We describe properties of Hadamard products of algebraic varieties. We show any Hadamard power of a line is a linear space, and we construct star configurations from products of collinear points. Tropical geometry is used to find the degree of Hadamard products of other linear spaces
Single-particle reconstruction in cryo-electron microscopy (cryo-EM) is an increasingly popular technique for determining the 3D structure of a molecule from several noisy 2D projections images taken at unknown viewing angles. Most reconstruction algorithms require a low-resolution initialization for the 3D structure, which is the goal of ab initio modeling. Suggested by Zvi Kam in 1980, the method of moments (MoM) offers one approach, wherein loworder statistics of the 2D images are computed and a 3D structure is estimated by solving a system of polynomial equations. Unfortunately, Kam's method suffers from restrictive assumptions, most notably that viewing angles should be distributed uniformly. Often unrealistic, uniformity entails the computation of higher-order correlations, as in this case first and second moments fail to determine the 3D structure. In the present paper, we remove this hypothesis, by permitting an unknown, non-uniform distribution of viewing angles in MoM. Perhaps surprisingly, we show that this case is statistically easier than the uniform case, as now first and second moments generically suffice to determine
We present a new insight into the systematic generation of minimal solvers in computer vision, which leads to smaller and faster solvers. Many minimal problem formulations are coupled sets of linear and polynomial equations where image measurements enter the linear equations only. We show that it is useful to solve such systems by first eliminating all the unknowns that do not appear in the linear equations and then extending solutions to the rest of unknowns. This can be generalized to fully non-linear systems by linearization via lifting. We demonstrate that this approach leads to more efficient solvers in three problems of partially calibrated relative camera pose computation with unknown focal length and/or radial distortion. Our approach also generates new interesting constraints on the fundamental matrices of partially calibrated cameras, which were not known before.
Motivated by geometric problems in signal processing, computer vision, and structural biology, we study a class of orbit recovery problems where we observe very noisy copies of an unknown signal, each acted upon by a random element of some group (such as Z/p or SO(3)). The goal is to recover the orbit of the signal under the group action in the high-noise regime. This generalizes problems of interest such as multi-reference alignment (MRA) and the reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain matching lower and upper bounds on the sample complexity of these problems in high generality, showing that the statistical difficulty is intricately determined by the invariant theory of the underlying symmetry group.In particular, we determine that for cryo-EM with noise variance σ 2 and uniform viewing directions, the number of samples required scales as σ 6 . We match this bound with a novel algorithm for ab initio reconstruction in cryo-EM, based on invariant features of degree at most 3. We further discuss how to recover multiple molecular structures from heterogeneous cryo-EM samples.
Abstract. The recovery of three calibrated cameras from image data is investigated using tools from computational algebraic geometry. We determine the algebraic degree for various minimal problems. Our formulation is based on the calibrated trifocal variety in computer vision, which is the configuration space for three calibrated cameras. Some of our calculations are done using homotopy continuation software, and so they rely on pseudo-randomness and numerical accuracy.
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