Courant Institute of Mathematical Sciences

About: Courant Institute of Mathematical Sciences is a education organization based out in New York, New York, United States. It is known for research contribution in the topics: Nonlinear system & Boundary value problem. The organization has 2414 authors who have published 7759 publications receiving 439773 citations. The organization is also known as: CIMS & New York University Department of Mathematics.

Space Variant Image Processing

Abstract: This paper describes a graph-based approach to image processing, intended for use with images obtained from sensors having space variant sampling grids. The connectivity graph (CG) is presented as a fundamental framework for posing image operations in any kind of space variant sensor. Partially motivated by the observation that human vision is strongly space variant, a number of research groups have been experimenting with space variant sensors. Such systems cover wide solid angles yet maintain high acuity in their central regions. Implementation of space variant systems pose at least two outstanding problems. First, such a system must be active, in order to utilize its high acuity region; second, there are significant image processing problems introduced by the non-uniform pixel size, shape and connectivity. Familiar image processing operations such as connected components, convolution, template matching, and even image translation, take on new and different forms when defined on space variant images. The present paper provides a general method for space variant image processing, based on a connectivity graph which represents the neighbor-relations in an arbitrarily structured sensor. We illustrate this approach with the following applications: (1) Connected components is reduced to its graph theoretic counterpart. We illustrate this on a logmap sensor, which possesses a difficult topology due to the branch cut associated with the complex logarithm function. (2) We show how to write local image operators in the connectivity graph that are independent of the sensor geometry. (3) We relate the connectivity graph to pyramids over irregular tessalations, and implement a local binarization operator in a 2-level pyramid. (4) Finally, we expand the connectivity graph into a structure we call a transformation graph, which represents the effects of geometric transformations in space variant image sensors. Using the transformation graph, we define an efficient algorithm for matching in the logmap images and solve the template matching problem for space variant images. Because of the very small number of pixels typical of logarithmic structured space variant arrays, the connectivity graph approach to image processing is suitable for real-time implementation, and provides a generic solution to a wide range of image processing applications with space variant sensors.

Large manifolds with positive Ricci curvature

Abstract: The main purpose of this paper is to show that an n-dimensional manifold with Ricci curvature greater or equal to (n−1) which is close (in the Gromov– Hausdor topology) to the unit n-sphere has volume close to that of the sphere. This shows the converse of the theorem in [C1]. Namely together with [C1] it shows that an n-manifold with Ricci curvature greater or equal to (n − 1) is close to the sphere if and only if the volume is close to that of the sphere. In particular, by [P], such a manifold is homeomorphic to a sphere. Further, as an application of this and the result of [C1], we prove a Radius Theorem saying that if an n-manifold with Ricci curvature greater or equal to (n − 1) has radius almost equal to ; then the volume is close to that of the sphere. In order to obtain these results we further develop and apply the estimates of [C1]. Whereas the main concern in [C1] were with the large scale geometry the main concern of this paper is with the small scale geometry. Let !n be the volume of the round n-sphere, Sn; with sectional curvature one.

Properties of Pseudoholomorphic Curves in Symplectizations III: Fredholm Theory

Abstract: We shall study smooth maps ũ: S → ℝ x M of finite energy defined on the punctured Riemann surface S = S\Γ and satisfying a Cauchy-Riemann type equation Tũ ∘ j = Jũ ∘ Tũ for special almost complex structures J, related to contact forms A on the compact three manifold M. Neither the domain nor the target space are compact. This difficulty leads to an asymptotic analysis near the punctures. A Fredholm theory determines the dimension of the solution space in terms of the asymptotic data defined by non-degenerate periodic solutions of the Reeb vector field associated with λ on M, the Euler characteristic of S, and the number of punctures. Furthermore, some transversality results are established.

Quantitative Steinitz's theorems with applications to multifingered grasping

Abstract: We prove the following quantitative form of a classical theorem of Steintiz: Letm be sufficiently large. If the convex hull of a subsetS of Euclideand-space contains a unit ball centered on the origin, then there is a subset ofS with at mostm points whose convex hull contains a solid ball also centered on the origin and havingresidual radius $$1 - 3d\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ The casem=2d was first considered by Baranyet al. [1]. We also show an upper bound on the achievable radius: the residual radius must be less than $$1 - \frac{1}{{17}}\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ These results have applications in the problem of computing the so-calledclosure grasps by anm-fingered robot hand. The above quantitative form of Steinitz's theorem gives a notion ofefficiency for closure grasps. The theorem also gives rise to some new problems in computational geometry. We present some efficient algorithms for these problems, especially in the two-dimensional case.

Molecular surface recognition by a computer vision-based technique

Abstract: Correct docking of a ligand onto a receptor surface is a complex problem, involving geometry and chemistry. Geometrically acceptable solutions require close contact between corresponding patches of surfaces of the receptor and of the ligand and no overlap between the van der Waals spheres of the remainder of the receptor and ligand atoms. In the quest for favorable chemical interactions, the next step involves minimization of the energy between the docked molecules. This work addresses the geometrical aspect of the problem. It is assumed that we have the atomic coordinates of each of the molecules. In principle, since optimally matching surfaces are sought, the entire conformational space needs to be considered. As the number of atoms residing on molecular surfaces can be several hundred, sampling of all rotations and translations of every patch of a surface of one molecule with respect to the other can reach immense proportions. The problem we are faced with here is reminiscent of object recognition problems in computer vision. Here we borrow and adapt the geometric hashing paradigm developed in computer vision to a central problem in molecular biology. Using an indexing approach based on a transformation invariant representation, the algorithm efficiently scans groups of surface dots (or atoms) and detects optimally matched surfaces. Potential solutions displaying receptor--ligand atomic overlaps are discarded. Our technique has been applied successfully to seven cases involving docking of small molecules, where the structures of the receptor--ligand complexes are available in the crystallographic database and to three cases where the receptors and ligands have been crystallized separately. In two of these three latter tests, the correct transformations have been obtained.