Curves and Singularities

Bruce, J. W.; Giblin, Peter

doi:10.1017/cbo9781139172615

Cited by 486 publications

(450 citation statements)

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“…Borrowing to the jargon of singularity theory [BG92] , the truncated Taylor expansion J B,n (x) or J B,n (x, y) is called a degree n jet, or n-jet. Since the differential properties of a n-jet match those of its defining curve/surface up to order n, the jet is said to have a n order contact with its defining curve or surface.…”

Section: Curves and Surfaces Height Functions And Jetsmentioning

confidence: 99%

Estimating differential quantities using polynomial fitting of osculating jets

Cazals

Pouget

2005

Computer Aided Geometric Design

323

282

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This paper addresses the point-wise estimation of differential properties of a smooth manifold S -a curve in the plane or a surface in 3D-assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, and either interpolation or approximation. A jet is a truncated Taylor expansion, and the incentive for using jets is that they encode all local geometric quantities -such as normal, curvatures, extrema of curvature.On the way to using jets, the question of estimating differential properties is recasted into the more general framework of multivariate interpolation / approximation, a well-studied problem in numerical analysis. On a theoretical perspective, we prove several convergence results when the samples get denser. For curves and surfaces, these results involve asymptotic estimates with convergence rates depending upon the degree of the jet used. For the particular case of curves, an error bound is also derived. To the best of our knowledge, these results are among the first ones providing accurate estimates for differential quantities of order three and more. On the algorithmic side, we solve the interpolation/approximation problem using Vandermonde systems. Experimental results for surfaces of R 3 are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.

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Section: Curves and Surfaces Height Functions And Jetsmentioning

confidence: 99%

Estimating differential quantities using polynomial fitting of osculating jets

Cazals

Pouget

2005

Computer Aided Geometric Design

323

282

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“…By Zakalyukin's theorem [11] (see also the appendix of [7]), Im(f ) is locally diffeomorphic to the standard A k+1 -singularity if and only if it is K-rightleft equivalent to (1.1), whenever the regular points of f are dense. Thus, it is sufficient to show the versality of Φ (which implies that D Φ is locally diffeomorphic to the standard A k+1 -singularity, see [4]). In fact, this is evident by Lemma 4.3.…”

Section: Proof Of the Criteriamentioning

confidence: 99%

A k singlarities of wave fronts

Saji¹,

Umehara²,

Yamada³

2008

Math. Proc. Camb. Phil. Soc.

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A k SINGLARITIES OF WAVE FRONTS.KENTARO SAJI, MASAAKI UMEHARA, AND KOTARO YAMADA Dedicated to Professor Yoshiaki Maeda on the occasion of his sixtieth birthday.Abstract. In this paper, we discuss the recognition problem for A k -type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors' previous work "the geometry of fronts" for surfaces. The crucial point to prove our criteria for A k -singularities is to introduce a suitable parametrization of the singularities called the "k-th KRSUY-coordinates" (see Section 3). Using them, we can directly construct a versal unfolding for a given singularity. As an application, we prove that a given nondegenerate singular point p on a real (resp. complex) hypersurface (as a wave front) in R n+1 (resp. C n+1 ) is differentiably (resp. holomorphically) right-left equivalent to the A k+1 -type singular point if and only if the linear projection of the singular set around p into a generic hyperplane R n (resp. C n ) is right-left equivalent to the A k -type singular point in R n (resp. C n ). Moreover, we show that the restriction of a C ∞ -map f : R n → R n to its Morin singular set gives a wave front consisting of only A k -type singularities. Furthermore, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts.

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“…For example, the symmetry set of an ellipse contains segments on both symmetry axes of the ellipse, while the skeleton or medial axis corresponds only to the segment on the principal axis. In the Euclidean case, one possible way to simplify the symmetry set into a skeleton, inspired by original work of Giblin and colleagues (Bruce and Giblin, 1992;Wright et al, 1995), is to require that the distance to the curve is a global minimum. The affine definition is analogous: Definition 2.4.…”

Section: Affine Area Symmetry Set and Affine Invariant Skeletonsmentioning

confidence: 99%

“…Thus, the computation of skeletons and symmetry sets of planar shapes is a subject that received a great deal of attention from the mathematical (see Bruce et al, 1985;Bruce and Giblin, 1992 and references therein), computational geometry (Preparata and Shamos, 1990), biological vision (Kovács and Julesz, 1994;Lee et al, 1995;Leyton, 1992), and computer vision communities (see for example Ogniewicz, 1993;Serra, 1982 and references therein) since the original work by Blum (1967Blum ( , 1973.…”

Section: Introductionmentioning

confidence: 99%

Area-Based Medial Axis of Planar Curves

Niethammer

Betelú

Sapiro

et al. 2004

International Journal of Computer Vision

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A new definition of affine invariant medial axis of planar closed curves is introduced. A point belongs to the affine medial axis if and only if it is equidistant from at least two points of the curve, with the distance being a minimum and given by the areas between the curve and its corresponding chords. The medial axis is robust, eliminating the need for curve denoising. In a dynamical interpretation of this affine medial axis, the medial axis points are the affine shock positions of the affine erosion of the curve. We propose a simple method to compute the medial axis and give examples. We also demonstrate how to use this method to detect affine skew symmetry in real images.

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Curves and Singularities

Cited by 486 publications

References 0 publications

Estimating differential quantities using polynomial fitting of osculating jets

Estimating differential quantities using polynomial fitting of osculating jets

A k singlarities of wave fronts

Area-Based Medial Axis of Planar Curves

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