Shape-from-image via cross-sections

International Conference on Pattern Recognition (ICPR), 2000, Barcelona (España)Using structural geometry, Whiteley (1991) showed that a line drawing is a correct projection of a spherical polyhedron if and only if it has a cross-section compatible with it. We extend the class of drawings to which this test applies, including those of polyhedral disks. Our proof is constructive, showing how to derive all spatial interpretations; it relies on elementary synthetic geometric arguments, and, as a by-product, it yields a simpler and shorter proof of Whiteley's result. Moreover, important properties of line drawings are visually derived as corollaries: realizability is independent of the adopted projection, it is an invariant projective property, and for trihedral drawings it can be checked with a pencil and an unmarked ruler alone.Peer Reviewe

Towards shape representation using trihedral mesh projections

This paper explores the possibility of approximating a surface by a trihedral polygonal mesh plus some triangles at strategic places. The presented approximation has attractive properties. It turns out that the Z-coordinates} of the vertices are completely governed by the Z-coordinates assigned to four selected ones. This allows describing the spatial polygonal mesh with just its 2D projection plus the heights of four vertices. As a consequence, these projections essentially capture the “spatial meaning” of the given surface, in the sense that, whatever spatial interpretations are drawn from them, they all exhibit essentially the same shape.This work was supported by the project 'Resolución de sistemas de ecuaciones cinemáticas para la simulación de mecanismos, posicionado interactivo de objetos y conformación de moléculas' (070-722).Peer Reviewe

Lifting symmetric pictures to polyhedral scenes

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

Shape-from-Image via Cross-Sections

Abstrac

Characterizing minimally flat symmetric hypergraphs

In Kaszanitzky and Schulze (2017) we gave necessary conditions for a symmetric d-picture (i.e., a symmetric realization of an incidence structure in Rd) to be minimally flat, that is, to be non-liftable to a polyhedral scene without having redundant constraints. These conditions imply very simply stated restrictions on the number of those structural components of the picture that are fixed by the elements of its symmetry group. In this paper we show that these conditions on the fixed structural components, together with the standard non-symmetric counts, are also sufficient for a plane picture which is generic with three-fold rotational symmetry C3 to be minimally flat. This combinatorial characterization of minimally flat C3-generic pictures is obtained via a new inductive construction scheme for symmetric sparse hypergraphs. We also give a sufficient condition for sharpness of pictures with C3 symmetry

Automatic creation of boundary-representation models from single line drawings

This thesis presents methods for the automatic creation of boundary-representation models of polyhedral objects from single line drawings depicting the objects. This topic is important in that automated interpretation of freehand sketches would remove a bottleneck in current engineering design methods. The thesis does not consider conversion of freehand sketches to line drawings or methods which require manual intervention or multiple drawings. The thesis contains a number of novel contributions to the art of machine interpretation of line drawings. Line labelling has been extended by cataloguing the possible tetrahedral junctions and by development of heuristics aimed at selecting a preferred labelling from many possible. The ”bundling” method of grouping probably-parallel lines, and the use of feature detection to detect and classify hole loops, are both believed to be original. The junction-line-pair formalisation which translates the problem of depth estimation into a system of linear equations is new. Treating topological reconstruction as a tree-search is not only a new approach but tackles a problem which has not been fully investigated in previous work