Certifying the Existence of Epipolar Matrices

Given a set of point correspondences in two images, the existence of a fundamental matrix is a necessary condition for the points to be the images of a 3-dimensional scene imaged with two pinhole cameras. If the camera calibration is known then one requires the existence of an essential matrix. We present an efficient algorithm, using exact linear algebra, for testing the existence of a fundamental matrix. The input is any number of point correspondences. For essential matrices, we characterize the solvability of the Demazure polynomials. In both scenarios, we determine which linear subspaces intersect a fixed set defined by non-linear polynomials. The conditions we derive are polynomials stated purely in terms of image coordinates. They represent a new class of two-view invariants, free of fundamental (resp.~essential)~matrices

On the local stability of semidefinite relaxations

We consider a parametric family of quadratically constrained quadratic programs (QCQP) and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework captures a wide array of statistical estimation problems including tensor principal component analysis, rotation synchronization, orthogonal Procrustes, camera triangulation and resectioning, essential matrix estimation, system identification, and approximate GCD. Our results can also be used to analyze the stability of SOS relaxations of general polynomial optimization problems.Comment: 23 pages, 3 figure

Transition Metal-Mediated Syntheses of Yohimbane and Indolizidine Alkaloids

Polycyclic nitrogen containing heterocycles form the basic skeleton of numerous alkaloids and physiologically active drugs. Alloyohimbane was obtained from 3,4-dihydro-â-carboline using an iron-mediated [2+2+1] cycloaddition as the key-step. The bis-TMS-diyne was conveniently obtained by the C-alkylation of 3,4-dihydro-â-carboline followed by N-alkylation. Demetalation of the iron-complex followed by hydrogenation, E-ring expansion, and reduction provided alloyohimbane, a structurally and biologically interesting substance, via a linear eight-step sequence in 7% overall yield based on 3,4-dihydro-â-carboline. Another sequence provided (±)-alloyohimbane and (±)-3-epi-alloyohimbane in nine steps. The pyrrole unit occurs in a variety of naturally occurring compounds, pharmaceutical products and polymers. A novel two-step procedure for the synthesis of pyrroles by addition of a propargyl Grignard reagent to a Schiff base and subsequent silver(I)-promoted oxidative cyclization of the resulting homopropargylamine has been developed. The generality of this reaction was proven by the synthesis of a broad variety of substituted pyrroles using silver(I)-promoted cyclization. A three-step synthesis of (±)-harmicine, a natural product isolated from the Malaysian plant Kopsia griffithii having strong anti-leishmania activity, from 3,4-dihydro-â-carboline is achieved by addition of 3-trimethylsilylpropargyl Grignard reagent, Ag(I)-promoted oxidative cyclization to a pyrrole, and chemoselective hydrogenation of pyrrole ring. Total synthesis of anti-tumor active crispine A and biologically active 1,2,3,5,6,10b-hexahydropyrrolo[2,1-a]isoquinoline have been achieved in three steps using silver(I)-promoted oxidative cyclization as key step

Beyond pairwise clustering

We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a two-step algorithm for solving this problem. In the first step we use a novel scheme to approximate the hypergraph using a weighted graph. In the second step a spectral partitioning algorithm is used to partition the vertices of this graph. The algorithm is capable of handling hyperedges of all orders including order two, thus incorporating information of all orders simultaneously. We present a theoretical analysis that relates our algorithm to an existing hypergraph partitioning algorithm and explain the reasons for its superior performance. We report the performance of our algorithm on a variety of computer vision problems and compare it to several existing hypergraph partitioning algorithms

An Atlas for the Pinhole Camera

We introduce an atlas of algebro-geometric objects associated with image formation in pinhole cameras. The nodes of the atlas are algebraic varieties or their vanishing ideals related to each other by projection or elimination and restriction or specialization respectively. This atlas offers a unifying framework for the study of problems in 3D computer vision. We initiate the study of the atlas by completely characterizing a part of the atlas stemming from the triangulation problem. We conclude with several open problems and generalizations of the atlas.Comment: 47 pages with references and appendices, final versio