Bezier Spline Curve Research Articles

Real-Time Lane Detection on Suburban Streets Using Visual Cue Integration

The detection of lane boundaries on suburban streets using images obtained from video constitutes a challenging task. This is mainly due to the difficulties associated with estimating the complex geometric structure of lane boundaries, the quality of lane markings as a result of wear, occlusions by traffic, and shadows caused by road-side trees and structures. Most of the existing techniques for lane boundary detection employ a single visual cue and will only work under certain conditions and where there are clear lane markings. Also, better results are achieved when there are no other on-road objects present. This paper extends our previous work and discusses a novel lane boundary detection algorithm specifically addressing the abovementioned issues through the integration of two visual cues. The first visual cue is based on stripe-like features found on lane lines extracted using a two-dimensional symmetric Gabor filter. The second visual cue is based on a texture characteristic determined using the entropy measure of the predefined neighbourhood around a lane boundary line. The visual cues are then integrated using a rule-based classifier which incorporates a modified sequential covering algorithm to improve robustness. To separate lane boundary lines from other similar features, a road mask is generated using road chromaticity values estimated from CIE L*a*b*colour transformation. Extraneous points around lane boundary lines are then removed by an outlier removal procedure based on studentized residuals. The lane boundary lines are then modelled with Bezier spline curves. To validate the algorithm, extensive experimental evaluation was carried out on suburban streets and the results are presented.

Interpolating Rational Bézier Spline Curves with Local Shape Control

The paper presents a technique for constructionof C n interpolating rational Bezier spline curves by means of blending rational quadric Bezier curves. A class of polynomials which satisfy special boundary conditions is used for blending. Properties of the polynomialsare considered. The constructed spline curves have local shape control that make them useful in such geometric applications a s real-time trajectory generation and fast curve sketching .

3D Reconstruction Using Cubic Bezier Spline Curves and Active Contours (Case Study)

Introduction 3D reconstruction of an object from its 2D cross-sections (slices) has many applications in different fields of sciences such as medical physics and biomedical engineering. In order to perform 3D reconstruction, at first, desired boundaries at each slice are detected and then using a correspondence between points of successive slices surface of desired object is reconstructed. Materials and Methods In this study, Gradient Vector Flow (GVF) was used in order to trace the boundaries at each slice. Then, cubic Bezier Spline curves were used to approximate each of obtained contours and to approximate the corresponding points of different contours at successive slices. The reconstructed surface was a bi-cubic Bezier Spline surface which was smooth with G2 continuity. Results Our presented method was tested on SPECT data of JASZCZAK phantom and human's left ventricle. The results confirmed that the presented method was accurate, promising, applicable, and effective. Conclusion Using GVF algorithm to trace boundaries at each slice, and cubic Bezier Spline curves to approximate the obtained rough contours yield to the procedure of reconstruction which was fast and also the final surface was smooth with G2 continuity. So far, some mathematical curves such as spline, cubic spline, and B-spline curves were used to approximate the computed contour during a time consuming procedure. This study presented a 3D reconstruction method based on a combination of GVF algorithm and cubic Bezier Spline curves. There was a good trade-off between speed and accuracy in using cubic Bezier Spline curves which is especially useful for training students.

Conic-like subdivision curves on surfaces

In this paper, we introduce a novel nonlinear curve subdivision scheme, suitable for designing curves on surfaces. The scheme is based on the concept of geodesic conic Bezier curves, which represents a natural extension of geodesic Bezier curves for the rational quadratic case. Given a set of points on a surface S, the scheme generates a sequence of geodesic polygons that converges to a continuous curve on S. If the surface S is C2-continuous, then the subdivision curve is C1-continuous and if S is a plane, then the limit curve is a conic Bezier spline curve. Each section of the subdivision curve depends on a free parameter that may be used to obtain a local control of the shape of the subdivision curve. Extending these results to triangulated surfaces, it is shown that the scheme has the convex hull property and that it is suitable for free-form design on triangulations.

Paths of algebraic hyperbolic curves

Cubic algebraic hyperbolic (AH) Bezier curves and AH spline curves are defined with a positive parameter α in the space spanned by {1, t, sinht, cosht}. Modifying the value of α yields a family of AH Bezier or spline curves with the family parameter α. For a fixed point on the original curve, it will move on a defined curve called “path of AH curve” (AH Bezier and AH spline curves) when α changes. We describe the geometric effects of the paths and give a method to specify a curve passing through a given point.

Reconstruction from contour lines based on bi-cubic Bézier spline surface

A novel reconstruction method from contours lines is provided. First, we use a simple method to get rid of redundant points on every contour, then we interpolate them by using cubic Bezier spline curve. For corresponding points of different contours, we interpolate them by the cubic Bezier spline curve too, so the whole surface can be reconstructed by the bi-cubic Bezier spline surface. The reconstructed surface is smooth because every Bezier surface is patched with G2 continuity, the reconstruction speed is fast because we can use the forward elimination and backward substitution method to solve the system of tridiagonal equations. We give some reconstruction examples at the end of this paper. Experiments showed that our method is applicable and effective.

OnG 2 continuous cubic spline interpolation

In this paper the interpolation byG 2 continuous planar cubic Bezier spline curves is studied. The interpolation is based upon the underlying curve points and the end tangent directions only, and could be viewed as an extension of the cubic spline interpolation to the curve case. Two boundary, and two interior points are interpolated per each spline section. It is shown that under certain conditions the interpolation problem is asymptotically solvable, and for a smooth curvef the optimal approximation order is achieved. The practical experiments demonstrate the interpolation to be very satisfactory.