[...]... end control points in such a way that that they lie on a straight line Hence, two pieces of curves can be easily drawn to maintain continuity at their joining point, and as a result, this provides effectively a single spline curve For the B-B basis function in the model, the spline curve so obtained is known as B-B spline curve and the underlying spline function is known as the B-B spline or simply... that in the analytical plane Detection of in ection points also helps in maintaining the curvature of the contour during reconstruction and, as a result, the reconstruction quality is improved P3 / / P3 P1 P / 2 P2 / Pi = G (Pi ) P1 Fig 1.9 Gaussian circle and its image detecting points of in ection 22 1 Bernstein Polynomial and B´zier-Bernstein Spline e Gaussian Circle Consider a unit circle in the... Fig 1.2 Cubic B´zier-Bernstein curves e 1.4.4 Approximation of Binary Images Data approximation, for binary images, based on B´zier-Bernstein spline e model is the inverse of drawing mechanism used in computer graphics So, instead of supplying the control points from outside, they are extracted from within images The extraction, in general, uses the local geometry As the control points are viewed as... points or the guiding points The line joining these control points is called the control line of the polynomial It reflects the shape of the curve that one wants to draw or generate Such curves have the following attractive properties: • They always interpolate the end control points, and the line joining two consecutive points at either end is a tangent to the curve at that end point • They remain... again be of two types, with all the pixels either lying on both sides (Figure 1.6(a)) or lying on the same side (Figure 1.6(b)) of the line joining the key pixels We denote the GE in Figure 1.6(c) by L (line) and that in Figure 1.6(b) by CC (curve) GE in Figure 1.6(a), therefore, is nothing but a combination of two CCs meeting at a point Q (point of in ection) Key pixels on the contour of a two-tone... large error depending on the nature of applications Such an approximation of image data points is useful in compression and feature extraction The concept of control points in B´zier-Bernstein spline is implicit in the e definition of the Bernstein polynomial and it was B´zier who made it explicit e Later on, the concept of control points was generalized to knots in B-spline to keep the interaction locally... at the end points, t = 0 and t = 1, is determined by the values of f (t) at the respective end point and at the r points nearest to that end point Specifically, the first derivatives are equal to the slope of the straight line joining the end point and the adjacent interior point Bernstein polynomials satisfy the Weierstrass approximation theorem, i.e., they converge uniformly, with increasing p, to the... drawing a curve is easy and straightforward It is always one less than the number of vertices of the control polygon 1.4 Use in Computer Graphics and Image Data Approximation Due to the attractive properties of the B´zier-Bernstein polynomial, one can e successfully use them in both computer graphics and image data approxima- 10 1 Bernstein Polynomial and B´zier-Bernstein Spline e tion Their use in. .. became widely popular in many industries e In order to design the body of an automobile, B´zier developed a spline model e that became the first widely accepted spline model in computer graphics and computer-aided design, due to its flexibility and ease over the then-used drawing and design techniques Since B´zier used the Bernstein polynomial basis e as the basis function in his spline model, the justification... Cardinal Splines 164 8.2.1 Cardinal B-Spline Basis and Riesz Basis 167 8.2.2 Scaling and Cardinal B-Spline Functions 170 8.3 Wavelets 172 8.3.1 Continuous Wavelet Transform 172 8.3.2 Properties of Continuous Wavelet Transform 173 8.4 A Glimpse of Continuous . B ´ ezier and Splines in Image Processing and Machine Vision Sambhunath Biswas • Brian C. Lovell B ´ ezier and Splines in Image Processing and Machine Vision Sambhunath Biswas Brian C. Lovell Indian. motivation for writing this book on splines, with special attention to applications in image processing and machine vision. The philosophy behind writing this book lies in the fact that splines are effective,. application in image processing and machine vision, and this justifies the title of the book. In writing this book, therefore, we introduce the Bernstein polynomial at the very beginning, since its