10,254 research outputs found
A sparse resultant based method for efficient minimal solvers
Many computer vision applications require robust and efficient estimation of
camera geometry. The robust estimation is usually based on solving camera
geometry problems from a minimal number of input data measurements, i.e.
solving minimal problems in a RANSAC framework. Minimal problems often result
in complex systems of polynomial equations. Many state-of-the-art efficient
polynomial solvers to these problems are based on Gr\"obner bases and the
action-matrix method that has been automatized and highly optimized in recent
years. In this paper we study an alternative algebraic method for solving
systems of polynomial equations, i.e., the sparse resultant-based method and
propose a novel approach to convert the resultant constraint to an eigenvalue
problem. This technique can significantly improve the efficiency and stability
of existing resultant-based solvers. We applied our new resultant-based method
to a large variety of computer vision problems and show that for most of the
considered problems, the new method leads to solvers that are the same size as
the the best available Gr\"obner basis solvers and of similar accuracy. For
some problems the new sparse-resultant based method leads to even smaller and
more stable solvers than the state-of-the-art Gr\"obner basis solvers. Our new
method can be fully automatized and incorporated into existing tools for
automatic generation of efficient polynomial solvers and as such it represents
a competitive alternative to popular Gr\"obner basis methods for minimal
problems in computer vision
Sparse resultant based minimal solvers in computer vision and their connection with the action matrix
Many computer vision applications require robust and efficient estimation of
camera geometry from a minimal number of input data measurements, i.e., solving
minimal problems in a RANSAC framework. Minimal problems are usually formulated
as complex systems of sparse polynomials. The systems usually are
overdetermined and consist of polynomials with algebraically constrained
coefficients. Most state-of-the-art efficient polynomial solvers are based on
the action matrix method that has been automated and highly optimized in recent
years. On the other hand, the alternative theory of sparse resultants and
Newton polytopes has been less successful for generating efficient solvers,
primarily because the polytopes do not respect the constraints on the
coefficients. Therefore, in this paper, we propose a simple iterative scheme to
test various subsets of the Newton polytopes and search for the most efficient
solver. Moreover, we propose to use an extra polynomial with a special form to
further improve the solver efficiency via a Schur complement computation. We
show that for some camera geometry problems our extra polynomial-based method
leads to smaller and more stable solvers than the state-of-the-art Grobner
basis-based solvers. The proposed method can be fully automated and
incorporated into existing tools for automatic generation of efficient
polynomial solvers. It provides a competitive alternative to popular Grobner
basis-based methods for minimal problems in computer vision. We also study the
conditions under which the minimal solvers generated by the state-of-the-art
action matrix-based methods and the proposed extra polynomial resultant-based
method, are equivalent. Specifically we consider a step-by-step comparison
between the approaches based on the action matrix and the sparse resultant,
followed by a set of substitutions, which would lead to equivalent minimal
solvers.Comment: arXiv admin note: text overlap with arXiv:1912.1026
Large Scale SfM with the Distributed Camera Model
We introduce the distributed camera model, a novel model for
Structure-from-Motion (SfM). This model describes image observations in terms
of light rays with ray origins and directions rather than pixels. As such, the
proposed model is capable of describing a single camera or multiple cameras
simultaneously as the collection of all light rays observed. We show how the
distributed camera model is a generalization of the standard camera model and
describe a general formulation and solution to the absolute camera pose problem
that works for standard or distributed cameras. The proposed method computes a
solution that is up to 8 times more efficient and robust to rotation
singularities in comparison with gDLS. Finally, this method is used in an novel
large-scale incremental SfM pipeline where distributed cameras are accurately
and robustly merged together. This pipeline is a direct generalization of
traditional incremental SfM; however, instead of incrementally adding one
camera at a time to grow the reconstruction the reconstruction is grown by
adding a distributed camera. Our pipeline produces highly accurate
reconstructions efficiently by avoiding the need for many bundle adjustment
iterations and is capable of computing a 3D model of Rome from over 15,000
images in just 22 minutes.Comment: Published at 2016 3DV Conferenc
Robust Principal Component Analysis?
This paper is about a curious phenomenon. Suppose we have a data matrix,
which is the superposition of a low-rank component and a sparse component. Can
we recover each component individually? We prove that under some suitable
assumptions, it is possible to recover both the low-rank and the sparse
components exactly by solving a very convenient convex program called Principal
Component Pursuit; among all feasible decompositions, simply minimize a
weighted combination of the nuclear norm and of the L1 norm. This suggests the
possibility of a principled approach to robust principal component analysis
since our methodology and results assert that one can recover the principal
components of a data matrix even though a positive fraction of its entries are
arbitrarily corrupted. This extends to the situation where a fraction of the
entries are missing as well. We discuss an algorithm for solving this
optimization problem, and present applications in the area of video
surveillance, where our methodology allows for the detection of objects in a
cluttered background, and in the area of face recognition, where it offers a
principled way of removing shadows and specularities in images of faces
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