Thus, the surface normal vector of w 0 is The edge image e for a given threshold T 0 is given by Image Algebra Formulation Let be the source image. The edge image e is defined as where = g i (w - w 0 ) if w M(w 0 ). For example, consider M(w 0 ) a 3 × 3 × 3 domain. The figure below (Figure 3.13.1) shows the domain of the g i ’s. Figure 3.13.1 Illustration of a three-dimensional 3 × 3 × 3 neighborhood. Fixing z to have value z 0 , we obtain Fixing z to have value z 0 + 1, we obtain Fixing z to have value z 0 - 1, we obtain Previous Table of Contents Next Thus, the surface normal vector of w 0 is The edge image e for a given threshold T 0 is given by Image Algebra Formulation Let be the source image. The edge image e is defined as where = g i (w - w 0 ) if w M(w 0 ). For example, consider M(w 0 ) a 3 × 3 × 3 domain. The figure below (Figure 3.13.1) shows the domain of the g i ’s. Figure 3.13.1 Illustration of a three-dimensional 3 × 3 × 3 neighborhood. Fixing z to have value z 0 , we obtain Fixing z to have value z 0 + 1, we obtain Fixing z to have value z 0 - 1, we obtain Previous Table of Contents Next Products | Contact Us | About Us | Privacy | Ad Info | Home Use of this site is subject to certain Terms & Conditions, Copyright © 1996-2000 EarthWeb Inc. All rights reserved. Reproduction whole or in part in any form or medium without express written permission of EarthWeb is prohibited. Read EarthWeb's privacy statement. 2j), (2i + 1, 2j), (2i, 2j + 1), (2i + 1, 2j + 1) in level k + 1. All other points at level k + 1 are assigned value 0. Boundary formation is repeated until level l - 1 is reached. Image Algebra Formulation For any non-negative integer k, let X k denote the set {(i, j) : 0 d i, j d 2 k - 1}. Given a source image construct a pyramid of images for k {0, 1, …, l - 2} as follows: where For , construct the boundary images b k ,…, b l - 1 using the statement: b k := abs(a k • t 1 ) + abs(a k • t 2 ), where To construct the boundary images at level m where k < m d l - 1, first define Next let be defined by g(i, j) = {(2i, 2j), (2i + 1, 2j), (2i, 2j + 1), (2i + 1, 2j + 1)} and be defined by The effect of this mapping is shown in Figure 3.14.1. Figure 3.14.1 Illustration of the effect of the pyramidal map . The boundary image b m is computed by where 0 denotes the zero image on the point set X m . Alternate Image Algebra Formulation The above image algebra formulation gives rise to a massively parallel process operating at each point in the pyramid. One can restrict the operator to only those points in levels m > k where the boundary threshold is satisfied as described below. Let g 1 (i, j) = {(i, j), (i - 1, j), (i + 1, j), (i, j - 1), (i, j + 1)}. This maps a point (i, j) into its von Neumann neighborhood. Let be defined by Note that in contrast to the functions g and , g 1 and are functions defined on the same pyramid level. We now compute b m by Comments and Observations This technique ignores edges that appear at high-resolution levels which do not appear at lower resolutions. The method of obtaining intermediate-resolution images could easily blur, or wipe out edges at high resolutions. 3.15. Edge Detection Using K-Forms The K-forms technique encodes the local intensity difference information of an image. The codes are numbers expressed either in ternary or decimal form. Local intensity differences are calculated and are labeled 0, 1, or 2 depending on the value of the difference. The K-form is a linear function of these labels, where k denotes the neighborhood size. The specific formulation, first described in Kaced [22], is as follows. Let a 0 be a pixel value and let a 1 , …, a 8 be the pixels values of the 8-neighbors of a 0 represented pictorially as Set e l = a l - a 0 for all l {1, …, 8} and for some positive threshold number T define p : as Some common K-form neighborhoods are pictured in Figure 3.15.1. We use the notation Figure 3.15.1 Common K-forms in a 3 × 3 window. to denote the K-form with representation in base b, having a neighborhood containing n pixel neighbors of the reference pixel, and having orientation characterized by the roman letter c. Thus the horizontal decimal 2-form shown in Figure 3.15.1 is denoted , having base 10, 2 neighbors, and horizontal shape. Its value is given by Previous Table of Contents Next Products | Contact Us | About Us | Privacy | Ad Info | Home Use of this site is subject to certain Terms & Conditions, Copyright © 1996-2000 EarthWeb Inc. All rights reserved. Reproduction whole or in part in any form or medium without express written permission of EarthWeb is prohibited. Read EarthWeb's privacy statement. (a) An edge in relief is associated with any pixel whose K-form value is 0, 1, or 3. (b) An edge in depth is associated with any pixel whose K-form value is 5, 7, or 8. (c) An edge by gradient threshold is associated with a pixel whose K-form is not 0, 4, or 8. Image Algebra Formulation Let be the source image. We can compute images for l {1, …, 8}, representing the fundamental edge images from which the K-forms are computed, as follows: where the are defined as shown in Figure 3.15.3. Figure 3.15.3 K-form fundamental edge templates. Thus, for example, t(1) is defined by Let the function p be defined as in the mathematical formulation above. The decimal horizontal 2-form is defined by Other K-forms are computed in a similar manner. Edge in relief is computed as Edge in depth is computed as Edge in threshold is computed as Comments and Observations The K-forms technique captures the qualitative notion of topographical changes in an image surface. The use of K-forms requires more computation than gradient formation to yield the same information as multiple forms must be computed to extract edge directions. 3.16. Hueckel Edge Operator The Hueckel edge detection method is based on fitting image data to an ideal two-dimensional edge model [1, 23, 24]. In the one-dimensional case, the image a is fitted to a step function If the fit is sufficiently accurate at a given location, an edge is assumed to exist with the same parameters as the ideal edge model. (See Figure 3.16.1.) An edge is assumed if Figure 3.16.1 One-dimensional edge fitting. the mean-square error is below some threshold value. In the two-dimensional formulation the ideal step edge is defined as where Á represents the distance from the center of a test disk D of radius R to the ideal step edge (Á d R), and ¸ denotes the angle of the normal to the edge as shown in Figure 3.16.2. Figure 3.16.2 Two-dimensional edge fitting. The edge fitting error is In Hueckel’s method, both image data and the ideal edge model are expressed as vectors in the vector space of continuous functions over the unit disk D = {(x, y) : x 2 + y 2 d 1}. A basis for this vector space — also known as a Hilbert space — is any complete sequence of orthonormalized continuous functions {h i : i = 0, 1, …} with domain D. For such a basis, a and s have vector form a = (a 1 , a 2 , …) and s = (s 1 , s 2 , …), where and For application purposes, only a finite number of basis functions can be used. Hueckel truncates the infinite Hilbert basis to only eight functions, h 0 , h 1 , …, h 7 . This provides for increased computational efficiency. Furthermore, his sub-basis was chosen so as to have a lowpass filtering effect for inherent noise smoothing. Although Hueckel provides an explicit formulation for his eight basis functions, their actual derivation has never been published! Having expressed the signal a and edge model s in terms of Hilbert vectors, minimization of the mean-square error of Equation 3.16.1 can be shown to be equivalent to minimization of . Hueckel has performed this minimization by using some simplifying approximations. Also, although a is expressed in terms of vector components a 0 , …, a 7 , s(x, y) is defined parametrically in terms of the parameters (b, h, Á, ¸). The exact discrete formulation is given below. Definition of the eight basis functions. Let D (x 0 , y 0 ) be a disk with center (x 0 , y 0 ) and radius R. For each (x, y) D(x 0 , y 0 ), define and Then Note that when , then Q(r) = 0. Thus, on the boundary of D each of the functions h i is 0. In fact, the functions h i intersect the disk D as shown in the following figure: Figure 3.16.3 The intersection D graph(h i ). In his algorithm, however, Hueckel uses the functions H i (i = 0, 1, …, 7) instead of h i (i = 0, 1, …, 7) in order to increase computation efficiency. This is allowable since the H i ’s are also linearly independent and span the same eight-dimensional subspace. Previous Table of Contents Next Products | Contact Us | About Us | Privacy | Ad Info | Home Use of this site is subject to certain Terms & Conditions, Copyright © 1996-2000 EarthWeb Inc. All rights reserved. Reproduction whole or in part in any form or medium without express written permission of EarthWeb is prohibited. Read EarthWeb's privacy statement. [...]... sequence of boundary points detected by this technique will often be inaccurate 3.18 Edge Following as Dynamic Programming The purpose of this particular technique is to extract one high-intensity line or curve of fixed length and locally low-curvature from an image using dynamic programming Dynamic programming in this particular case involves the definition and evaluation of a figure -of- merit (FOM) function... card(Y) = 0, the line segment joining p and q defines the boundary between p and q, and our work is finished If card(Y) > 0, define image by b(x) = d(x, L(p, q)) Let r = choice(domain ) Thus, as shown in Figure 3.17.3, r will be an edge point in set S farthest from line segment L(p, q) The algorithm can be repeated with points p and r, and with points r and q Figure 3.17.3 Choice of edge point r Comments... useful in the case that a low curvature boundary is known to exist between edge elements and the noise levels in the image are low It could be used for filling in gaps left by an edge detecting-thresholding-thinning operation Difficulties arise in the use of this technique when there is more than one candidate point for the new edge point, or when these points do not lie on the boundary of the object being... that f is partial in that it is defined only if there is a point on L(x, y) with second coordinate j Furthermore, let F(j, w, x, y, z) denote the set points with second coordinate j bounded by L(w, x) and L(y, z) ; that is, the set of Figure 3.17.2 Variables characterizing the region of search We can then compute the set S containing all integral points in the rectangle described by corner points s,... represents the optimal curve of length n of this process Image Algebra Formulation The input consists of an edge magnitude image e, the corresponding direction image d, and a positive integer n representing the desired curve length The edge magnitude/direction image (e, d) could be provided by the directional edge detection algorithm of Section 3.10 Suppose and a function Define a parameterized template... | Ad Info | Home Use of this site is subject to certain Terms & Conditions, Copyright © 19 96- 2000 EarthWeb Inc All rights reserved Reproduction whole or in part in any form or medium without express written permission of EarthWeb is prohibited Read EarthWeb's privacy statement and Image Processing, vol 49, pp 297-331, Mar 1990 17 A Rosenfeld, “A nonlinear edge detection technique,” Proceedings of the... which recognizes edges and lines,” Journal of the ACM, vol 20, pp 63 4 -64 7, Oct 1973 Previous Table of Contents Next Products | Contact Us | About Us | Privacy | Ad Info | Home Use of this site is subject to certain Terms & Conditions, Copyright © 19 96- 2000 EarthWeb Inc All rights reserved Reproduction whole or in part in any form or medium without express written permission of EarthWeb is prohibited... expressed as a sum of functions then a multistage optimization procedure using the following recursion process can be applied: 1 Define a sequence of functions recursively by setting for k = 1,…, n - 1 2 For each y X, determine the point mk+1(y) = xk X such that that is, find the point xk for which the maximum of the function is achieved The n-tuple 3 Find can now be computed by using the recursion... segmented In our presentation of the algorithm, we make a non-deterministic choice of the new edge point if there are several candidates We assume that the edge point selection method — which is used as a preprocessing step to boundary detection — is robust and accurate, and that all edge points fall on the boundary of the object of interest In practice, this does not usually occur, thus the sequence of boundary... order to incorporate the constraints (i), (ii), and (iii) into the process, we define two step functions s1 and s2 by and Redefining g by where for k = 1, …, n - 1, results in a function which has constraints (i), (ii), and (iii) incorporated into its definition and has the required format for the optimization process The algorithm reduces now to applying steps 1 through 4 of the optimization process to . from an image using dynamic programming. Dynamic programming in this particular case involves the definition and evaluation of a figure -of- merit (FOM) function that embodies a notion of “best. points detected by this technique will often be inaccurate. 3.18. Edge Following as Dynamic Programming The purpose of this particular technique is to extract one high-intensity line or curve of. model s in terms of Hilbert vectors, minimization of the mean-square error of Equation 3. 16. 1 can be shown to be equivalent to minimization of . Hueckel has performed this minimization by using some