20,776 research outputs found
Estimation under group actions: recovering orbits from invariants
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
and uniform viewing directions, the number of samples required scales as
. 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.Comment: 54 pages. This version contains a number of new result
A New Method for Fast Computation of Moments Based on 8-neighbor Chain CodeApplied to 2-D Objects Recognition
2D moment invariants have been successfully applied in pattern recognition tasks. The main difficulty of using moment invariants is the computational burden. To improve the algorithm of moments computation through an iterative method, an approach for fast computation of moments based on the 8-neighbor chain code is proposed in this paper. Then artificial neural networks are applied for 2D shape recognition with moment invariants. Compared with the method of polygonal approximation, this approach shows higher accuracy in shape representation and faster recognition speed in experiment
Hyperkahler quotients and algebraic curves
We develop a graphical representation of polynomial invariants of unitary
gauge groups, and use it to find the algebraic curve corresponding to a
hyperkahler quotient of a linear space. We apply this method to four
dimensional ALE spaces, and for the A_k, D_k, and E_6 cases, derive the
explicit relation between the deformations of the curves away from the orbifold
limit and the Fayet-Iliopoulos parameters in the corresponding quotient
construction. We work out the orbifold limit of E_7, E_8, and some higher
dimensional examples.Comment: Two typos corrected--Journal version; 23 pages, 13 figures, harvma
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