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Smooth Interpolating curves with Local Control and Monotone Alternating Curvature

COMPUTER GRAPHICS FORUM 2022年第5期41卷 25-38页

作者： Binninger, Alexandre Sorkine-Hornung, Olga Swiss Fed Inst Technol Zurich Switzerland

We propose a method for the construction of a planar curve based on piecewise clothoids and straight lines that intuitively interpolates a given sequence of control points. Our method has several desirable properties that are not simultaneously fulfilled by previous approaches: Our interpolating curves are C-2 continuous, their computation does not rely on global optimization and has local support, enabling fast evaluation for interactive modeling. Further, the sign of the curvature at control points is consistent with the control polygon;the curvature attains its extrema at control points and is monotone between consecutive control points of opposite curvature signs. In addition, we can ensure that the curve has self-intersections only when the control polygon also self-intersects between the same control points. For more fine-grained control, the user can specify the desired curvature and tangent values at certain control points, though it is not required by our method. Our local optimization can lead to discontinuity w.r.t. the locations of control points, although the problem is limited by its locality. We demonstrate the utility of our approach in generating various curves and provide a comparison with the state of the art.

关键词： CCS Concepts • computing methodologies → parametric curve and surface models

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A Hybrid Parametrization Method for B-Spline curve Interpolation via Supervised Learning

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COMPUTER GRAPHICS FORUM 2024年第7期43卷

作者： Song, Tianyu Shen, Tong Ge, Linlin Feng, Jieqing Zhejiang Univ State Key Lab CAD&CG Hangzhou Peoples R China

B-spline curve interpolation is a fundamental algorithm in computer-aided geometric design. Determining suitable parameters based on data points distribution has always been an important issue for high-quality interpolation curves generation. Various parameterization methods have been proposed. However, there is no universally satisfactory method that is applicable to data points with diverse distributions. In this work, a hybrid parametrization method is proposed to overcome the problem. For a given set of data points, a classifier via supervised learning identifies an optimal local parameterization method based on the local geometric distribution of four adjacent data points, and the optimal local parameters are computed using the selected optimal local parameterization method for the four adjacent data points. Then a merging method is employed to calculate global parameters which align closely with the local parameters. Experiments demonstrate that the proposed hybrid parameterization method well adapts the different distributions of data points statistically. The proposed method has a flexible and scalable framework, which can includes current and potential new parameterization methods as its components.

关键词： CCS Concepts • computing methodologies → parametric curve and surface models Supervised learning by classification

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Disk B-spline on S²: A Skeleton-based Region Representation on 2-Sphere

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COMPUTER GRAPHICS FORUM 2024年第7期43卷

作者： Zheng, Chunhao Zhao, Yuming Wu, Zhongke Wang, Xingce Beijing Normal Univ Beijing Peoples R China

Due to the widespread applications of 2-dimensional spherical designs, there has been an increasing requirement of modeling on the S-2 manifold in recent years. Due to the non-Euclidean nature of the sphere, it has some challenges to find a method to represent 2D regions on S-2 manifold. In this paper, a skeleton-based representation method of regions on S-2, disk B-spline(DBSC) on S-2 is proposed. Firstly, we give the definition and basic algorithms of DBSC on S-2. Then we provide the calculation method of DBSC on S-2, which includes calculating the boundary points, internal points and their corresponding derivatives. Based on that, we give some modeling methods of DBSC on S-2, including approximation, deformation. In the end, some stunning application examples of DBSC on S-2 are shown. This work lays a theoretical foundation for further applications of DBSC on S-2.

关键词： CCS Concepts • computing methodologies → parametric curve and surface models

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Implicit Modeling of Patient-Specific Aortic Dissections with Elliptic Fourier Descriptors

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COMPUTER GRAPHICS FORUM 2021年第3期40卷 423-434页

作者： Mistelbauer, G. Rossl, C. Baeumler, K. Preim, B. Fleischmann, D. Otto von Guericke Univ Dept Simulat & Graph Magdeburg Germany Stanford Univ Sch Med Dept Radiol 3D & Quantitat Imaging Lab Stanford CA 94305 USA

Aortic dissection is a life-threatening vascular disease characterized by abrupt formation of a new flow channel (false lumen) within the aortic wall. Survivors of the acute phase remain at high risk for late complications, such as aneurysm formation, rupture, and death. Morphologic features of aortic dissection determine not only treatment strategies in the acute phase (surgical vs. endovascular vs. medical), but also modulate the hemodynamics in the false lumen, ultimately responsible for late complications. Accurate description of the true and false lumen, any communications across the dissection membrane separating the two lumina, and blood supply from each lumen to aortic branch vessels is critical for risk prediction. Patient-specific surface representations are also a prerequisite for hemodynamic simulations, but currently require time-consuming manual segmentation of CT data. We present an aortic dissection cross-sectional model that captures the varying aortic anatomy, allowing for reliable measurements and creation of high-quality surface representations. In contrast to the traditional spline-based cross-sectional model, we employ elliptic Fourier descriptors, which allows users to control the accuracy of the cross-sectional contour of a flow channel. We demonstrate (i) how our approach can solve the requirements for generating surface and wall representations of the flow channels, (ii) how any number of communications between flow channels can be specified in a consistent manner, and (iii) how well branches connected to the respective flow channels are handled. Finally, we discuss how our approach is a step forward to an automated generation of surface models for aortic dissections from raw 3D imaging segmentation masks.

关键词： CCS Concepts • Human-centered computing → Scientific visualization • Applied computing → Health informatics • computing methodologies → parametric curve and surface models

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Maximum Likelihood Coordinates

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COMPUTER GRAPHICS FORUM 2023年第5期42卷 e14908-e14908页

作者： Chang, Q. Deng, C. Hormann, K. Univ Svizzera italiana Lugano Switzerland Hangzhou Dianzi Univ Hangzhou Zhejiang Peoples R China

Any point inside a d-dimensional simplex can be expressed in a unique way as a convex combination of the simplex's vertices, and the coefficients of this combination are called the barycentric coordinates of the point. The idea of barycentric coordinates extends to general polytopes with n vertices, but they are no longer unique if n > d+ 1. Several constructions of such generalized barycentric coordinates have been proposed, in particular for polygons and polyhedra, but most approaches cannot guarantee the non-negativity of the coordinates, which is important for applications like image warping and mesh deformation. We present a novel construction of non-negative and smooth generalized barycentric coordinates for arbitrary simple polygons, which extends to higher dimensions and can include isolated interior points. Our approach is inspired by maximum entropy coordinates, as it also uses a statistical model to define coordinates for convex polygons, but our generalization to non-convex shapes is different and based instead on the project-and-smooth idea of iterative coordinates. We show that our coordinates and their gradients can be evaluated efficiently and provide several examples that illustrate their advantages over previous constructions.

关键词： CCS Concepts • computing methodologies → parametric curve and surface models • Mathematics of computing → Convex optimization

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Integer-Grid Sketch Simplification and Vectorization

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COMPUTER GRAPHICS FORUM 2020年第5期39卷 149-161页

作者： Stanko, Tibor Bessmeltsev, Mikhail Bommes, David Bousseau, Adrien Univ Cote Azur Nice France Univ Montreal Montreal PQ Canada Univ Bern Bern Switzerland

A major challenge in line drawing vectorization is segmenting the input bitmap into separate curves. This segmentation is especially problematic for rough sketches, where curves are depicted using multiple overdrawn strokes. Inspired by feature-aligned mesh quadrangulation methods in geometry processing, we propose to extract vector curve networks by parametrizing the image with local drawing-aligned integer grids. The regular structure of the grid facilitates the extraction of clean line junctions;due to the grid's discrete nature, nearby strokes are implicitly grouped together. We demonstrate that our method successfully vectorizes both clean and rough line drawings, whereas previous methods focused on only one of those drawing types.

关键词： CCS Concepts • computing methodologies → parametric curve and surface models Reconstruction

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