revolved around the solving of partial differential equations in 2 and 3 dimensions, using the adaptive mesh refinement method.
An interesting problem which arises there is that of embedding grid points (where the solution needs further refinement) in rectangles in a manner that minimizes the computational effort that is needed to improve the solution at those points. Working with Marsha Berger I developed a very efficient gridding algorithm currently in use in software packages that implement adaptive mesh refinement -- the approach is now known as the "Berger-Rigoutsos algorithm."
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Databases
Together with Professor Alex Delis from the Polytechnic University of Brooklyn, I have been studying issues affecting the performance of hash-based storage/retrieval systems when the underlying databases are very large. See Publications for more details.