On the critical set for photogrammetric reconstruction using line tokens in P 3(?)

Buchanan, Thomas

doi:10.1007/bf00182950

Cited by 17 publications

(18 citation statements)

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“…Buchanan [2,3] pointed out that (3, 6, 5)-congruence is a critical line set, and suggested a few line sets that defeat the LiuHuang algorithm [9]. Maybank [11] used Semple's representation to describe lines so as to let line congruence be parameterized by surfaces of lower degree.…”

Section: Previous Workmentioning

confidence: 98%

Critical configurations of lines to geometry determination of three cameras

Zhao

Chung

2008

2008 19th International Conference on Pattern Recognition

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We address the question of, what structure of a set of lines in space constitutes a critical configuration to three generally positioned cameras. By critical configuration, it is meant a set of lines in space whose image projections to the cameras do not allow unique determination of the cameras' relative positions in space. We approach the question by looking into the trifocal tensor of the cameras, a quantity tightly related to the camera geometry. We focus on structures of lines in space that are common in reality -the linear structures -and examine which of them would not allow the tensor to be uniquely determined. Linear structures of lines include linear ruled surface, linear line congruence, and linear line complex, and more specifically line pencil, point star, and ruled plane. We offer a summary of by how much such families of line set are shy of the general enough structure for a unique determination of the tensor. We also point out how many lines need be visible minimally in each of them, for the full information of the associated structure to be revealed in the image data. We present synthetic and real image results to confirm the findings. The findings are important to the validity and stability of algorithms related to structure from motion and projective reconstruction using line correspondences.

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Section: Previous Workmentioning

confidence: 98%

Critical configurations of lines to geometry determination of three cameras

Zhao

Chung

2008

2008 19th International Conference on Pattern Recognition

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“…A different reconstruction problem arises when one considers projections of lines in P 3 instead of projections of points. This set-up has been considered and studied by various authors, in particular T.Buchanan [14] and S.J.Maybank [30]. Given a set of lines in P 3 and n projections of these lines to P 2 , T.Buchanan [14] shows that n = 3 is the minimum n such that it is possible to reconstruct the set from their images, up to projective transformations in P 3 .…”

Section: Critical Loci For Projective Reconstruction Of Lines and Thementioning

confidence: 99%

“…This set-up has been considered and studied by various authors, in particular T.Buchanan [14] and S.J.Maybank [30]. Given a set of lines in P 3 and n projections of these lines to P 2 , T.Buchanan [14] shows that n = 3 is the minimum n such that it is possible to reconstruct the set from their images, up to projective transformations in P 3 . Of course, also in this context, there is a natural notion of critical locus, consisting of lines in P 3 .…”

Section: Critical Loci For Projective Reconstruction Of Lines and Thementioning

confidence: 99%

The Bordiga surface as critical locus for 3-view reconstructions

Bertolini

Notari

Turrini

2019

Journal of Symbolic Computation

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In Computer Vision, images of dynamic or segmented scenes are modeled as linear projections from P k to P 2. The reconstruction problem 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 variety in P k containing the objects for which the reconstruction fails. In this paper, we deal with projections both of points from P 4 → P 2 and of lines from P 3 → P 2. In both cases, we consider 3 projections, minimal number for a uniquely determined reconstruction. In the case of projections of points, we declinate the Grassmann tensors introduced in [21] in our context, and we use them to compute the equations of the critical locus. Then, given the ideal that defines this locus, we prove that, in the general case, it defines a Bordiga surface, or a scheme in the same irreducible component of the associated Hilbert scheme. Furthermore, we prove that every Bordiga surface is actually the critical locus for the reconstruction for suitable projections. In the case of projections of lines, we compute the defining ideal of the critical locus, that is the union of 3 α-planes and a line congruence of bi-degree (3, 6) and sectional genus 5 in the Grassmannian G(1, 3) ⊂ P 5. This last surface is biregular to a Bordiga surface [40]. We use this fact to link the two reconstruction problems by showing how to compute the projections of one of the two settings, given the projections of the other one. The link is effective, in the sense that we describe an algorithm to compute the projection matrices.

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“…The curve lies on the variety defined by (1). The order of p is defined to be the number of lines which meet a general given line, where again lines are counted properly in the space of complex numbers.…”

Section: Definitions From Line Geometrymentioning

confidence: 99%

“…The proof of this theorem is given in [1]. Essentially, the proof determines ~'s order and class and ~'s singular points and planes.…”

Section: Theorem 31 With Respect To Images From Three Cameras the Gementioning

confidence: 99%

Critical sets for 3D reconstruction using lines

Buchanan¹

1992

Computer Vision — ECCV'92

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A b s t r a c t . This paper describes the geometrical limitations of algorithms for 3D reconstruction which use corresponding line tokens. In addition to announcing a description of the general critical set, we analyse the configurations defeating the Liu-Huang algorithm and study the relations between these sets. I n t r o d u c t i o nThe problem of 3D reconstruction is to determine the geometry of a three-dimensional scene on the basis of two-dimensional images. In computer vision it is of utmost importance to develop robust algorithms for solving this problem. It is also of importance to understand the limitations of the algorithms, which are presently available, because knowledge of such limitations guides their improvement or demonstrates their optimality.From a theoretical point of view there are two types of limitations. The first type involves sets of images, where there exist more than one essentially distinct 3D scene, each giving rise to the images. The superfluous reconstructions in this case can be thought of as "optical illusions". This type of limitation describes the absolute "bottom line" of the problem, because it involves scenes where the most optimal algorithm breaks down.The second type of limitation is specific to a given not necessarily optimal algorithm. It describes those scenes which "defeat" that particular algorithm.Currently, algorithms for 3D reconstruction are of two types. One type assumes a correspondence between sets of points in the images. We use projective geometry throughout the paper. Configurations in 3-space are considered to be distinct if they cannot be transformed into one another by a projective linear transformation. The use of the projective standpoint can be thought of as preliminary to studying the situation in euclidean space. But the projective situation is of interest in its own right, because some algorithms operate essentially within the projective setting. Generally, algorithms using a projective setting are easier to analyse and implement than algorithms which fully exploit the euclidean situation. This paper is organized as follows. In section 2 we collect some standard definitions from line geometry, which will allow us to describe the line sets in Sections 3 and 4. In Section 3 we describe line sets gr in 3-space and images of ~ which give rise to ambiguous reconstructions. In Section 4 we describe line sets F in 3-space which defeat the algorithm introduced in [7]. Essential properties of F were first noted in [10, p. 106] in the context of constructive geometry. In Section 5 we discuss the relationship between ~ and F.

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On the critical set for photogrammetric reconstruction using line tokens in P 3(?)

Cited by 17 publications

References 9 publications

Critical configurations of lines to geometry determination of three cameras

Critical configurations of lines to geometry determination of three cameras

The Bordiga surface as critical locus for 3-view reconstructions

Critical sets for 3D reconstruction using lines

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