where d is a specified nonnegative quantity which represents the cutoff frequency and . The transfer function of the Butterworth highpass filter of order k is given by where c is a scaling constant. Typical values for c are 1 and . The transfer function of the exponential highpass filter is given by Typical values for a are 1 and ln . Image Algebra Formulation Let denote the source image, where . Specify the point set , and define by Once the transfer function is specified, the remainder of the algorithm is analogous to the lowpass filter algorithm. Thus, one specification would be This pseudocode specification can also be used for the Butterworth and exponential highpass filter transfer functions which are described next. The Butterworth highpass filter transfer function of order k with scaling constant c is given by where The transfer function for exponential highpass filtering is given by Previous Table of Contents Next where d is a specified nonnegative quantity which represents the cutoff frequency and . The transfer function of the Butterworth highpass filter of order k is given by where c is a scaling constant. Typical values for c are 1 and . The transfer function of the exponential highpass filter is given by Typical values for a are 1 and ln . Image Algebra Formulation Let denote the source image, where . Specify the point set , and define by Once the transfer function is specified, the remainder of the algorithm is analogous to the lowpass filter algorithm. Thus, one specification would be This pseudocode specification can also be used for the Butterworth and exponential highpass filter transfer functions which are described next. The Butterworth highpass filter transfer function of order k with scaling constant c is given by where The transfer function for exponential highpass filtering 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. 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. image. Boundary transforms can be especially useful when used inside of other algorithms that require location of the boundary to perform their tasks. Many of the other thinning transforms in this chapter fall into this category. The techniques outlined below work by using the appropriate neighborhood to either enlarge or reduce the region with the or operation, respectively. After the object has been enlarged or reduced, it is intersected with its original complement to produce the boundary image. For a {0, 1} X , let A denote the support of a. The boundary image b {0, 1} X of a is classified by its connectivity and whether B 4 A or B 4 A2, where B denotes the support of b. (a) The image b is an exterior 8-boundary image if B is 8-connected, B 4 A2, and B is the set of points in A2 whose 4-neighborhoods intersect A. That is, (b) The image b is an interior 8-boundary image if B is 8-connected, B 4 A, and B is the set of points in A whose 4-neighborhoods intersect A2. The interior 8-boundary b can be expressed as (c) The image b is an exterior 4-boundary image if B is 4-connected, B 4 A2, and B is the set of points in A2 whose 8-neighborhoods intersect A. That is, the image b is defined by (d) The image b is an interior 4-boundary image if B is 4-connected, B 4 A, and B is the set of points in A whose 8-neighborhoods intersect A2. Thus, Figure 3.2.1 below illustrates the boundaries just described. The center image is the original image. The 8-boundaries are to the left, and the 4-boundaries are to the right. Exterior boundaries are black. Interior boundaries are gray. Figure 3.2.1 Interior and exterior 8-boundaries (left), original image (center), and interior and exterior 4-boundaries (right). Image Algebra Formulation The von Neumann neighborhood function N is used in the image algebra formulation for detecting 8-boundaries, while the Moore neighborhood function M is used for detection of 4-boundaries. These neighborhoods are defined below. Let a {0, 1} X be the source image. The boundary image will be denoted by b. (1) Exterior 8-boundary — (2) Interior 8-boundary — (3) Exterior 4-boundary — (4) Interior 4-boundary — Comments and Observations These transforms are designed for binary images only. More sophisticated algorithms must be used for gray level images. Noise around the boundary may adversely affect results of the algorithm. An algorithm such as the salt and pepper noise removal transform may be useful in cleaning up the boundary before the boundary transform is applied. 3.3. Edge Enhancement by Discrete Differencing Discrete differencing is a local edge enhancement technique. It is used to sharpen edge elements in an image by discrete differencing in either the vertical or horizontal direction, or in a combined fashion [1, 2, 3, 4]. Let be the source image. The edge enhanced image can be obtained by one of the following difference methods: (1) Horizontal differencing or (2) Vertical differencing or (3) Gradient approximation or Image Algebra Formulation Given the source image , the edge enhanced image is obtained by the appropriate image-template convolution below. (1) Horizontal differencing where the invariant enhancement template r is defined by or where (2) Vertical differencing where the template t is defined by or where (3) Gradient approximation or where the templates t and r are defined as above and v, w are defined by and The templates can be described pictorially as 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. Given the source image , the edge enhanced image is given by where the templates s and t are defined by The templates s and t can be represented pictorially as Comments and Observations Figure 3.4.1 shows the result of applying the Roberts edge detector to the image of a motorcycle. Figure 3.4.1 Motorcycle and its Roberts edge enhanced image. 3.5. Prewitt Edge Detector The Prewitt edge detector calculates an edge gradient vector at each point of the source image. An edge enhanced image is produced from the magnitude of the gradient vector. An edge angle, which is equal to the angle the gradient makes to the horizontal axis, can also be assigned to each point of the source image [6, 7, 1, 8, 4]. Two masks are convolved with the source image to approximate the gradient vector. One mask represents the partial derivative with respect to x and the other the partial derivative with respect to y. Let be the source image, and a 0 , a 1 , …, a 7 denote the pixel values of the eight neighbors of (i, j) enumerated in the counterclockwise direction as follows: Let u = (a 5 + a 6 + a 7 ) - (a 1 + a 2 + a 3 ) and v = (a 0 + a 1 + a 7 ) - (a 3 + a 4 + a 5 ). The edge enhanced image is given by and the edge direction image is given by Image Algebra Formulation Let be the source image and let s and t be defined as follows: Pictorially we have: The edge enhanced image is given by The edge direction image is given by Here we use the common programming language convention for the arctangent of two variables which defines Comments and Observations A variety of masks may be used to approximate the partial derivatives. Figure 3.5.1 Motorcycle and the image that represents the magnitude component of the Prewitt edge detector. 3.6. Sobel Edge Detector The Sobel edge detector is a nonlinear edge enhancement technique. It is another simple variation of the discrete differencing scheme for enhancing edges [9, 10, 1, 8, 4]. Let be the source image, and a 0 , a 1 , …, a 7 denote the pixel values of the eight neighbors of (i, j) enumerated in the counterclockwise direction as follows: [...]... scale appropriately to map them onto points with integral coordinates The transformation f : X ’ Z defined by is just such a function The effect of applying f to a set of edge points shown in Figure 3.12.1 below Figure 3.12.1 The effect of applying f to a set of edge points Comments and Observations Crack edge finding can be used to find edges using a variety of underlying techniques The primary benefit... maximum of h and v at each point in X Figure 3.11.1 compares the results of applying discrete differencing and the product of the difference of averages techniques The source image (labeled 90/10) in the top left corner of the figure is of a circular region whose pixel values have a 90% probability of being white against a background whose pixel values have a 10% probability of being white The corresponding... thresholding based on the statistic which is easier to compute In this case, a larger value indicates a stronger edge point The Frei-Chen method can also be used to detect lines Subimages v5, v6, v7, and v8 form the basis of the line subspace of V The Frei-Chen edge detection method bases its determination of lines on the size of the angle between the subimage b and its projection on the line subspace... for a point with integral second coordinate, t can be pictured as The crack edge image is given by Alternate Image Algebra Formulation Some implementations of image algebra will not permit one to specify points with non-integral coordinates as are required above In those cases, several different techniques can be used to represent the crack edges as images One may want to map each point (y1, y2) in Y... Thresholding for line detection is done using the statistic Larger values indicate stronger line points Image Algebra Formulation Let be the source image and let v(i)y denote the parameterized template whose values are defined by the image vi of Figure 3.8.2 The center cell of the image vi is taken to be the location of y for the template For a given threshold level Ä, the Frei-Chen edge image e is... regions of the image in the upper right corner have probabilities of white pixels equal to 70% and 30% The center images show the result of taking the maximum from discrete differencing along horizontal and vertical directions The images at the bottom of the figure show the results of the product of the difference of average algorithm Specifically, the figure’s edge enhanced image e produced by using the... point (y1, y2) in Y to the point (2y1, 2y2) In such a case, the domain of the crack edge image does not cover a rectangular subset of coordinates are missing , all points involving two odd or two even is to Another technique that can be used to transform the set of crack edges onto a less sparse subset of employ a spatial transformation that will rotate the crack edge points by angle À/4 and shift and... vertical differences in a single image with scalar edge values (This cannot be done using pixel edges since each pixel is associated with both a horizontal and vertical edge.) That is, given , we can compute where and Image Algebra Formulation Given source image , define spatial functions f1, f2, f3, f4 on Next construct the point set and define by For a point with integral first coordinate, t can be pictured... detect edges of features of single pixel width Representation of crack edges requires one to construct images over pointsets that are either sparse, rotated, or contain fractional coordinates Such images may be difficult to represent efficiently 3.13 Local Edge Detection in Three-Dimensional Images This technique detects surface elements in three-dimensional data It consists of approximating the surface... permission of EarthWeb is prohibited Read EarthWeb's privacy statement at the point, and points in the direction of the edge Figure 3.10.1 Source image of causeway with bridge Figure 3.10.2 Edge mask with their associated directions Figure 3.10.3 Edge points with their associated directions Image Algebra Formulation Let be the source image For , let where f(r) is defined as The result image b is given . set of points in A2 whose 8-neighborhoods intersect A. That is, the image b is defined by (d) The image b is an interior 4-boundary image if B is 4-connected, B 4 A, and B is the set of points in. y 2 ) in Y to the point (2y 1 , 2y 2 ). In such a case, the domain of the crack edge image does not cover a rectangular subset of , all points involving two odd or two even coordinates are missing. Another. below. Figure 3.12.1 The effect of applying f to a set of edge points. Comments and Observations Crack edge finding can be used to find edges using a variety of underlying techniques. The primary