Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F24%3A00380112" target="_blank" >RIV/68407700:21230/24:00380112 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1007/s10851-024-01182-1" target="_blank" >https://doi.org/10.1007/s10851-024-01182-1</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s10851-024-01182-1" target="_blank" >10.1007/s10851-024-01182-1</a>

Result language

angličtina

Original language name

Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix

Original language description

Many computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements. Minimal problems are usually formulated as complex systems of sparse polynomial equations. 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 based on the Newton polytopes has not been used so often for generating efficient solvers, primarily because the polytopes do not respect the constraints amongst the coefficients. In an attempt to tackle this challenge, here 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 Schur complement computation. We show that for some camera geometry problems our resultant-based method leads to smaller and more stable solvers than the state-of-the-art Gröbner basis-based solvers, while being significantly smaller than the state-of-the-art resultant-based methods. The proposed method can be fully automated and incorporated into existing tools for the automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Gröbner basis-based methods for minimal problems in computer vision. Additionally, we 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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

Result was created during the realization of more than one project. More information in the Projects tab.

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2024

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Journal of Mathematical Imaging and Vision

ISSN

0924-9907

e-ISSN

1573-7683

Volume of the periodical

66

Issue of the periodical within the volume

3

Country of publishing house

US - UNITED STATES

Number of pages

26

Pages from-to

335-360

UT code for WoS article

001190195900001

EID of the result in the Scopus database

2-s2.0-85188428706