Bezier Approximation Research Articles

Deep Neural Network with a Characteristic Analysis for Seal Stroke Recognition

Seal characters are derived from ancient Chinese pictographs, naturally inheriting pictographic characteristics and complex structures. As the essential components of seal characters, seal strokes play a vital role in seal character recognition, composition and writing, so accurate recognition of seal strokes can greatly promote the investigation of seal characters. Inspired by curve fitting, we propose a new model called the characteristic analysis neural network (CANN) for seal stroke recognition. Instead of indiscriminate grasping of feature information in regular neural networks, we design an efficient approximation technique based on the piecewise Bezier curves that can effectively facilitate structural compression and lossless feature extraction. The feature extraction capability of Bezier approximation helps the methodology achieve impressive recognition accuracy not only on the seal strokes but also on any curve-based symbols. Furthermore, the hierarchical structure of the deep learning strategy is inherited and improved for better performance with high generalisation. Experiments conducted on different types of strokes verify that CANN obtains superior performance on both seal strokes and other smooth symbols. The robustness and the effectiveness of CANN are also demonstrated with minimal learning cost compared to other state-of-art models.

Adaptive Quadrature of Trimmed Finite Elements and Cells Based on Bezier Approximation

In this paper, a new boundary-conforming adaptive method for the numerical integration of trimmed elements is presented. The locations and weights of new integration points are determined based on special mapping formulations. The prerequisite of this technique is to describe the trimming curves/surfaces by parametric Bezier curves/surfaces within the parent element domain. The fitting error is under control, therefore the number of quadrature points for exact integration can be adjusted automatically based on the complexity of trimming curve and the corresponding integrand. The proposed method can be easily implemented into fictitious domain approaches. Different integration tasks as well as structural examples reveal that the proposed method delivers accurate and robust solutions for a wide variety of 2D/3D geometries with a rather low number of integration points.

Accurate integration of trimmed cells based on Bezier approximation

AbstractIn this work, a new adaptive integration method for simulation of two‐dimensional linear elasticity problems is presented. The main benefit of the proposed method is the reduction of the computational cost by lowering the number of integration points required to reach a certain level of accuracy. The main concept of the proposed method is to calculate new weights for trimmed cells employing the advantage of Bezier parametric curves. Within this concept, it is possible to map a square to a triangle with one curved edge where any curved edge is approximated by a parametric Bezier curve. In this way, a new set of Gaussian quadrature points is introduced for each trimmed cell in a fast and robust way. Besides main mapping cases, the integration method includes supplementary cases as well to increase the robustness and generality of the method. In the next step, the proposed method is implemented in a two‐dimensional fictitious domain code in MATLAB to solve structural problems. The results will be compared to those obtained through the commercial finite element code ABAQUS. It is shown that the proposed method is accurate and robust.

Microscope self-calibration based on micro laser line imaging and soft computing algorithms

A technique to perform microscope self-calibration via micro laser line and soft computing algorithms is presented. In this technique, the microscope vision parameters are computed by means of soft computing algorithms based on laser line projection. To implement the self-calibration, a microscope vision system is constructed by means of a CCD camera and a 38 µm laser line. From this arrangement, the microscope vision parameters are represented via Bezier approximation networks, which are accomplished through the laser line position. In this procedure, a genetic algorithm determines the microscope vision parameters by means of laser line imaging. Also, the approximation networks compute the three-dimensional vision by means of the laser line position. Additionally, the soft computing algorithms re-calibrate the vision parameters when the microscope vision system is modified during the vision task. The proposed self-calibration improves accuracy of the traditional microscope calibration, which is accomplished via external references to the microscope system. The capability of the self-calibration based on soft computing algorithms is determined by means of the calibration accuracy and the micro-scale measurement error. This contribution is corroborated by an evaluation based on the accuracy of the traditional microscope calibration.

Approximation of conic section by quartic Bézier curve with endpoints continuity condition

A new method for approximation of conic section by quartic Bezier curve is presented, based on the quartic Bezier approximation of circular arcs. Here we give an upper bound of the Hausdorff distance between the conic section and the approximation curve, and show that the error bounds have the approximation order of eight. Furthermore, our method yields quartic G 2 continuous spline approximation of conic section when using the subdivision scheme, and the effectiveness of this method is demonstrated by some numerical examples.

A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves

A new approximation method for conic section by quartic Bezier curves is proposed. This method is based on the quartic Bezier approximation of circular arcs. We give the upper bound of Hausdorff distance between the conic section and the quartic Bezier curve, and also show that the approximation order is eight. And we prove that our approximation method has a smaller upper bound than previous quartic Bezier approximation methods. A quartic G2-continuous spline approximation of conic sections is obtained by using the subdivision scheme at the shoulder point of the conic section.

B-spline Collocation with Domain Decomposition Method

A global B-spline collocation method has been previously developed and successfully implemented by the present authors for solving elliptic partial differential equations in arbitrary complex domains. However, the global B-spline approximation, which is simply reduced to Bezier approximation of any degree p with C0 continuity, has led to the use of B-spline basis of high order in order to achieve high accuracy. The need for B-spline bases of high order in the global method would be more prominent in domains of large dimension. For the increased collocation points, it may also lead to the ill-conditioning problem. In this study, overlapping domain decomposition of multiplicative Schwarz algorithm is combined with the global method. Our objective is two-fold that improving the accuracy with the combination technique, and also investigating influence of the combination technique to the employed B-spline basis orders with respect to the obtained accuracy. It was shown that the combination method produced higher accuracy with the B-spline basis of much lower order than that needed in implementation of the initial method. Hence, the approximation stability of the B-spline collocation method was also increased.

MRI-Based CFD Analysis of Flow in a Human Left Ventricle: Methodology and Application to a Healthy Heart

A three-dimensional computational fluid dynamics (CFD) method has been developed to simulate the flow in a pumping left ventricle. The proposed method uses magnetic resonance imaging (MRI) technology to provide a patient specific, time dependent geometry of the ventricle to be simulated. Standard clinical imaging procedures were used in this study. A two-dimensional time-dependent orifice representation of the heart valves was used. The location and size of the valves is estimated based on additional long axis images through the valves. A semi-automatic grid generator was created to generate the calculation grid. Since the time resolution of the MR scans does not fit the requirements of the CFD calculations a third order bezier approximation scheme was developed to realize a smooth wall boundary and grid movement. The calculation was performed by a Navier-Stokes solver using the arbitrary Lagrange-Euler (ALE) formulation. Results show that during diastole, blood flow through the mitral valve forms an asymmetric jet, leading to an asymmetric development of the initial vortex ring. These flow features are in reasonable agreement with in vivo measurements but also show an extremely high sensitivity to the boundary conditions imposed at the inflow. Changes in the atrial representation severely alter the resulting flow field. These shortcomings will have to be addressed in further studies, possibly by inclusion of the real atrial geometry, and imply additional requirements for the clinical imaging processes.

Shape detection using light line and Bezier approximation network

A technique for measuring the object shape is presented. In this technique, the object is scanned by means of a light line. From the scanning, a set of images is captured by a CCD camera. The object surface is recovered by processing these images. To determine the surface dimensions, a Bezier approximation network is applied. This approximation network is constructed using information on images of a light line projected onto the objects, whose dimensions are known. The image data are extracted by applying the Gaussian approximation method. Using the approximation network in this technique, the surface measurement is determined by image processing, and the parameters of the set-up are avoided. In this manner, the accuracy of the techniques of light line projection for shape detection is improved, because errors in the parameters of the set-up are not introduced into the system. This technique has great potential, because it is a very simple experimental set-up and inexpensive. The experimental results obtained by this technique are verified with a contact method.

Binocular imaging of a laser stripe and approximation networks for shape detection

AbstractA technique for shape detection based on binocular imaging of a laser stripe is presented. In this technique, the object shape is recovered by means of laser stripe projection and binocular imaging. The approach of the binocular imaging in this technique is to avoid stripe occlusions, which appear due to the variation to the object surface. Based on the behavior of the stripe displacement, the object topography is computed by a Bezier approximation network. By means of this network, the measurements of the binocular geometry are avoided. The parameters of the binocular imaging are computed based on the Bezier approximation network. To reconstruct the topography, the object is scanned by a laser stripe. From the scanning, a set of binocular images of the stripe are processed to compute the object dimensions by means of the network. By applying Bezier approximation networks, the performance of the binocular imaging and the accuracy are improved. It is because the errors of the measurement are not added to the computational procedure, which performs the shape reconstruction. This procedure represents a contribution for the stripe projection methods and the binocular imaging. To describe the accuracy a root mean square of error is calculated. This technique is tested with real objects and its experimental results are presented. Also, the time processing is described. © 2007 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 17, 62–74, 2007

Trajectory Extrapolation for Mobile Robots Using Dual-Rate High-Order Holds

This paper presents Dual-Rate High-Order Holds (DR-HOH) based on primitive functions, where conventional DR-HOH, such as DR-ZOH, DR-FOH and DR-SOH, are obtained using a zero, first and second order interpolation functions, respectively. Similarly, holds based on Bezier approximation functions are also obtained. They are used for signal estimation at high-frequency sampling rate from low-frequency input values. DR-HOH are validated through simulations by applying them to trajectory extrapolation in mobile robots. Copyright This paper presents Dual-Rate High-Order Holds (DR-HOH) based on primitive functions, where conventional DR-HOH, such as DR-ZOH, DR-FOH and DR-SOH, are obtained using a zero, first and second order interpolation functions, respectively. Similarly, holds based on Bezier approximation functions are also obtained. They are used for signal estimation at high-frequency sampling rate from low-frequency input values. DR-HOH are validated through simulations by applying them to trajectory extrapolation in mobile robots.

Optimal control of hybrid systems in manufacturing

Hybrid systems combine time-driven and event-driven dynamics. This is a natural framework for manufacturing processes: The physical characteristics of production parts undergo changes at various operations described by time-driven models, while the timing control of operations is described by event-driven models. Accordingly, in the framework we propose, manufactured parts are characterized by physical states (e.g. temperature, geometry) subject to time-driven dynamics and by temporal states (e.g., operation start and stop times) subject to event-driven dynamics. We first provide a tutorial introduction to this hybrid system framework and associated optimal control problems through a single-stage manufacturing process model. We then show how the structure of the problem can be exploited to decompose what is a hard nonsmooth, nonconvex optimization problem into a collection of simpler problems. Next, we present extensions to multistage manufacturing processes for which we develop solution algorithms that make use of Bezier approximation techniques. Emphasis is given to the issue of deriving solutions through efficient algorithms, and some explicit numerical results are included.

Shape description using cubic polynomial Bezier curves

In this paper we present a new method for shape description consisting of an approximation of a shape by a variable number of Bezier curve segments. In our method we can control the accuracy of the Bezier approximation by a parameter thus controlling the complexity and resolution of the approximation process. The technique described here is suited to a variety of shape-based image retrieval applications and matching processes.

Error analysis of the Bezier approximation of algebraic surfaces.

For computer-aided geometric design or for computer-aided geometric representation, parametric curve segments and parametric surface patches such as the Bezier form or B-spline form have been frequently used. It is very convenient if we can use the same parametric representations for the approximation of algebraic curves or surfaces, such as circles, ellipses, spheres and tori. Then we can perform unified treatment for all free-form and nonfree-form curves and surfaces. For the design, machining and inspection of accurate geometry such as precision machine parts, it is very important to know the qualitative and quantitative error of approximation from exact geometry. But up to the present, little work has been done on this error. In this report, we treated the error of approximation and examined analytically and numerically the positions of control points and the magnitudes of the errors to approximate optimally algebraic curves and surfaces by cubic Bezier curves and by bicubic Bezier surfaces.

Approximate coding of digital contours

Two methods are proposed for coding the discrete contour of binary images. A set of key pixels (guiding pixels) on the contour is defined for this purpose. The decoding schemes approximate the contour through the key pixels using the quadratic Bezier approximation technique. The amount of deviation of the decoded image from the original image is studied using the objective measures of percentage error and shape compactness. The decoded images are found to be faithful reproductions of the original image. A set of cleaning operations is also introduced as an intermediate step before final reproduction. It was found that the bit requirements and the compression ratios are also improved significantly as compared to the contour run length coding and discrete line segment coding techniques. >

Interactive curve drawing by segmented Bezier approximation with a control parameter

The Bezier approximation technique is considered for curve drawing and curve fitting in computer graphics. Segmented Bezier approximation with a control parameter is proposed for fast and efficient curve approximation. The control parameter is very useful in interactive graphics. Interesting properties of the proposed technique are proved and experimental results are presented. The results are compared with those using the B-spline approximation technique.