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1

Critical hypersurfaces and instability for reconstruction of scenes in high dimensional projective spaces

Bertolini Marina, Magri Luca

Machine Graphics and Vision

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2020

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Vol. 29, No. 1/4

3--20

EN

In the context of multiple view geometry, images of static scenes are modeled as linear projections from a projective space ℙ3 to a projective plane ℙ2 and, similarly, videos or images of suitable dynamic or segmented scenes can be modeled as linear projections from ℙk to ℙh, with k > h ≥ 2. In those settings, the projective reconstruction of a scene consists in recovering the position of the projected objects and the projections themselves from their images, after identifying many enough correspondences between the images. A critical locus for the reconstruction problem is a configuration of points and of centers of projections, in the ambient space, where the reconstruction of a scene fails. Critical loci turn out to be suitable algebraic varieties. In this paper we investigate those critical loci which are hypersurfaces in high dimension complex projective spaces, and we determine their equations. Moreover, to give evidence of some practical implications of the existence of these critical loci, we perform a simulated experiment to test the instability phenomena for the reconstruction of a scene, near a critical hypersurface.

2

An iterative method based on 1D subspace for projective reconstruction

Liu S., Peng Y., Zeng Z., Han C.

Opto-Electronics Review

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2011

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Vol. 19, No. 1

89-94

EN

Heyden et al. introduced an iterative factorization method for projective reconstruction from image sequences. In their formulation, the projective structure and motion are computed by using an iterative factorization based on 4D subspace. In this paper, the problem is reformulated based on fact that the x, y, and z coordinates of each feature in projective space are known from their projection. The projective reconstruction, i.e., the relative depths w and the 3D motion, is obtained by a simple iterative factorization based on 1D subspace. This allows the use of very fast algorithms even when using a large number of features and large number of frames. The experiments with both simulate and real data show that the method presented in the paper is efficient and has good convergency.

3

Projective reconstruction with occlusions

Peng Y., Liu S., Liu F.

Opto-Electronics Review

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2010

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Vol. 18, No. 2

150-154

EN

An iteration method for projective reconstruction from uncalibrated image sequence with occlusions is presented in this paper. The reprojection points replace all the occlusions and projective reconstruction is obtained from all the points alternately, and the real positions of occlusions and the accurate projective reconstruction are finally obtained. The experiments with both synthetic and real data show that the method is effective and accurate.

4

3D reconstruction of parametric curves: recovering the control points

Bhuiyan M. A., Hama H.

Machine Graphics and Vision

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2004

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Vol. 13, No. 4

307-328

EN

This article introduces a new curve reconstruction method based on recovering the control points of parametric cubic curves. The method developed here has two stages: finding the 3D control points of parametric curves and reconstruction of free curves. The 3D control points of curves are computed from 2D image sequences by using projective reconstruction of the 3D control points and the bundle adjustment algorithm. The relationships among parametric curves, such as Hermite curves, Bézier curves and B-spline curves, are established so that a curve of any model can be achieved for best fitting. Some experiments are performed to show the performance and effectiveness of the algorithm. The method is based on the slope following and learning algorithm, which provides an efficient way of finding the 3D control points of any type of cubic Bézier curves. This method, which is an extension of our previous work on recovering control points of 2D Bézier curves, can automatically fit a set of data points with piecewise geometrically continuous cubic parametric curves. The experimental results demonstrate that our method is a fast and efficient way of recovering 3D control points of parametric curves, matching free curves and shape reforming.