A Bézier model with shape parameters is one of the momentous research topics in geometric modeling and computer-aided geometric design. In this study, a new recursive formula in explicit expression ...is constructed that produces the generalized blended trigonometric Bernstein (or GBT-Bernstein, for short) polynomial functions of degree
m
. Using these basis functions, generalized blended trigonometric Bézier (or GBT-Bézier, for short) curves with two shape parameters are also constructed, and their geometric features and applications to curve modeling are discussed. The newly created curves share all geometric properties of Bézier curves except the shape modification property, which is superior to the classical Bézier. The
C
3
and
G
2
continuity conditions of two pieces of GBT-Bézier curves are also part of this study. Moreover, in contrast with Bézier curves, our generalization gives more shape adjustability in curve designing. Several examples are presented to show that the proposed method has high applied values in geometric modeling.
The subject of this paper is C1 sign, monotonicity and convexity preserving spline interpolation to a set of ordered points from a real function of one real variable. Two solutions are proposed ...constructing, respectively, a Hermite parametric polynomial Cubic Spline (CS), and a Hermite Cubic Rational polynomial Spline (CRS). Both curves are based on the shape preserving Hermite Variable Degree Spline (VDS) Gabrielides and Sapidis (2018) (first introduced in Kaklis and Pandelis (1990)) and they use the Bézier representation of polynomials. Since the CS curve is parametric, the present problem also requires calculation of the y-component of CS for any specific x-value; a robust solution to this problem is discussed in detail. The CRS is non-parametric and it does solve the given interpolation-problem with its weights (which play the role of tension parameters) being directly computed using the properties of the VDS segments.
This paper introduces a special class of 3D Bézier curves that are defined by their degree, a starting point, the first leg of their control polygons, and a 3D affine transformation composing of a ...uniform scaling and a rotation. We present new formulas for the curvature of such Bézier curves and based on the new formulas we derive sufficient conditions for the curves to have monotonic curvature. The conditions are expressed by a simple constraint on the rotation angle and the scaling factor. This facilitates constructing 3D Class A Bézier curves that are a generalization of planar typical curves proposed by Mineur et al. (1998), which are often used in automotive and other design applications. Some examples are provided to demonstrate the effectiveness of our construction.
•This paper presents a class of 3D Bézier curves of arbitrary degree with monotone curvature.•It is a generalization of planar typical curves proposed by Mineur et al. (1998).•We derive a sufficient condition for a 3D Bézier curve to belong to this class .
•We construct 2D bézier curves with monotone curvature using class a matrices.•A new condition for class a matrices based on its singular values is proved.•An algorithm is provided utilizing the ...condition for easier class a curve creation.•Several aesthetic curves are given to demonstrate the effectiveness of our approach.
In this paper, we construct 2D Bézier curves with monotone curvature using Class A matrices. A new sufficient condition for Class A matrices based on its singular values, is provided and proved, generalizing the 2D typical curves proposed by Mineur et al. (1998). An algorithm is provided utilizing the condition for easier Class A curve creation. Several 2D aesthetic curve examples are constructed to demonstrate the effectiveness of our approach.
Fidelity vs. simplicity Favreau, Jean-Dominique; Lafarge, Florent; Bousseau, Adrien
ACM transactions on graphics,
07/2016, Volume:
35, Issue:
4
Journal Article
Peer reviewed
Open access
Vector drawing is a popular representation in graphic design because of the precision, compactness and editability offered by parametric curves. However, prior work on line drawing vectorization ...focused solely on faithfully capturing input bitmaps, and largely overlooked the problem of producing a compact and editable curve network. As a result, existing algorithms tend to produce overly-complex drawings composed of many short curves and control points, especially in the presence of thick or sketchy lines that yield spurious curves at junctions. We propose the first vectorization algorithm that explicitly balances fidelity to the input bitmap with simplicity of the output, as measured by the number of curves and their degree. By casting this trade-off as a global optimization, our algorithm generates few yet accurate curves, and also disambiguates curve topology at junctions by favoring the simplest interpretations overall. We demonstrate the robustness of our algorithm on a variety of drawings, sketchy cartoons and rough design sketches.
•A bidirectional alternating search (BAS) strategy is introduced into the A* algorithm which improves the search efficiency.•The improved heuristic function overcomes the shortcomings of the BAS-A* ...algorithm.•The filtering function and Bézier curves is employed to reduce the turning angle, path length and smooth the path.•The practicability of the proposed algorithm is validated on the TurtleBot3 Waffle Pi mobile robot.
While the A* algorithm has been widely investigated and applied in path planning problems, it has outstanding issues such as long calculation time, large turning angles, and the unsmoothed path in large task spaces. Aiming at overcoming these problems, an improved A* algorithm is proposed in this work. First, a bidirectional alternating search (BAS) strategy is introduced into the A* algorithm; the forward and backward path lists use the current path node in the opposite list as the target node to alternately search for paths until the paths meet, which improves the search efficiency of the algorithm. Then, the heuristic function is weighted via exponential attenuation, which overcomes the shortcomings of the BAS-A* algorithm. Finally, the filtering function of path nodes is introduced to reduce redundant nodes in the path, thereby effectively reducing the turning angle. Moreover, the use of Bézier curves fulfills the requirements of smooth path planning, which is critical for the motion control of mobile robots. Simulations of path planning on maps of different sizes were performed, and the proposed algorithm was compared with that of the A* algorithm, the genetic algorithm (GA) and the simulated annealing algorithm (SA) in terms of the computation time, path length, and turning angle. The results reveal that the proposed algorithm can solve the robot path planning problem more efficiently and smoothly than the other algorithms mentioned above. Additionally, the practicability of the proposed algorithm is validated on the TurtleBot3 Waffle Pi mobile robot.
Geometry calibration of microphone arrays is an essential preprocessing step for many applications, e.g., beamforming in deformable arrays. However, most of existing geometry calibration methods ...involve the use of non-convex cost functions, suffering from the local minimum problem and low stability. To overcome these drawbacks, we introduce a novel approach that leverages the geometry feature of deformable linear arrays (DLAs) as an additional constraint. The proposed method employs Bézier curve fitting, utilizing the characteristics of Bézier curves to model the geometry feature. Specifically, we first introduce the general form of the geometry calibration problem, and an alternative approach is then proposed for a specific scenario where quadratic Bźier curves are used to fit the array shape. Finally, an additional scale modification is adopted to improve the performance of the proposed method in real scenarios. Simulations and real experiments validate the effectiveness of the proposed method for geometry calibration of DLAs.
The paper presents a new procedure for finding the minimum radius of a Bézier Curves. The authors choose to use the theoretical procedure for a practical application. For this purpose, the authors ...find the minimum radii of a Bézier Curve consisting of 16 Cubic Bézier Curves on a bike trail situated in Mălini, Romania. In the design of the bike route, the authors consider elevations to design a bike trail on level curves. This decision is made to obtain a slope of less than 20%, for a relaxing route for the family. The route is created on a length of about 5 km completely respecting the design norms of a cycling track, according to the Methodological Guide for regulating the design. Authors choose to present a comparison for the procedure between hand-drawn horizontal curves, and approximating the trail using Cubic Bézier Curves and Python
•We present Adversarial Scratches, a powerful attack to CNN classifiers.•Adversarial Scratches are designed to be deployable over a target image region.•We adopt Bezier Curves to reduce the ...dimensionality of the search space.•Adversarial Scratches yield state-of-the-art performance amongst deployable attacks.•We propose image filtering defenses and investigate their impact on healthy images.
A growing body of work has shown that deep neural networks are susceptible to adversarial examples. These take the form of small perturbations applied to the model’s input which lead to incorrect predictions. Unfortunately, most literature focuses on visually imperceivable perturbations to be applied to digital images that often are, by design, impossible to be deployed to physical targets.
We present Adversarial Scratches: a novel L0 black-box attack, which takes the form of scratches in images, and which possesses much greater deployability than other state-of-the-art attacks. Adversarial Scratches leverage Bézier Curves to reduce the dimension of the search space and possibly constrain the attack to a specific location.
We test Adversarial Scratches in several scenarios, including a publicly available API and images of traffic signs. Results show that our attack achieves higher fooling rate than other deployable state-of-the-art methods, while requiring significantly fewer queries and modifying very few pixels.
Monotone curvature plays an important role on curve design with aesthetic shapes. In this paper, we analyze the curvature distributions of Bézier curves and show that if three conditions are ...satisfied, then the curvature of the Bézier curve is monotone. We also give a sufficient geometric condition of Bézier curves with monotone curvature. Furthermore, we present a designing approach of monotone curvature variation (MCV) Bézier curves. Experiments show that curves constructed by the new method perform well not only in terms of aesthetically pleasing shape, but also in monotone curvature distribution.
•A sufficient condition of Bézier curves with monotone curvature is given.•A new designing approach of monotone curvature variation Bézier curves is presented.•The effectiveness of the new designing method is verified by some examples.
PDF
