Degenerate Quadrics Research Articles

On a classical correspondence of real surfaces

This paper is devoted to the classical correspondence between real surfaces of degree 8 which are complete intersections of three real quadrics, and real surfaces which are two-sheeted coverings of the corresponding pencils of quadrics with branching along the curves of degenerate quadrics.

Simultaneous and Sequential Reconstruction of Visual Primitives with Bundle Adjustment

In this paper, we compare the various methods for the simultaneous and sequential reconstruction of points, lines, planes, quadrics, plane conics and degenerate quadrics using Bundle Adjustment, both in projective and metric space. In contrast, most existing work on projective reconstruction focuses mainly on one type of primitive. We also compare the simultaneous refinement of all primitives through Bundle Adjustment with various sequential methods were only certain primitives are refined together. We found that even though the sequential methods may seem somewhat arbitrary on the choice of which primitives are refined together, a higher precision and speed is achieved in most cases.

Projective reconstruction of all visual primitives

We present a method to simultaneously reconstruct cameras, points, lines, degenerate quadrics and full-rank quadrics from a set of n uncalibrated views of a static scene. Our method is based on the use of Bundle Adjustment to refine an initial reconstruction. During the initialization stage, a novel algorithm was proposed for the conic outline correction in n views. Constrains derived from this algorithm were also integrated in the cost function of Bundle Adjustment to help preserve the right topology of the reconstructed quadrics.

The projective reconstruction of points, lines, quadrics, plane conics and degenerate quadrics using uncalibrated cameras

In this paper we present an algorithm for the simultaneous projective reconstruction of points, lines, quadrics, plane conics and degenerate quadrics using Bundle Adjustment. In contrast, most existing work on projective reconstruction focuses mainly on one type of primitive. Furthermore, for the reconstruction of quadrics (both full-rank and degenerate) and plane conics, a novel algorithm for the rectification of the outlines projected in n views is presented. Finally, we found out that due to the noise in the data, the global minimum of the classic cost function used in Bundle Adjustment may unfortunately break the topology of the quadrics. Hence, during the iterations of this method, we need to add the constraints to keep the quadrics within acceptable limits to preserve their topology.

Weighted radial displacement: A geometric look at Bézier conics and quadrics

We describe a simple geometric construction for Bézier conics and quadrics, based on a tool called Weighted Radial Displacement (WRD). The shape of a rational Bézier curve or surface is modified via WRD by choosing an arbitrary point O and displacing the control points along radial directions through O , changing simultaneously the weights. To construct a conic through O , take an arbitrary segment representing a curve of degree n=1, degree raise it to n=2 and apply a WRD. Analogously, if a degree-elevated triangle is modified using a WRD, we get a quadric through O . Any quadratic Bézier patch on a nondegenerate quadric, which defines a stereographic projection, can be obtained through this method. We present a practical algorithm to detect such quadratic Bézier patches lying on quadrics. Bézier patches on degenerate quadrics are also derived via a WRD.

The weight hierarchies and generalized weight spectra of the projective codes from degenerate quadrics

The weight hierarchies and generalized weight spectra of the projective codes from degenerate quadrics in projective spaces over finite fields are determined. These codes satisfy also the chain conditions.

The geometry of quadrics and correlations of sequences (Corresp.)

Nondegenerate quadrics of PG <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2l, 2^{s})</tex> have been used to construct ternary sequences of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2^{2sl+1} - 1)/(2^{s} - 1)</tex> with perfect autocorrelation function. The same construction can be used for degenerate quadrics for this case as well as quadrics of PG <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(N,q)</tex> , with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> arbitrary and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q = p^{s}</tex> , for any prime <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> . This is possible because it is shown that if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q \subseteq {\rm PG} (N, q)</tex> is a quadric, possibly degenerate, that has the same size as a hyperplane, then, provided <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> itself is not a hyperplane, the hyperplanes of PG <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(N,q)</tex> intersect <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> in three sizes. These sizes depend on whether <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is even or odd and the degeneracy of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> . Finally, a connection to maximum period linear recursive sequences is made.