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Lifting symmetric pictures to polyhedral scenes

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Lifting symmetric pictures to polyhedral scenes. / Kaszanitzky, Viktoria Eszter; Schulze, Bernd.
In: Ars Mathematica Contemporanea, Vol. 13, No. 1, 2017, p. 31-47.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Harvard

Kaszanitzky, VE & Schulze, B 2017, 'Lifting symmetric pictures to polyhedral scenes', Ars Mathematica Contemporanea, vol. 13, no. 1, pp. 31-47.

APA

Kaszanitzky, V. E., & Schulze, B. (2017). Lifting symmetric pictures to polyhedral scenes. Ars Mathematica Contemporanea, 13(1), 31-47.

Vancouver

Kaszanitzky VE, Schulze B. Lifting symmetric pictures to polyhedral scenes. Ars Mathematica Contemporanea. 2017;13(1):31-47. Epub 2016 Oct 6.

Author

Kaszanitzky, Viktoria Eszter ; Schulze, Bernd. / Lifting symmetric pictures to polyhedral scenes. In: Ars Mathematica Contemporanea. 2017 ; Vol. 13, No. 1. pp. 31-47.

Bibtex

@article{065cd50acfa444bcac273d56cfa68ef6,
title = "Lifting symmetric pictures to polyhedral scenes",
abstract = "Scene analysis is concerned with the reconstruction of d-dimensional objects, such as polyhedral surfaces, from (d − 1)-dimensional pictures (i.e., projections of the objects onto a hyperplane). In this paper we study the impact of symmetry on the lifting properties of pictures. We first use methods from group representation theory to show that the lifting matrix of a symmetric picture can be transformed into a block-diagonalized form. Using this result we then derive new symmetry-extended counting conditions for a picture with a non-trivial symmetry group in an arbitrary dimension to be minimally flat (i.e., {\textquoteleft}non-liftable{\textquoteright}). These conditions imply very simply stated restrictions on the number of those structural components of the picture that are fixed by the various symmetry operations of the picture. We then also transfer lifting results for symmetric pictures from Euclidean (d − 1)-space to Euclidean d-space via the technique of coning. Finally, we offer some conjectures regarding sufficient conditions for a picture realized generically for a symmetry group to be minimally flat.",
keywords = "Incidence structure, picture, polyhedral scene, lifting, symmetry, coning",
author = "Kaszanitzky, {Viktoria Eszter} and Bernd Schulze",
year = "2017",
language = "English",
volume = "13",
pages = "31--47",
journal = "Ars Mathematica Contemporanea",
issn = "1855-3966",
publisher = "DMFA Slovenije",
number = "1",

}

RIS

TY - JOUR

T1 - Lifting symmetric pictures to polyhedral scenes

AU - Kaszanitzky, Viktoria Eszter

AU - Schulze, Bernd

PY - 2017

Y1 - 2017

N2 - Scene analysis is concerned with the reconstruction of d-dimensional objects, such as polyhedral surfaces, from (d − 1)-dimensional pictures (i.e., projections of the objects onto a hyperplane). In this paper we study the impact of symmetry on the lifting properties of pictures. We first use methods from group representation theory to show that the lifting matrix of a symmetric picture can be transformed into a block-diagonalized form. Using this result we then derive new symmetry-extended counting conditions for a picture with a non-trivial symmetry group in an arbitrary dimension to be minimally flat (i.e., ‘non-liftable’). These conditions imply very simply stated restrictions on the number of those structural components of the picture that are fixed by the various symmetry operations of the picture. We then also transfer lifting results for symmetric pictures from Euclidean (d − 1)-space to Euclidean d-space via the technique of coning. Finally, we offer some conjectures regarding sufficient conditions for a picture realized generically for a symmetry group to be minimally flat.

AB - Scene analysis is concerned with the reconstruction of d-dimensional objects, such as polyhedral surfaces, from (d − 1)-dimensional pictures (i.e., projections of the objects onto a hyperplane). In this paper we study the impact of symmetry on the lifting properties of pictures. We first use methods from group representation theory to show that the lifting matrix of a symmetric picture can be transformed into a block-diagonalized form. Using this result we then derive new symmetry-extended counting conditions for a picture with a non-trivial symmetry group in an arbitrary dimension to be minimally flat (i.e., ‘non-liftable’). These conditions imply very simply stated restrictions on the number of those structural components of the picture that are fixed by the various symmetry operations of the picture. We then also transfer lifting results for symmetric pictures from Euclidean (d − 1)-space to Euclidean d-space via the technique of coning. Finally, we offer some conjectures regarding sufficient conditions for a picture realized generically for a symmetry group to be minimally flat.

KW - Incidence structure

KW - picture

KW - polyhedral scene

KW - lifting

KW - symmetry

KW - coning

M3 - Journal article

VL - 13

SP - 31

EP - 47

JO - Ars Mathematica Contemporanea

JF - Ars Mathematica Contemporanea

SN - 1855-3966

IS - 1

ER -