【正文】 espectively.It is obvious from Figure 3, that an image of the gradient magnitudes often indicate the edges quite clearly. However, the edges are typically broad and thus do not indicate exactly where the edges are. To make it possible to determine this (see Section ), the direction of the edges must be determined and stored as shown in Equation (5). (5) (a) Smoothed (b) Gradient magnitudesFigure 3: The gradient magnitudes in the smoothed image shown in 3b as well as their directions are determined by applying . the Sobeloperator. Nonmaximum suppressionThe purpose of this step is to convert the “blurred” edges in the image of the gradient magnitudes to “sharp” edges. Basically this is done by preserving all local maxima in the gradient image, and deleting everything else. The algorithm is for each pixel in the gradient image:1. Round the gradient direction  to nearest 45?, corresponding to the use of an 8connected neighbourhood.2. Compare the edge strength of the current pixel with the edge strength of the pixel in the positive and negative gradient direction. . if the gradient direction is north (theta = 90?), pare with the pixels to the north and south.3. If the edge strength of the current pixel is largest。 preserve the value of the edge strength. If not, suppress (. remove) the value.A simple example of nonmaximum suppression is shown in Figure 4. Almost all pixels have gradient directions pointing north. They are therefore pared with the pixels above and below. The pixels that turn out to be maximal in this parison are marked with white borders. All other pixels will be suppressed. Figure 5 shows the effect on the test image.Figure 4: Illustration of nonmaximum suppression. The edge strengths are indicated both as colors and numbers, while the gradient directions are shown as arrows. The resulting edge pixels are marked with white borders. (a) Gradient values (b) Edges after nonmaximum suppressionFigure 5: Nonmaximum suppression. Edgepixels are only preserved where the gradient has local Maxima. Double thresholdingThe edgepixels remaining after the nonmaximum suppression step are (still) marked with their strength pixelbypixel. Many of these will probably be true edges in the image, but some may be caused by noise or color variations for instance due to rough surfaces. The simplest way to discern between these would be to use a threshold, so that only edges stronger that a certain value would be preserved. The Canny edge detection algorithm uses double thresholding. Edge pixels stronger than the high threshold are marked as strong。 edge pixels weaker than the low threshold are suppressed and edge pixels between the two thresholds are marked as weak. The effect on the test image with thresholds of 20 and 80 is shown in Figure 6. (a) Edges after nonmaximum suppression (b) Double thresholdingFigure 6: Thresholding of edges. In the second image strong edges are white, while weak edges are with a strength below both thresholds are suppressed. Edge tracking by hysteresisStrong edges are interpreted as “certain edges”, and can immediately be included in the final edge image. Weak edges are included if and only if they are connected to strong edges. The logic is of course that noise and other small variations are unlikely to result in a strong edge (with proper adjustment of the threshold levels). Thus strong edges will (almost) only be due to true edges in the original image. The weak edges can either be due to true edges or noise/color variations. The latter type will probably be distributed independently of edges on the entire image, and thus only a small amount will be located adjacent to strong edges. Weak edges due to true edges are much more likely to be connected directly to strong edges.Edge tracking can be implemented by BLOBanalysis (Binary Large OBject). The edge pixels are divided into connected BLOB’s using 8connected neighbourhood. BLOB’s containing at least one strong edge pixel are then preserved, while other BLOB’s are suppressed. The effect of edge tracking on the test image is shown in Figure 7. (a) Double thresholding (b) Edge tracking by hysteresis (c) Final outputFigure 7: Edge tracking and final output. The middle image shows strong edges in white, weak edges connected to strong edges in blue, and other weak edges in red.3 Implementation of Canny Edge DetectionAs noted in Section 1, all images in this worksheet (except the original) are produced by our implementation. A few things should be noted with regards to this:1. The (source) image and the thresholds can be chosen arbitrarily.2. Only a smoothing filter with a standard deviation of  = is supported (the one shown in Equation 1).3. The implementation uses the “correct” Euclidean measure for the edge strengths, described in Section .4. The different filters cannot be applied to edge pixels. This causes the output image to be 8 pixels smaller in each direction.The last step in the algorithm known as edge tracking can be implemented as either iterative or recursive BLOB analysis [4]. A recursive implementation can use the grassfire algorithm. However, our implementation uses the iterative approach. First all weak edges are scanned for neighbour edges and joined into groups. At the same time it is marked which groups are adjacent. Then all of these markings are examined to determine which groups of weak edges are connected to strong edges (directly or indirectly). All weak edges that are connected to strong edges are marked as strong edges themselves. The rest of the weak edges are suppressed. This can be interpreted as BLOB analysis where only BLOB’s containing strong edges are preserved (and considered as one BLOB).Figure 8 shows the plete edge detection process on the test image including all intermediate results. (a) Origi