Conic Spline Research Articles

A new approach to adaptive spline threshold autoregression by using Tikhonov regularization and continuous optimization

In this study adaptive spline threshold autoregression and conic quadratic programming is used to develope conic adaptive spline threshold autoregression. With the introduced approach the second stepwise algorithm of adaptive spline threshold autoregression model turned to the Tikhonov regularization problem which was transformed into conic quadratic programming problem. The aim is to attain an optimum solution chosen in many solutions obtained by determining the bounds of the optimization problem using multiobjective optimization approach. Furthermore, in application part we used two different data set to compare performances of linear regression, adaptive spline threshold autoregression and conic adaptive spline threshold autoregression approaches.

Spline regression models for complex multi-modal regulatory networks

Complex regulatory networks often have to be further expanded and improved with regard to the unknown effects of additional parameters and factors that can emit a disturbing influence on the key variables under consideration. The concept of target-environment (TE) networks provides a holistic framework for the analysis of such parameter-dependent multi-modal systems. In this study, we consider time-discrete TE regulatory systems with spline entries. We introduce a new regression model for these particular two-modal systems that allows us to determine the unknown system parameters by applying the multivariate adaptive regression spline (MARS) technique and the newly developed conic multivariate adaptive regression spline (CMARS) method. We obtain a relaxation by means of continuous optimization, especially, conic quadratic programming (CQP) that could be conducted by interior point methods. Finally, a numerical example demonstrates the efficiency of the spline-based approach.

Intelligent Approaches for Vectorizing Image Outlines

Two computing approaches, based on linear and conic splines, are proposed here in reviewed and extended for vectorizing image outlines. Both of the approaches have various phases including extracting outlines of images, detecting corner points from the detected outlines, and curve fitting. Interpolation splines are the bases of the two approached. Linear spline approach is straight forward as it does not have a degree of freedom.in terms of some shape controller in its description. However, the idea of the soft computing approach, namely simulated annealing, has been utilized for conic splines. This idea has been incorporated to optimize the shape parameters in the description of the generalized conic spline. Both of the linear and conic approaches ultimately produce optimal results for the approximate vectorization of the digital contours obtained from the generic shapes.

Two Approaches for Vectorizing Image Outlines

Two approaches, based on linear and conic splines, are proposed here for vectorizing image outlines. Both of the approaches have various phases including extracting outlines of images, detecting corner points from the detected outlines, and curve fitting. Interpolation splines are the bases of the two approached. Linear spline approach is straight forward as it does not have a degree of freedom.in terms of some shape controller in its description. However, the idea of the soft computing approach, namely simulated annealing, has been utilized for conic splines. This idea has been incorporated to optimize the shape parameters in the description of the generalized conic spline. Both of the linear and conic approaches ultimately produce optimal results for the approximate vectorization of the digital contours obtained from the generic shapes. Demonstrations and a comparative study also make the essential parts of the paper.

Approximation by Conic Splines

We show that the complexity of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature to within Hausdorff distance ɛ is c1ɛ−1/4 + O(1), if the spline consists of parabolic arcs, and c2ɛ−1/5 + O(1), if it is composed of general conic arcs of varying type. The constants c1 and c2 are expressed in the Euclidean and affine curvature of the curve. We also show that the Hausdorff distance between a curve and an optimal conic arc tangent at its endpoints is increasing with its arc length, provided the affine curvature along the arc is monotone. This property yields a simple bisection algorithm for the computation of an optimal parabolic or conic spline.

Cone spline approximation via fat conic spline fitting

Fat conic section and fat conic spline are defined. With well established properties of fat conic splines, the problem of approximating a ruled surface by a tangent smooth cone spline can then be changed as the problem of fitting a plane fat curve by a fat conic spline. Moreover, the fitting error between the ruled surface and the cone spline can be estimated explicitly via fat conic spline fitting. An efficient fitting algorithm is also proposed for fat conic spline fitting with controllable tolerances. Several examples about approximation of general developable surfaces or other types of ruled surfaces by cone spline surfaces are presented.

Helix approximations with conic and quadratic Bézier curves

In this paper we present the error analysis for the approximation of a cylindrical helix by conic and quadratic Bézier curves. The approximation method yields G 1 conic spline and G 1 quadratic spline, respectively. We give a sharp upper bound of the Hausdorff distance between the helix and each approximation curve. We also show that the error bound has the approximation order three and monotone increases as the angle subtended to helix increases. Furthermore, using the error bound analysis for the helix approximation by conic and quadratic Bézier curves, we present the error bounds for the torus-like helicoid approximations by quadric surfaces and quadratic Bézier tensor product surfaces.

Curve fitting and fairing using conic splines

We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bézier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bézier spline curve. For parts of the G 1 conic spline without an inflection, we can obtain a curvature continuous conic spline by adjusting the tangent direction at the joint point and scaling the weights for every two adjacent rational Bézier curves. The unwanted curvature extrema within conic segments or at some joint points can be removed efficiently by scaling the weights of the conic segments or moving the joint points along the normal direction of the curve at the point. In the end, a fair conic spline curve is obtained that is G 2 continuous at convex or concave parts and G 1 continuous at inflection points. The main advantages of the method lies in two aspects, one advantage is that we can construct a curvature continuous conic spline by a local algorithm, the other one is that the curvature plot of the conic spline can be controlled efficiently. The method can be used in the field where fair shape is desired by interpolating or approximating a given point set. Numerical examples from simulated and real data are presented to show the efficiency of the new method.

Function plotting using conic splines

A method is presented whereby, given a mathematical description of a function, a conic spline approximating the plot of the function is produced. Conic arcs were selected as the primitive curves because there are simple incremental plotting algorithms for conics already included in some device drivers, and there are simple algorithms for local approximations by conics. A split-and-merge algorithm for choosing the knots adaptively, according to shape analysis of the original function based on its first-order derivatives, is introduced. >

Pixel-planes 5: a heterogeneous multiprocessor graphics system using processor-enhanced memories

This paper introduces the architecture and initial algorithms for Pixel-Planes 5, a heterogeneous multi-computer designed both for high-speed polygon and sphere rendering (1M Phong-shaded triangles/second) and for supporting algorithm and application research in interactive 3D graphics. Techniques are described for volume rendering at multiple frames per second, font generation directly from conic spline descriptions, and rapid calculation of radiosity form-factors. The hardware consists of up to 32 math-oriented processors, up to 16 rendering units, and a conventional 1280 × 1024-pixel frame buffer, interconnected by a 5 gigabit ring network. Each rendering unit consists of a 128 × 128-pixel array of processors-with-memory with parallel quadratic expression evaluation for every pixel. Implemented on 1.6 micron CMOS chips designed to run at 40MHz, this array has 208 bits/pixel on-chip and is connected to a video RAM memory system that provides 4,096 bits of off-chip memory. Rendering units can be independently reasigned to any part of the screen or to non-screen-oriented computation. As of April 1989, both hardware and software are still under construction, with initial system operation scheduled for fall 1989.