Bézier Method For Image Processing

This project concerns about Bézier method in Computer Aided GeometricDesign (CAGD) involving Bézier Curve and Bézier Surface which widely related to the other theorem and method. The aim of this project is to introduce the basic of Bézier method and then generate the Bézier curves, Bézier surfaces, theory and properties and develop Bézier method in image processing application specifically image compression by using MATLAB

A Robust Approach to Find the Control Points for Wide Variety of 3rd Order Bzier Curves

This paper represents a new approach that can recover the control points for wide variety of 3rd order BE9;zier curves. In this regards, the two stage approximation learning algorithm is adopted with some modifications. At 1st stage our key feature is segmentation of the curve which can determine intermediate points of the wide variety of curves. In this respect, an efficient recursive algorithm is used to find out the height of the curve (h) with less iteration. The proposed approach introduced horizontal segmentation rather than vertical segmentation. Different height (H), where the 2nd and 3rd control point are assumed, and also the step-size (2202;), at which the control points are moved toward the actual direction, are used to find out the exact location of the control points. Experimental results demonstrate that our proposing method can recover control points for wide variety of curves with minimum error level and less iteration. Wide variety of curve shapes are used to test the proposing approach and results are presented to prove its effectivenes

Visualization Of Curve And Surface Data Using Rational Cubic Ball Functions

This study considered the problem of shape preserving interpolation through regular data using rational cubic Ball which is an alternative scheme for rational Bézier functions. A rational Ball function with shape parameters is easy to implement because of its less degree terms at the end polynomial compared to rational Bézier functions. In order to understand the behavior of shape parameters (weights), we need to discuss shape control analysis which can be used to modify the shape of a curve, locally and globally. This issue has been discovered and brought to the study of conversion between Ball and Bézier curve

A global search algorithm for phase transition pathways in computer-aided nano-design

One of the most important design issues for phase change materials is to engineer the phase transition process. The challenge of accurately predicting a phase transition is estimating the true value of transition rate, which is determined by the saddle point with the minimum energy barrier between stable states on the potential energy surface (PES). In this thesis, a new algorithm for searching the minimum energy path (MEP) is presented. The new algorithm is able to locate both the saddle point and local minima simultaneously. Therefore no prior knowledge of the precise positions for the reactant and product on the PES is needed. Unlike existing pathway search methods, the algorithm is able to search multiple transition paths on the PES simultaneously, which gives us a more comprehensive view of the energy landscape than searching individual ones. In this method, a Bézier curve is used to represent each transition path. During the searching process, the reactant and product states are located by minimizing the two end control points of the curve, while the shape of the transition pathway is refined by moving the intermediate control points of the curve in the conjugate directions. A curve subdivision scheme is developed so that multiple transitions paths can be located. The algorithm is demonstrated by examples of LEPS potential, LEPS plus harmonic oscillator potential, and PESs defined by Rastrigin function and Schwefel function.M.S

Subdivision and manifold techniques for isogeometric design and analysis of surfaces

Design of surfaces and analysis of partial differential equations defined on them are of great importance in engineering applications, e.g., structural engineering, automotive and aerospace. This thesis focuses on isogeometric design and analysis of surfaces, which aims to integrate engineering design and analysis by using the same representation for both. The unresolved challenge is to develop a desirable surface representation that simultaneously satisfies certain favourable properties on meshes of arbitrary topology around the extraordinary vertices (EVs), i.e., vertices not shared by four quadrilaterals or three triangles. These properties include high continuity (geometric or parametric), optimal convergence in finite element analysis as well as simplicity in terms of implementation. To overcome the challenge, we further develop subdivision and manifold surface modelling techniques, and explore a possible scheme to combine the distinct appealing properties of the two. The unique advantages of the developed techniques have been confirmed with numerical experiments. Subdivision surfaces generate smooth surfaces from coarse control meshes of arbitrary topology by recursive refinement. Around EVs the optimal refinement weights are application-dependent. We first review subdivision-based finite elements. We then proceed to derive the optimal subdivision weights that minimise finite element errors and can be easily incorporated into existing implementations of subdivision schemes to achieve the same accuracy with much coarser meshes in engineering computations. To this end, the eigenstructure of the subdivision matrix is extensively used and a novel local shape decomposition approach is proposed to choose the optimal weights for each EV independently. Manifold-based basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. This thesis derives novel manifold-based basis functions with arbitrary prescribed smoothness using quasi-conformal maps, enabling us to model and analyse surfaces with sharp features, such as creases and corners. Their practical utility in finite element simulation of hinged or rigidly joined structures is demonstrated with Kirchhoff-Love thin shell examples. We also propose a particular manifold basis reproducing subdivision surfaces away from EVs, i.e., B-splines, providing a way to combine the appealing properties of subdivision (available in industrial software) for design and manifold basis (relatively new) for analysis.Cambridge International Scholarship Scheme (CISS) by Cambridge Trus

Dynamic Bezier curves for variable rate-distortion

Bezier curves (BC) are important tools in a wide range of diverse and challenging applications, from computer-aided design to generic object shape descriptors. A major constraint of the classical BC is that only global information concerning control points (CP) is considered, consequently there may be a sizeable gap between the BC and its control polygon (CtrlPoly), leading to a large distortion in shape representation. While BC variants like degree elevation, composite BC and refinement and subdivision narrow this gap, they increase the number of CP and thereby both the required bit-rate and computational complexity. In addition, while quasi-Bezier curves (QBC) close the gap without increasing the number of CP, they reduce the underlying distortion by only a fixed amount. This paper presents a novel contribution to BC theory, with the introduction of a dynamic Bezier curve (DBC) model, which embeds variable localised CP information into the inherently global Bezier framework, by strategically moving BC points towards the CtrlPoly. A shifting parameter (SP) is defined that enables curves lying within the region between the BC and CtrlPoly to be generated, with no commensurate increase in CP. DBC provides a flexible rate-distortion (RD) criterion for shape coding applications, with a theoretical model for determining the optimal SP value for any admissible distortion being formulated. Crucially DBC retains core properties of the classical BC, including the convex hull and affine invariance, and can be seamlessly integrated into both the vertex-based shape coding and shape descriptor frameworks to improve their RD performance. DBC has been empirically tested upon a number of natural and synthetically shaped objects, with qualitative and quantitative results confirming its consistently superior shape approximation performance, compared with the classical BC, QBC and other established BC-based shape descriptor techniques

Bezier Curve Interpolation On Road Design

This research focuses on reconstructing the road curve by using Bezier curve fitting on a map. The usual way of constructing the Bezier curve is by using control points which can be very tedious. Since Bezier curves do not interpolate the control points, designers need to estimates the position of control points so that the curve fits well. In order to ease up the process, we will construct the Bezier curve by using the parameterization method where the data point information is required instead of the usual way of using control points. However, this method does not work on Bezier curve of high degree as the curves tend to become perturbed. One way of solving the problem is by using piecewise Bezier curve made up of several parameterized Bezier curves of lower degree. We propose a method to satisfy the continuity properties along this piecewise parameterized Bezier curve. The method had been implemented on two-dimensional model and spatial model. By using this method, we manage to construct a Bezier curve that can interpolate high number of data points while satisfying the continuity properties along the curve