TY - JOUR

T1 - Multiple view geometry of non-planar algebraic curves

AU - Kaminski, J. Yermiyahu

AU - Fryers, Michael

AU - Shashua, Amnon

AU - Teicher, Mina

PY - 2001

Y1 - 2001

N2 - We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general (non-planar) algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus. We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation for which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d - 1), (ii) the dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of the curve degree alone.

AB - We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general (non-planar) algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus. We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation for which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d - 1), (ii) the dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of the curve degree alone.

UR - http://www.scopus.com/inward/record.url?scp=0034849810&partnerID=8YFLogxK

U2 - 10.1109/iccv.2001.937622

DO - 10.1109/iccv.2001.937622

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AN - SCOPUS:0034849810

SN - 1550-5499

VL - 2

SP - 181

EP - 186

JO - Proceedings of the IEEE International Conference on Computer Vision

JF - Proceedings of the IEEE International Conference on Computer Vision

ER -