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Measures for evaluating the performance of edge detectors have been formulated by Abdou and Pratt [1] and DeMicheli, Caprile, Ottonello, and Torre [66] The criteria to consider in evaluating the performance of an edge detector include 1 Probability of false edges 2 Probability of missing edges 3 Error in estimation of the edge angle 4 Mean square distance of the edge estimate from the true edge 5 Tolerance to distorted edges and other features such as corners and junctions
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58 EDGE DETECTOR PERFORMANCE
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The first two criteria concern the performance of an algorithm as a detector of edges The second two criteria concern the performance of an algorithm as an estimator of the edge location and orientation The last criterion concerns the tolerance of the edge algorithm to edges that depart from the ideal model used to formulate the algorithm
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The performance of an edge detector can be evaluated in two stages: count the number of false and missing edges and measure the variance (or error distribution) for the estimated location and orientation For a test case, select a synthetic image where the true edges are known to lie along a contour that can be modeled by a curve with a simple mathematical formula-for example, a filled rectangle where the boundary contour can be modeled by line segments or two filled rectangles where the gap between them is known Count the number of correct, missing, and false edges by comparing the results of the edge detector with the original (synthetic) image This is a harder task than it appears to be The results vary with the threshold, smoothing filter size, interactions between edges, and other factors If you run an edge detector Qverthe test image with no added noise, no smoothing, and no interactions between edges, then you should get a perfect set of edges (no missing or false edges) Use this set of edges as the standard for comparison Now consider the edges obtained from a test case that has added noise, or other distortions in the image that create missing or false edges Compute a one-to-one match of edges in the test image to edges in the standard, based on the criterion of Euclidean distance Ideally, we should use a proper matching algorithm such as the method for the disparity analysis of images presented
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in Section 143 Edges too far fromthe edgesin the standard are false edges;
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edges that pair closely with one edge in the standard are correct After this procedure, the edges in the standard that are not paired with one edge in the test case are missing edges This procedure tests an edge detector based only on its ability to indicate the presence or absence of edges, but says nothing about how accurately the edge locations or orientations are estimated Compare the locations and orientations of edges in the set of correct edges (computed above) with the original test image This comparison requires that the model of the test
CHAPTER 5 EDGE DETECTION
case be available For the filled rectangle, the model is the line segments that make up the sides of the rectangle The edge locations and orientations must be compared with a mathematical description of the model of the scene contours For each edge with location (x, y), how far is this location from the true location What is the difference between the orientation of the edge and the orientation of the true curve The edge location (x, y) could correspond to any point along the contour, but the closest point along the contour is used as the corresponding point, and the distance between the edge point and the closest point is computed For a line segment, use the formulas in Section 64 Estimate the error distribution from a histogram of location errors, or tabulate the sum of the squared error and divide by n - 1, where n is the number of edges, to estimate the variance (refer to the formula in Appendix B) The orientation error of an edge is measured by comparing the orientation of the edge fragment with the angle of the normal to the curve that models the scene contour, evaluated at the closest point to the edge point 582 Figure of Merit
One method to judge the performance of edge detectors is to look at an edge image and subjectively evaluate the performance However, this does not provide an objective measure of performance To quantitatively evaluate the performance of various edge detectors, we should formulate a criterion that may help in judging the relative performance under controlled conditions We observe that in the response of an edge detector, there can be three types of errors:
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