Bivariate B-Splines from Convex Pseudo-circle Configurations

Schmitt, Dominique

doi:10.1007/978-3-030-25027-0_23

Cited by 4 publications

(2 citation statements)

References 13 publications

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“…Generally, the subdivision-based approaches may generate an excess amount of primitives, and can be strongly dependent on the feature extraction output [146]. Recently, Zhu et al [146] proposed to use quadratic triangle configuration B-splines (TCBs) [19], [112] to faithfully depict complex color variations while remaining C 1 continuous. An image can be treated as a surface in 5D (geometry and color) and can therefore be approximated and refined as a TCB-spline surface.…”

Section: Triangular Meshesmentioning

confidence: 99%

A Survey of Smooth Vector Graphics: Recent Advances in Repr esentation, Creation, Rasterization, and Image Vectorization

Tian

Günther

2024

IEEE Trans. Visual. Comput. Graphics

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The field of smooth vector graphics explores the representation, creation, rasterization, and automatic generation of light-weight image representations, frequently used for scalable image content. Over the past decades, several conceptual approaches on the representation of images with smooth gradients have emerged that each led to separate research threads, including the popular gradient meshes and diffusion curves. As the computational models matured, the mathematical descriptions diverged and papers started to focus more narrowly on subproblems, such as on the representation and creation of vector graphics, or the automatic vectorization from raster images. Most of the work concentrated on a specific mathematical model only. With this survey, we describe the established computational models in a consistent notation to spur further knowledge transfer, leveraging the recent advances in each field. We therefore categorize vector graphics papers from the last decades based on their underlying mathematical representations as well as on their contribution to the vector graphics content creation pipeline, comprising representation, creation, rasterization, and automatic image vectorization. This survey is meant as an entry point for both artists and researchers. We conclude this survey with an outlook on promising research directions and challenges to overcome in the future.

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Section: Triangular Meshesmentioning

confidence: 99%

A Survey of Smooth Vector Graphics: Recent Advances in Repr esentation, Creation, Rasterization, and Image Vectorization

Tian

Günther

2024

IEEE Trans. Visual. Comput. Graphics

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“…A first algorithm to construct a polynomial-reproducing simplex spline space over conv(A), valid in the case d = 2 and without repeated or affinely independent points, was provided by Liu and Snoeyink [14,15] and proved to be valid at all degrees k by Schmitt [18]. This approach relies on the construction and subsequent triangulation, without added points, of a series of two-dimensional (non-convex) polygonal regions, called vertex links.…”

Section: Simplex Spline Space Constructionmentioning

confidence: 99%

Practical unstructured splines: Algorithms, multi-patch spline spaces, and some applications to numerical analysis

Frambati

Barucq

Calandra

et al. 2022

Journal of Computational Physics

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Separation by Convex Pseudo-Circles

Chevallier

Fruchard

Schmitt

et al. 2020

Discrete Comput Geom

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Let S be a finite set of n points in the plane in general position. We prove that every inclusion-maximal family of subsets of S separable by convex pseudo-circles has the same cardinal n 0 + n 1 + n 2 + n 3. This number does not depend on the configuration of S and is the same as the number of subsets of S separable by true circles. Buzaglo, Holzman, and Pinchasi showed that it is an upper bound for the number of subsets separable by (non necessarily convex) pseudo-circles. Actually, we first count the number of elements in a maximal family of k-subsets of S separable by convex pseudo-circles, for a given k. We show that Lee's result on the number of k-subsets separable by true circles still holds for convex pseudo-circles. In particular, this means that the number of k-subsets of S separable by a maximal family of convex pseudo-circles is an invariant of S: It does not depend on the choice of the maximal family. To prove this result, we introduce a graph that generalizes the dual graph of the order-k Voronoi diagram, and whose vertices are the k-subsets of S separable by a maximal family of convex pseudo-circles. In order to count the number of vertices of this graph, we first show that it admits a planar realization which is a triangulation. It turns out (but is not detailed in the present paper) that these triangulations are the centroid triangulations Liu and Snoeyink conjectured to construct. Theorem 1.2. Let S be a set of n points in the plane in general position and k ∈ {1,. .. , n}. Every inclusion-maximal family of k-subsets of S separable by convex pseudo-circles admits 2kn − n − k 2 + 1 − k−1 i=1 a i (S) elements. To prove these results, we first characterize convex pseudo-circle separability in terms of convex hulls (Section 2). We show that a family F of subsets of S is separable by convex pseudo-circles if conv(T) ∩ S = T for all T ∈ F, and conv(T \ T ′) ∩ conv(T ′ \ T) = ∅ for all T, T ′ ∈ F. We also characterize in a similar way families of pairs (P, Q) of subsets of S determined by convex pseudo-circles passing through the points of Q, containing P , and excluding S \ (P ∪ Q). Such pairs are called convex pairs. We then introduce a graph that generalizes the dual of the order-k Voronoi diagram (Sections 3 and 4). The vertices of this graph are the elements of size k in a family of subsets of S separable by a maximal family C of convex pseudo-circles. The edges of the graph are the convex pairs of the form (P, {s, t}), with |P | = k − 1, determined by convex pseudo-circles compatible with C. The edge (P, {s, t}) connects the two vertices P ∪ {s} and P ∪ {t}. The key result of the article, Theorem 3.9, is that this graph admits a planar geometric realization which induces a triangulation. Proposition 3.4 already shows that the edges of this realization are disjoint. The main difficulty is to show that every edge is incident to a triangle, see Theorem 3.7. The long and tricky proof of this result can be found in Section 5. Two important contributions of our work are, firstly, the notion of convex p...

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Bivariate B-Splines from Convex Pseudo-circle Configurations

Cited by 4 publications

References 13 publications

A Survey of Smooth Vector Graphics: Recent Advances in Repr esentation, Creation, Rasterization, and Image Vectorization

A Survey of Smooth Vector Graphics: Recent Advances in Repr esentation, Creation, Rasterization, and Image Vectorization

Practical unstructured splines: Algorithms, multi-patch spline spaces, and some applications to numerical analysis

Separation by Convex Pseudo-Circles

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