A Novel Contour Tracing Algorithm for Object Shape Reconstruction Using Parametric Curves

Parametric curves such as Bézier and B-splines, originally developedfor the design of automobile bodies, are now also used in image processing andcomputer vision. For example, reconstructing an object shape in an image,including different translations, scales, and orientations, can be performedusing these parametric curves. For this, Bézier and B-spline curves can be generatedusing a point set that belongs to the outer boundary of the object. Theresulting object shape can be used in computer vision fields, such as searchingand segmentation methods and training machine learning algorithms. Theprerequisite for reconstructing the shape with parametric curves is to obtainsequentially the points in the point set. In this study, a novel algorithm hasbeen developed that sequentially obtains the pixel locations constituting theouter boundary of the object. The proposed algorithm, unlike the methods inthe literature, is implemented using a filter containing weights and an outercircle surrounding the object. In a binary format image, the starting point ofthe tracing is determined using the outer circle, and the next tracing movementand the pixel to be labeled as the boundary point is found by the filter weights.Then, control points that define the curve shape are selected by reducing thenumber of sequential points. Thus, the Bézier and B-spline curve equationsdescribing the shape are obtained using these points. In addition, differenttranslations, scales, and rotations of the object shape are easily provided bychanging the positions of the control points. It has also been shown that themissing part of the object can be completed thanks to the parametric curves.

Obstacle Avoidance and Multitarget Tracking of a Super Redundant Modular Manipulator Based on Bezier Curve and Particle Swarm Optimization

A super redundant serpentine manipulator has slender structure and multiple degrees of freedom.It can travel through narrow spaces and move in complex spaces.This manipulator is composed of many modules that can form different lengths of robot arms for different application sites.The increase in degrees of freedom causes the inverse kinematics of redundant manipulator to be typical and immensely increases the calculation load in the joint space.This paper presents an integrated optimization method to solve the path planning for obstacle avoidance and discrete trajectory tracking of a super redundant manipulator.In this integrated optimization,path planning is established on a Bezier curve,and particle swarm optimization is adopted to adjust the control points of the Bezier curve with the kinematic constraints of manipulator.A feasible obstacle avoidance path is obtained along with a discrete trajectory tracking by using a follow-the-leader strategy.The relative distance between each two discrete path points is limited to reduce the fitting error of the connecting rigid links to the smooth curve.Simulation results show that this integrated optimization method can rapidly search for the appropriate trajectory to guide the manipulator in obtaining the target while achieving obstacle avoidance and meeting joint constraints.The proposed algorithm is suitable for 3D space obstacle avoidance and multitarget path tracking.

Efficient Construction of B-Spline Curves with Minimal Internal Energy

In this paper,we propose an efficient method to construct energy-minimizing B-spline curves by using discrete mask method.The linear relations between control points are firstly derived for different energy-minimization problems,then the construction of B-spline curve with minimal internal energy can be addressed by solving a sparse linear system.The existence and uniqueness of the solution for the linear system are also proved.Experimental results show the efficiency of the proposed approach,and its application in 1 G blending curve construction is also presented.

Approximate merging of B-spline curves and surfaces

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.

One Fairing Method of Cubic B-spline Curves Based on Weighted Progressive Iterative Approximation

A new method to the problem of fairing planar cubic B-spline curves is introduced in this paper. The method is based on weighted progressive iterative approximation （WPIA for short） and consists of following steps： finding the bad point which needs to fair, deleting the bad point, re-inserting a new data point to keep the structm-e of the curve and applying WPIA method with the new set of the data points to obtain the faired curve. The new set of the data points is formed by the rest of the original data points and the new inserted point. The method can be used for shape design and data processing. Numerical examples are provided to demonstrate the effectiveness of the method.

A transition method based on Bezier curve for trajectory planning in cartesian space

In order to smooth the trajectory of a robot and reduce dwell time,a transition curve is introduced between two adjacent curves in three-dimensional space.G2 continuity is guaranteed to transit smoothly.To minimize the amount of calculation,cubic and quartic Bezier curves are both analyzed.Furthermore,the contour curve is characterized by a transition parameter which defines the distance to the corner of the deviation.How to define the transition points for different curves is presented.A general move command interface is defined for receiving the curve limitations and transition parameters.Then,how to calculate the control points of the cubic and quartic Bezier curves is analyzed and given.Different situations are discussed separately,including transition between two lines,transition between a line and a circle,and transition between two circles.Finally,the experiments are carried out on a six degree of freedom(DOF) industrial robot to validate the proposed method.Results of single transition and multiple transitions are presented.The trajectories in the joint space are also analyzed.The results indicate that the method achieves G2 continuity within the transition constraint and has good efficiency and adaptability.

Graphics Evolutionary Computations in Higher Order Parametric Bezier Curves

This work demonstrates in practical terms the evolutionary concepts and computational applications of Parametric Curves.Specific cases were drawn from higher order parametric Bezier curves of degrees 2 and above.Bezier curves find real life applications in diverse areas of Engineering and Computer Science,such as computer graphics,robotics,animations,virtual reality,among others.Some of the evolutionary issues explored in this work are in the areas of parametric equations derivations,proof of related theorems,first and second order calculus related computations,among others.A Practical case is demonstrated using a graphical design,physical hand sketching,and programmatic implementation of two opposite-faced handless cups,all evolved using quadratic Bezier curves.The actual drawing was realized using web graphics canvas programming based on HTML 5 and JavaScript.This work will no doubt find relevance in computational researches in the areas of graphics,web programming,automated theorem proofs,robotic motions,among others.

Shape Control and Modification of Rational Bezier Curve and Surface

A Bezier interpolation approach is proposed which uses local generation of endpoint slopes and forces the curve and the surface to pass through an arbitrarily specified point to control and modify the shape of curve and surface, making the result satisfactory.

RECONSTRUCTION OF SYMMETRIC B-SPLINE CURVES AND SURFACES

A method to reconstruct symmetric B-spline curves and surfaces is presented. The symmetry property is realized by using symmetric knot vector and symmetric control points. Firstly, data points are divided into two parts based on the symmetry axis or symmetry plane extracted from data points. Then the divided data points are parameterized and a symmetric knot vector is selected in order to get symmetric B-spline basis functions. Constraint equations regarding the control points are deduced to keep the control points of the B-spline curve or surface to be symmetric with respect to the extracted symmetry axis or symmetry plane. Lastly, the constrained least squares fitting problem is solved with the Lagrange multiplier method. Two examples from industry are given to show that the proposed method is efficient, robust and able to meet the general engineering requirements.

Development of Cubic Bezier Curve and Curve-Plane Intersection Method for Parametric Submarine Hull Form Design to Optimize Hull Resistance Using CFD

优化分析和计算液体动力学(CFD ) 同时被使用了，在哪个一个参量的模型在发现最佳的答案起一个重要作用。然而，与不规则的曲线为复杂形状创造一个参量的模型是困难的，例如一种海底的壳形式。在这研究，立方的 Bezier 曲线和曲线飞机交叉方法被用来产生考虑三个输入参数的一种参量的海底的壳形式的一个稳固的模型：鼻子半径，尾巴半径，和长度高度壳比率(L/H ) 。应用程序接口(API ) 脚本也被用来在 ANSYS 设计 modeler 写代码。结果证明海底的形状能与输入参数的某变化被产生。一个例子被给那显示出建议方法怎么能成功地被用于一个壳抵抗优化盒子。中间的海底的类型的参量的设计被选择被修改。首先，预先，原来的海底的模型用 CFD 被分析。然后，使用反应表面图，某候选人有一个最小的壳抵抗系数的最佳的图案被获得。进一步，在目标驱动的优化(GDO ) 的优化方法被实现与最小的壳抵抗系数发现海底的壳形式(C t ) 。最小的 C t 被获得。在在起始的潜水艇和最佳潜水艇之间的 C t 价值的计算差别在 0.26% 附近，与起始的潜水艇和是的最佳潜水艇的 C t 0.001 508 26 和 0.001 504 29 分别地。结果证明最佳潜水艇壳形式显示出更高的鼻子半径(r n ) 和更高的 L/H 起始的潜水艇比那些塑造，当时尾巴的半径(r t ) 比起始的形状的小。

THE RELATIONSHIP BETWEEN PROJECTIVE GEOMETRIC AND RATIONAL QUADRATIC B-SPLINE CURVES

Abstract For two rational quadratic B spline curves with same control vertexes, the cross ratio of four collinear points are represented: which are any one of the vertexes, and the two points that the ray initialing from the vertex intersects with the corresponding segments of the two curves, and the point the ray intersecting with the connecting line between the two neighboring vertexes. Different from rational quadratic Bézier curves, the value is generally related with the location of the ray, and the necessary and sufficient condition of the ratio being independent of the ray's location is showed. Also another cross ratio of the following four collinear points are suggested, i.e. one vertex, the points that the ray from the initial vertex intersects respectively with the curve segment, the line connecting the segments end points, and the line connecting the two neighboring vertexes. This cross ratio is concerned only with the ray's location, but not with the weights of the curve. Furthermore, the cross ratio is projective invariant under the projective transformation between the two segments.

Judging or setting weight steady-state of rational Bézier curves and surfaces

Many works have investigated the problem of reparameterizing rational B^zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.

Constrained multi-degree reduction of rational Bézier curves using reparameterization

Applying homogeneous coordinates, we extend a newly appeared algorithm of best constrained multi-degree reduc- tion for polynomial Bézier curves to the algorithms of constrained multi-degree reduction for rational Bézier curves. The idea is introducing two criteria, variance criterion and ratio criterion, for reparameterization of rational Bézier curves, which are used to make uniform the weights of the rational Bézier curves as accordant as possible, and then do multi-degree reduction for each component in homogeneous coordinates. Compared with the two traditional algorithms of "cancelling the best linear common divisor" and "shifted Chebyshev polynomial", the two new algorithms presented here using reparameterization have advantages of simplicity and fast computing, being able to preserve high degrees continuity at the end points of the curves, do multi-degree reduction at one time, and have good approximating effect.

Degree elevation from Bzier curve to C-Bzier curve with corner cutting form

The existing results of curve degree elevation mainly focus on the degree of algebraic polynomials. The paper considers the elevation of degree of the trigonometric polynomial, from a Bezier curve on the algebraic polynomial space, to a C-Bezier curve on the algebraic and trigonometric polynomial space. The matrix of degree elevation is obtained by an operator presentation and a derivation pyramid. It possesses not a recursive presentation but a direct expression. The degree elevation process can also be represented as a corner cutting form.

Direction monotonicity for a rational Bézier curve

The monotonicity of a rational Bezier curve, usually related to an explicit function, is determined by the used coordinate system. However, the shape of the curve is independent of the coordinate system. To meet the affine invariant property, a kind of generalized mono- tonicity, called direction monotonicity, is introduced for rational Bezier curves. The direction monotonicity is applied to both planar and space curves and to both Cartesian and affine co- ordinate systems, and it includes the traditional monotonicity as a subcase. By means of it, proper affine coordinate systems may be chosen to make some rational Bezier curves monotonic. Direction monotonic interpolation may be realized for some of the traditionally nonmonotonic data as well.

Algorithms and Identities for(q, h)-Bernstein Polynomials and(q, h)-Bézier Curves–A Non-Blossoming Approach

We establish several fundamental identities, including recurrence relations, degree elevation formulas, partition of unity and Marsden identity, for quantum Bernstein bases and quantum Bezier curves. We also develop two term recurrence relations for quantum Bernstein bases and recursive evaluation algorithms for quantum Bezier curves. Our proofs use standard mathematical induction and other elementary techniques.

Re-parameterization of Bézier Curves by Square Approximation with Endpoint Constraints

In order to get an approximation with better effect of pararneterization of Bezier curves, we proposed a method for arc-length parameterization and the corresponding algorithms by square approximation for the discrete even de-parameterization of the curves. This method is simple and easy to implement, and the property of the approximation has no change compared with the original curve. A quantitative criterion for estimating the effect of parameterization is also built to quantitatively characterize the parameterization effect of the algorithms. As a result, the nearly arc-length parameterized curve has a smaller relative deviation using either the algorithm with point constraint at endpoints or the algorithm with point constraint plus the first derivative constraint at endpoints. Experiments show that after re-parameterization with our algorithms, the relative deviation will have at least a 20% reduction.