Parametric Spline Curves Research Articles

COMPUTER REALIZATIONS OF THE CUBIC PARAMETRIC SPLINE CURVE OF BÈZIER TYPE

This paper presents a new method for constructing a third degree parametric spline curve of C1 continuity. Like the Bèzier curve, the proposed curve is constructed and operated by control points. The peculiarity of the proposed algorithm is the assignment of some unknown values of the spline in the control points abscissas, which are based on the conditions of the first derivative continuity of the curve at these points. The position of the touch points, as well as the control points, can be set interactively. Changing of these points positions leads to a change in the curve shape. This allows the user to flexibly adjust the shape of the curve. Systems of algebraic equations with tridiagonal matrix for calculating the coefficients of a spline curve are constructed. Conditions for the existence and uniqueness of such a curve are presented. Examples of the use of the proposed curve, in particular, for monotone data sets, approximation the ellipse and constructing the letter "S" are given.

A two-step method for interpolating interval data based on cubic hermite polynomial models

We propose a two-step modelling method to build a smoothing curve or surface representation model for interpolating given interval data. Due to the difficulties of building interpolation models with explicit expressions directly on given multivariable interval data in existing studies, the first step is about building a cubic Hermite spline model to construct a smoothing piecewise parametric curve of Ck(k=1,2) continuity to show the implicit relationship between variables. Thereinto, we formulate the Hermite spline model as a finite convex optimization problem. Then in the second step, sample points are taken from the optimal parametric spline curve in the first step as the interpolation points to establish the explicit expressions and illustrate the direct relationship between variables. Some numerical examples are provided to compare with other interpolation methods and show the feasibility and advantages of the proposed method.

Jump Motion Planning for Shrouded Blisks Multi-Axis EDM with Short Line Segments

As key components of aero and rocket engines, shrouded blisks are made of difficult-to-cut materials and have complex structures. Multi-axis electrical discharge machining (EDM) which involves rotary axes is an effective approach for machining shrouded blisks. Most EDM machines provide only linear or circular interpolations. To machine a workpiece with a complex structure, a large number of short line segments are used to represent a feeding path for EDM, which is expressed in terms of G01 codes in a machine coordinate system (MCS). Discontinuities along a feeding path consisting of short line segments can cause feedrate fluctuations in jump motions. To eliminate velocity discontinuities, smoothening a multi-axis jump path which involves rotary axes needs to be addressed. In this paper, a transition algorithm and a jump motion velocity planning for multi-axis EDM of shrouded blisks are proposed to smoothen multi-axis jump paths. Considering the differences between translational and rotational movements, one multi-axis jump path can be divided into two subpaths: a linear subpath and a rotary subpath. For each subpath, a parametric spline curve is used to smoothen adjacent short line segments. Depending on the computer aided design (CAD) models of both an electrode and a shrouded blisk, approximation error constraints for corner smoothening in the MCS can be converted into the approximation error constraints in the workpiece coordinate system (WCS). When designing a transition curve, discharge gap is used as the approximation error tolerance in WCS for jump motions, and the multi-axis jump path can be smoothened. Then, a jerk-limited S-shaped feedrate planning algorithm is used for jump motion planning. Simulation results show that the velocity profile of the proposed motion planning is smoother.

A look-ahead transition algorithm for jump motions with short line segments in EDM

Most electrical discharge machining (EDM) machines provide only linear or circular interpolations. To machine a workpiece with a complex geometry, a large number of line segments in terms of G01 codes are used to represent a feeding path in EDM. Periodic jump motions of an electrode are adopted such that the debris concentration in the discharge gap can be reduced and clean dielectric can enter into the discharge gap, thus increasing the effective discharge ratio and improving the machining stability. In the conventional way, the electrode will experience a deceleration to zero, and then an acceleration at junctions connecting two adjacent line segments. To solve this problem, a real-time look-ahead transition algorithm and a motion planning scheme for jump motions with short line segments are proposed. A parametric spline curve is used to blend adjacent short line segments. A transition curve can use the discharge gap as the approximation error tolerance for jump motions in EDM. Analytical calculations for the curvature extrema, real-time performance are also considered in the transition scheme. The motion planning with a look-ahead function for jump motions is developed to obtain a feedrate profile of jump motion. This motion planning scheme includes a path-smoothing, a bidirectional scanning, and a jerk-limited feedrate scheduling. Simulation results show that the velocity profile of the proposed motion planning is smoother. Machining tests for a workpiece with complex curves shows that the proposed method can reduce the machining time by 12.16 % as compared with the traditional method.

Landmark-Based Shape Encoding and Sparse-Dictionary Learning in the Continuous Domain.

We provide a generic framework to learn shape dictionaries of landmark-based curves that are defined in the continuous domain. We first present an unbiased alignment method that involves the construction of a mean shape as well as training sets whose elements are subspaces that contain all affine transformations of the training samples. The alignment relies on orthogonal projection operators that have a closed form. We then present algorithms to learn shape dictionaries according to the structure of the data that needs to be encoded: 1) projection-based functional principal-component analysis for homogeneous data and 2) continuous-domain sparse shape encoding to learn dictionaries that contain imbalanced data, outliers, or different types of shape structures. Through parametric spline curves, we provide a detailed and exact implementation of our method. We demonstrate that it requires fewer parameters than purely discrete methods and that it is computationally more efficient and accurate. We illustrate the use of our framework for dictionary learning of structures in biomedical images as well as for shape analysis in bioimaging.

Shape Projectors for Landmark-Based Spline Curves

We present a generic method to construct orthogonal projectors for two-dimensional landmark-based parametric spline curves. We construct vector spaces that define a geometric transformation (e.g., affine, similarity, and scaling) that is applied to a reference curve. These vector spaces contain all parametric curves up to the chosen transformation. We define the vector spaces implicitly through an orthogonal projection operator and present a theorem that characterizes the projector for landmark-based spline curves, which are popular for the user-interactive analysis of biomedical images. Finally, we show how shape priors are constructed with the spline projector and provide an example of application for the segmentation of microscopy images in biology.

Learning Convex Regularizers for Optimal Bayesian Denoising

We propose a data-driven algorithm for the Bayesian estimation of stochastic processes from noisy observations. The primary statistical properties of the sought signal are specified by the penalty function (i.e., negative logarithm of the prior probability density function). Our alternating direction method of multipliers (ADMM) based approach translates the estimation task into successive applications of the proximal mapping of the penalty function. Capitalizing on this direct link, we define the proximal operator as a parametric spline curve and optimize the spline coefficients by minimizing the average reconstruction error for a given training set. The key aspects of our learning method are that the associated penalty function is constrained to be convex and the convergence of the ADMM iterations is proven. As a result of these theoretical guarantees, adaptation of the proposed framework to different levels of measurement noise is extremely simple and does not require any retraining. We apply our method to estimation of both sparse and nonsparse models of Lévy processes for which the minimum mean square error (MMSE) estimators are available. We carry out a single training session for a fixed level of noise and perform comparisons at various signal-to-noise ratio values. Simulations illustrate that the performance of our algorithm are practically identical to the one of the MMSE estimator irrespective of the noise power.

An Evolution-Based Approach for Approximate Parameterization of Implicitly Defined Curves by Polynomial Parametric Spline Curves

We propose a novel approach for the approximate parameterization of an implicitly defined curve in the plane by polynomial parametric spline curves. The method generates the parameterization of the curve (which may consist of several open and closed branches) without using any a priori information about its topology. If needed the topology of the approximate parameterization can be certified against the initial curve in a simple way.

The transition algorithm based on parametric spline curve for high-speed machining of continuous short line segments

A numerical control (NC) program generated by a complex CAD/CAM system mainly composes a number of short line segments. For continuous short line segment machining, the traditional machining approach and the direct transitional approach cause the oscillations of the federate and affect machining efficiency. Moreover, the arc transitional approach enhances machining efficiency, but it brings about the greater machining error. To increase the smoothness of velocity profile and machining efficiency of continuous short line segment machining, this paper presents a parametrical cubic spline curve transitional approach with the S-shaped jerk-limited acceleration and deceleration with look-ahead. The trajectory error of the transitional curve is deduced according to the correlative factors, the preconditions of the transitional curve are put forward and analyzed, the mathematical model of the parametrical cubic spline curve is constructed, and the constraint conditions of the transforming velocity are built. In addition, the velocity control method is adopted for the smoother velocity profile. Thus, the machining productivity is dramatically increased and the velocity profile is more smoothness. The simulative and experimental results verify the feasibility and validity of the proposed approach.

Velocity Control Method for Continuous Short Line Segment High-Speed Machining Based on Transition of Parametric Spline

To enhance the velocity profile’s smoothness and machining efficiency of continuous short line segments, a velocity smooth control algorithm was presented based on the transition of cubic parametric spline curves. The transitional mathematical model was constructed, and an improved S-shaped jerk-limited acceleration/deceleration algorithm with look-ahead was adopted for smoother velocity profiles. The simulation and experiment results showed that machining efficiency was greatly improved, and smoother velocity profile was achieved, which approved the feasibility and validity of the proposed method.

Dual evolution of planar parametric spline curves and [formula omitted]-spline level sets

By simultaneously considering evolution processes for parametric spline curves and implicitly defined curves, we formulate the framework of dual evolution. This allows us to combine the advantages of both representations. On the one hand, the implicit representation is used to guide the topology of the parametric curve and to formulate additional constraints, such as range constraints. On the other hand, the parametric representation helps to detect and to eliminate unwanted branches of the implicitly defined curves. Moreover, it is required for many applications, e.g., in Computer Aided Design.

Chordal cubic spline interpolation is fourth-order accurate

It is well known that complete cubic spline interpolation of functions with four continuous derivatives is fourth-order accurate. In this paper we show that this kind of interpolation, when used to construct parametric spline curves through sequences of points in any space dimension, is again fourth-order accurate if the parameter intervals are chosen by chord length. We also show how such chordal spline interpolants can be used to approximate the arc-length derivatives of a curve and its length.

Threading splines through 3D channels

Given a polygonal channel between obstacles in the plane or in space, we present an algorithm for generating a parametric spline curve with few pieces that traverses the channel and stays inside. While the problem without emphasis on few pieces has trivial solutions, the problem for a limited budget of pieces represents a nonlinear and continuous (‘infinite’) feasibility problem. Using tight, two-sided, piecewise linear bounds on the potential solution curves, we reformulate the problem as a finite, linear feasibility problem whose solution, by standard linear programming techniques, is a solution of the channel-fitting problem. The algorithm allows the user to specify the degree and smoothness of the solution curve and to minimize an objective function, for example, to approximately minimize the curvature of the spline. We describe in detail how to formulate and solve the problem, as well as the problem of fitting parallel curves, for a spline in Bernstein-Bézier form.

Constructing parametric quadratic curves

Constructing a parametric spline curve to pass through a set of data points requires assigning a knot to each data point. In this paper we discuss the construction of parametric quadratic splines and present a method to assign knots to a set of planar data points. The assigned knots are invariant under affine transformations of the data points, and can be used to construct a parametric quadratic spline which reproduces parametric quadratic polynomials. Results of comparisons of the new method with several known methods are included.

COMPUTATIONAL METHODS FOR DISCRETE PARAMETRIC ℓ1AND ℓ∞CURVE FITTING

The paper is devoted to l1 and l∞ approximation with parametric spline curves. We discuss the questions of existence and uniqueness of solutions. With the help of a suitable linearization of the Euclidean norm, we derive a method for computing the approximating spline curves. The method uses linear and quadratic programming in order to find the solution.

Shape preserving least-squares approximation by polynomial parametric spline curves

This article presents a method for shape preserving least-squares approximation. This method generalizes an algorithm of Dierckx (1980, 1993) to the case of planar parametric curves. Using a reference curve we generate linear sufficient conditions for the convexity of the approximant. This leads us to a quadratic programming problem which can be solved exactly, e.g., with an active set strategy.

Chebyshev splines and shape parameters

A parametric spline curve is defined whose restriction to each sub-interval belongs to a 4-dimensional piecewise Chebyshev subspace depending on coefficients which play the role of shape parameters.

Axis location of a worn cylindrical mechanical object

In the refurbishing of mechanical parts, it is important to identify where the part is worn. If the perimeter of a crossection of a worn cylindrical object is approximated by a closed parametric spline curve, the radius and centre of curvature of the curve can be used to determine the original position of the axis of the object, i.e. its position in its unworn state, to a given tolerance. Such approximations by cubic and quintic B-splines with least-squares smoothing are discussed and compared.

Inflections on curves in two and three dimensions

It is shown that for parametric spline curves in two dimensions, such as B-spline curves, the number of inflections in the curve cannot exceed the number of inflections in its control polygon, provided that the control polygon does not turn too sharply. In order to consider shape properties of curves in three dimensions, we then define the inflection count to be the maximum number of inflections that the curve can appear to have when viewed from any direction. The inflection count of a curve is then classified in terms of the direction of its curvature vector. It is shown that the inflection count of a quadratic spline is equal to that of its control polygon and, with this motivation, we classify the inflection count of polygonal arcs.

Drawing antialiased cubic spline curves

Cubic spline curves have many nice properties that make them desirable for use in comptuer graphics, and the advantages of antialiasing have been known for some years. Yet, only recently has there been any attempt at directly antialiasing spline curves. Parametric spline curves have resisted antialiasing in several ways: single segments may cross or become tangent to themselves. Cusps and small loops are easily missed entirely. Thus, short pieces of the curve cannot necessarily be rendered in isolation. Finding the distance from a pixel center to the curve accurately and efficiently—usually an essential part of antialiasing—is an unsolved problem. The method presented by Lien, Shantz, and Pratt [21] is a good start, although it considers pixel-length pieces of the curve in isolation and lacks robustness in the handling of certain curves. This paper provides an improved method that is more robust, and is able to handle intersections and tangency.