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EDGE DETECTION. Edge Template Gradient Generation. Edge gradients are computed in two orthogonal directions, usually along rows and columns, the edge direction is inferred by computing the vector sum of the gradients.
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Edge Template Gradient Generation • Edge gradients are computed in two orthogonal directions, • usually along rows and columns, • the edge direction is inferred by computing the vector sum of the gradients. • Another approach is to compute gradients in a large number of directions by convolution of an image with a set of template gradient impulse response arrays.
Edge Template Gradient Generation • The edge template gradient is defined as:
Truncated Pyramid Operator • Gives a linearly decreasing weighting to pixels away from the center of an edge. • The row gradient impulse response array for a 7x7 truncated pyramid operator is given by:
Canny Edge Detector • (1) The image is smoothed by Gaussian convolution. • (2) A simple 2-D first derivative operator (Roberts Cross) is applied to highlight regions of the image with high first spatial derivatives. (Edges give rise to ridges in the gradient magnitude image.) • (3) Tracks along the top of these ridges and sets to zero all pixels that are not actually on the ridge top so as to give a thin line in the output, a process known as non-maximal suppression.
Canny Edge Detector • Tracking process exhibits hysteresis controlled by two thresholds: T1 and T2, with T1 > T2. • Tracking can only begin at a point on a ridge higher than T1. • Tracking then continues in both directions out from that point until the height of the ridge falls below T2. • This hysteresis helps to ensure that noisy edges are not broken up into multiple edge fragments.
The Effect of the Canny Operator • Determined by following parameters: • the width of the Gaussian kernel used in the smoothing phase, and • the upper and lower thresholds used by the tracker. • Increasing the width reduces the detector's sensitivity to noise, at the expense of losing some of the finer detail in the image. • The localization error in the detected edges also increases slightly as the Gaussian width is increased.
Tracking Threshold • For good results, the tracking threshold can be set: • The upper quite high • The lower quite low • Too high lower threshold will cause noisy edges to break up. (Why?) • Too low upper threshold increases the number of spurious and undesirable edge fragments appearing in the output. (Why?)
Using a Gaussian kernel with standard deviation 1.0 and upper and lower thresholds of 255 and 1 Most of the major edges are detected and lots of details have been picked out well
The same kernel size and upper threshold, but with the lower threshold increased to 220 The edges have become more broken up than in the previous image, which is likely to be bad for subsequent processing. Also, the vertical edges on the wall have not been detected, along their full length. 255 and 220 255 and 1
Obtained by lowering the upper threshold to 128. The lower threshold is kept at 1 and the Gaussian standard deviation remains at 1.0. Many more faint edges are detected along with some short `noisy' fragments. Notice that the detail in the clown's hair is now picked out.
upper threshold to 128 255 and 220 255 and 1
Obtained with the same thresholds as the previous image, but the Gaussian used has a standard deviation of 2.0. Much of the detail on the wall is no longer detected, but most of the strong edges remain. The edges also tend to be smoother and less noisy
Edges in Artificial Scenes • Often sharper and less complex than those in natural scenes. • This generally improves the performance of any edge detector.
How to control the details contained in the resulting edge image The result of applying the Canny edge detector using a standard deviation of 1.0 and an upper and lower threshold of 255 and 1 This image contains many details.
We might be interested to obtain only lines that correspond to the boundaries of the objects Increasing the standard deviation for the Gaussian smoothing to 1.8 Some edges corresponding to changes in the surface orientation remain.
Scaling down the image before the edge detection, we can use the upper threshold of the edge tracker to remove the weaker edges. The result of: 1. scaling the image with 0.25 2. applying the Canny operator (a standard deviation of 1.8) (upper threshold of 200) (lower threshold of 1 )
What effect does increasing the Gaussian kernel size have on the magnitudes of the gradient maxima at edges? What change does this imply has to be made to the tracker thresholds when the kernel size is increased? It is sometimes easier to evaluate edge detector performance after thresholding the edge detector output at some low gray scale value (e.g. 1) so that all detected edges are marked by bright white pixels. Try this out on the third and fourth example images of the clown. Comment on the differences between the two images. How does the Canny operator compare with the Roberts Cross and Sobel edge detectors in terms of speed? What do you think is the slowest stage of the process? How does the Canny operator compare in terms of noise rejection and edge detection with other operators such as the Roberts Cross and Sobel operators? How does the Canny operator compare with other edge detectors on simple artificial 2-D scenes? And on more complicated natural scenes? Under what situations might you choose to use the Canny operator rather than the Roberts Cross or Sobel operators? In what situations would you definitely not choose it?
Laplacian/Laplacian of Gaussian • The Laplacian is a 2-D isotropic measure of the 2nd spatial derivative of an image. • The Laplacian of an image highlights regions of rapid intensity change and is therefore often used for edge detection. • The Laplacian is often applied to an image that has first been smoothed with something approximating a Gaussian smoothing filter in order to reduce its sensitivity to noise.
How Laplacian/Laplacian of Gaussian Works The Laplacian L(x,y) of an image with pixel intensity values I(x,y) is given by: Image is represented as a set of discrete pixels, we have to find a discrete convolution kernel that can approximate the second derivatives in the definition of the Laplacian. Three commonly used small kernels are shown below:
Noise and Laplacian • Because kernels are approximating a second derivative measurement on the image, they are very sensitive to noise. • To counter this, the image is often Gaussian smoothed before applying the Laplacian filter. • This pre-processing step reduces the high frequency noise components prior to the differentiation step. • The convolution operation is associative, we can convolve the Gaussian smoothing filter with the Laplacian filter, and then convolve this hybrid filter LoG (`Laplacian of Gaussian')
2-D LoG Function The 2-D LoG function centered on zero and with Gaussian standard deviation has the form:
Discrete 2-D LoG Function A discrete kernel that approximates this function (for a Gaussian = 1.4) is shown below. A discrete domain version of the LOG operator can be obtained by sampling the continuous domain impulse response function of Eq. over a window. To avoid deleterious truncation effects, the size of the array should be set such that W = 3c, or greater, where is the width of the positive center lobe of the LOG function
The result of applying a LoG filter with Gaussian = 1.0. A 7×7 kernel for display purposes the image has been normalized to the range 0 - 255
If a portion of the filtered, or gradient, image is added to the original image, then the result will be to make any edges in the original image much sharper and give them more contrast. This is commonly used as an enhancement technique in remote sensing applications.
Zero Crossing Detector • Looks for places in the Laplacian of an image where the value of the Laplacian passes through zero. (i.e. points where the Laplacian changes sign) • Such points often occur at `edges' in images. • They also occur at places that are not as easy to associate with edges. • Zero crossing detector as some sort of • feature detector rather than • as a specific edge detector.
Zero Crossing Detector • The starting point is • an image filtered (Laplacian of Gaussian). • The results are strongly influenced by the size of the Gaussian • Increasing the smoothing • fewer and fewer zero crossing contours,
LoG filter with Gaussian standard deviation 1.0 The zero crossings
LoG filter with Gaussian standard deviation 2.0 The zero crossings There are far fewer detected crossings, and that those that remain are largely due to recognizable edges in the image
LoG filter with Gaussian standard deviation 3.0 The zero crossings Only the strongest contours remain, due to the heavy smoothing All edges detected are in the form of closed curves in the same way that contour lines on a map are always closed. The only exception is where the curve goes off the edge
