In developing an automatic system of a single tooth length measurement on x-ray image, since a tooth shape is assumed to be straight and curve, an algorithm which can accurately deal with straight and curve is required. This paper proposes an automatic algorithm for measuring the length of single straight and curve teeth. In the algorithm consisting of control point determination, curve fitting, and length measurement, PCA is employed to find the first and second principle axes as vertical and horizontal ones of the tooth, and two terminal points of vertical axis and the junction of those axes are determined as three first-order control points. Signature is then used to find a peak representing tooth root apex as the forth control point. Bezier curve, Euclidean distance, and perspective transform are finally applied with determined four control points in curve fitting and tooth length measurement. In the experiment, comparing with the conventional PCA-based method, the average mean square error (MSE) of the line points plotted by the expert is reduced from 7.548 pixels to 4.714 pixels for tooth image type-I, whereas the average MSE value is reduced from 7.713 pixels and 7.877 pixels to 4.809 pixels and 5.253 pixels for left side and right side of tooth image type-H, respectively.

We propose a practical local and global path-planning algorithm for an autonomous vehicle or a car-like robot in an unknown semi-structured (or unstructured) environment, where obstacles are detected online by the vehicle's sensors. The algorithm utilizes a probabilistic method based on particle filters to estimate the dynamic obstacles' locations, a support vector machine to provide the critical points and Bezier curves to smooth the generated path. The generated path safely travels through various static and moving obstacles and satisfies the vehicle's movement constraints. The algorithm is implemented and verified on simulation software. Simulation results demonstrate the effectiveness of the proposed method in complicated scenarios that posit the existence of multi moving objects.

We propose the extended Bezier spiral in this paper. The spiral is useful for both design purposes and improved aesthetics. This is because the spiral is one of the Bezier curves, which play an important role in interactive curve design, and because the assessment of the curve is based on the human reception of the curve. For the latter purpose we utilize the logarithmic distribution graph that quantifies the designers' preferences. This paper contributes the unification of the two different curve design objectives (the interactive operation and so called "eye pleasing" result generation); which have been independently investigated so far.

This paper presents a new method for embedding digital watermarks into Bezier polynomial patches. An object surface is supposed to be represented by multiple piecewise Bezier polynomial patches. A Bezier patch passes through its four-corner control points, which are called data points, and does not pass through the other control points. To embed a watermark, a Bezier patch is divided into two patches. Since each subdivided patch shares two data points of the original patch, the subdivision apparently generates two additional data points on the boundaries of the original patch. We can generate the new data points in any position on the boundaries by changing the subdivision parameters. The additional data points can not be removed without knowing some parameters for subdividing and deforming the patch, hence the patch subdivision enables us to embed a watermark into the surface.

This paper develops a simple algorithm for calculating a polynomial curve or surface in a parallel way. The number of arithmetic operations and the necessary time for the calculation are evaluated in terms of polynomial degree and resolution of a curve and the number of processors used. We made some comparisons between our method and a conventional method for generating polynomial curves and surfaces, especially in computation time and approximation error due to the reduction of the polynomial degree. It is shown that our method can perform fast calculation within tolerable error.

The act of finding or constructing a model for a portion of a given polynomial or Bezier parametric surface from the whole original one is an encountered problem in surface modeling. A new method is described for constructing polynomial or Bezier piecewise model from an original one. It is based on the "Parametric Piecewise Model," abbreviated to PPM, of curve representation. The PPM representation is given by explicit expressions in terms of only control points or polynomial coefficients. The generated piecewise model behaves completely as a normal, polynomial or Bezier model in the same way as the original one for the piece of region considered. Also it has all characteristics, i. e, order and number of control points as the original one, and satisfies at the boundaries all order continuities. The PPM representation permits normalization, piecewise modeling, PPM reduction and systematic processes.

In this paper, a new explicit transformation method between Bezier and polynomial representation is proposed. An expression is given to approximate (n + 1) Bezier control points by another of (m + 1), and to perform simple and sufficiently good approximation without any additional transformation, such as Chebyshev polynomial. A criterion of reduction is then deduced in order to know if the given number of control points of a Bezier curve is reducible without error on the curve or not. Also an error estimation is given only in terms of control points. This method, unlike previous works, is more transparent because it is given in form of explicit expressions. Finally, we discuss some applications of this method to curve-fitting, order decreasing and increasing number of control points.

In CAD and curve-fitting fields, we want to generate such rational cubic Bezier curve which is a unique curve passed given points and convenient to connect other curve segments with C1 connection. However, the method proposed in paper [1] can not meet above objective. in this paper, we propose a new method for generating a unique rational cubic Bezier curve which passed given points. The generated curve is with given tangent vectors at its two end points, and it is convenient to connect other curve segments with C1 connection. Also, some examples of curve generated by this method are given.