Canny edge detector

The Canny edge detector is an edge detection operator that uses a multi-stage algorithm to detect a wide range of edges in images. It was developed by John F. Canny in 1986. Canny also produced a "computational theory of edge detection" explaining why the technique works.
Development of the Canny algorithm.
Canny edge detection is a technique to extract useful structural information from different vision objects and dramatically reduce the amount of data to be processed. It has been widely applied in various computer vision systems. Canny has found that the requirements for the application of edge detection on diverse vision systems are relatively similar. Thus, an edge detection solution to address these requirements can be implemented in a wide range of situations. 
The general criteria for edge detection include:
To satisfy these requirements Canny used the calculus of variations – a technique which finds the function which optimizes a given functional. The optimal function in Canny's detector is described by the sum of four exponential terms, but it can be approximated by the first derivative of a Gaussian.
Among the edge detection methods developed so far, Canny edge detection algorithm is one of the most strictly defined methods that provides good and reliable detection. Owing to its optimality to meet with the three criteria for edge detection and the simplicity of process for implementation, it became one of the most popular algorithms for edge detection.
Process of Canny edge detection algorithm.
The Process of Canny edge detection algorithm can be broken down to 5 different steps:
Gaussian filter.
Since all edge detection results are easily affected by the noise in the image, it is essential to filter out the noise to prevent false detection caused by it. To smooth the image, a Gaussian filter kernel is convolved with the image. This step will slightly smooth the image to reduce the effects of obvious noise on the edge detector. The equation for a Gaussian filter kernel of size (2"k"+1)×(2"k"+1) is given by:
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Here is an example of a 5×5 Gaussian filter, used to create the adjacent image, with formula_2 = 1. (The asterisk denotes a convolution operation.)
It is important to understand that the selection of the size of the Gaussian kernel will affect the performance of the detector. The larger the size is, the lower the detector's sensitivity to noise. Additionally, the localization error to detect the edge will slightly increase with the increase of the Gaussian filter kernel size. A 5×5 is a good size for most cases, but this will also vary depending on specific situations.
Finding the intensity gradient of the image.
An edge in an image may point in a variety of directions, so the Canny algorithm uses four filters to detect horizontal, vertical and diagonal edges in the blurred image. The edge detection operator (such as Roberts, Prewitt, or Sobel) returns a value for the first derivative in the horizontal direction (G"x") and the vertical direction (G"y"). From this the edge gradient and direction can be determined:
where G can be computed using the hypot function and atan2 is the arctangent function with two arguments.
The edge direction angle is rounded to one of four angles representing vertical, horizontal and the two diagonals (0°, 45°, 90° and 135°). An edge direction falling in each color region will be set to a specific angle values, for instance θ in [0°, 22.5°] or [157.5°, 180°] maps to 0°.
Gradient magnitude thresholding or lower bound cut-off suppression.
Minimum cut-off suppression of gradient magnitudes, or lower bound thresholding, is an edge thinning technique.
Lower bound cut-off suppression is applied to find the locations with the sharpest change of intensity value. The algorithm for each pixel in the gradient image is:
In some implementations, the algorithm categorizes the continuous gradient directions into a small set of discrete directions, and then moves a 3x3 filter over the output of the previous step (that is, the edge strength and gradient directions). At every pixel, it suppresses the edge strength of the center pixel (by setting its value to 0) if its magnitude is not greater than the magnitude of the two neighbors in the gradient direction. For example,
In more accurate implementations, linear interpolation is used between the two neighbouring pixels that straddle the gradient direction. For example, if the gradient angle is between 89° and 180°, interpolation between gradients at the north and north-east pixels will give one interpolated value, and interpolation between the south and south-west pixels will give the other (using the conventions of the last paragraph). The gradient magnitude at the central pixel must be greater than both of these for it to be marked as an edge.
Note that the sign of the direction is irrelevant, i.e. north–south is the same as south–north and so on.
Double threshold.
After application of non-maximum suppression, remaining edge pixels provide a more accurate representation of real edges in an image. However, some edge pixels remain that are caused by noise and color variation. In order to account for these spurious responses, it is essential to filter out edge pixels with a weak gradient value and preserve edge pixels with a high gradient value. This is accomplished by selecting high and low threshold values. If an edge pixel’s gradient value is higher than the high threshold value, it is marked as a strong edge pixel. If an edge pixel’s gradient value is smaller than the high threshold value and larger than the low threshold value, it is marked as a weak edge pixel. If an edge pixel's gradient value is smaller than the low threshold value, it will be suppressed. The two threshold values are empirically determined and their definition will depend on the content of a given input image.
Edge tracking by hysteresis.
So far, the strong edge pixels should certainly be involved in the final edge image, as they are extracted from the true edges in the image. However, there will be some debate on the weak edge pixels, as these pixels can either be extracted from the true edge, or the noise/color variations. To achieve an accurate result, the weak edges caused by the latter reasons should be removed. Usually a weak edge pixel caused from true edges will be connected to a strong edge pixel while noise responses are unconnected. 
To track the edge connection, blob analysis is applied by looking at a weak edge pixel and its 8-connected neighborhood pixels. As long as there is one strong edge pixel that is involved in the blob, that weak edge point can be identified as one that should be preserved.
Walkthrough of Canny edge detection.
This section will show the progression of an image through each of the five steps.
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Improvement on Canny edge detection.
While traditional Canny edge detection provides a relatively simple but precise methodology for the edge detection problem, with more demanding requirements on the accuracy and robustness on the detection, the traditional algorithm can no longer handle the challenging edge detection task. The main defects of the traditional algorithm can be summarized as follows:
In order to address these defects, an improvement to the canny edge algorithm is presented in the following paragraphs.
Replace Gaussian filter.
As both edge and noise will be identified as a high frequency signal, a simple Gaussian filter will add a smooth effect on both of them. However, in order to reach high accuracy of detection of the real edge, it is expected that a more smooth effect should be applied to noise and a less smooth effect should be added to the edge. Bing Wang and Shaosheng Fan from Changsha University of Science and Technology developed an adaptive filter, where the filter will evaluate discontinuity between greyscale values of each pixel. The higher the discontinuity, the lower the weight value is set for the smooth filter at that point. Contrarily, the lower the discontinuity between the greyscale values, the higher the weight value is set to the filter. The process to implement this adaptive filter can be summarized in five steps:
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to smooth the image, where
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Improvement on gradient magnitude and direction calculation.
The gradient magnitude and direction can be calculated with a variety of different edge detection operators, and the choice of operator can influence the quality of results. A very commonly chosen one is the 3x3 Sobel filter. However, other filters may be better, such as a 5x5 Sobel filter which will reduce noise or the Scharr filter which has better rotational symmetry. Other common choices are Prewitt (used by Zhou ) and Roberts Cross.
Robust method to determine the dual-threshold value.
In order to resolve the challenges where it is hard to determine the dual-threshold value empirically, Otsu's method can be used on the non-maximum suppressed gradient magnitude image to generate the high threshold. The low threshold is typically set to 1/2 of the high threshold in this case. Since the gradient magnitude image is continuous-valued without a well-defined maximum, Otsu's method has to be adapted to use value/count pairs instead of a complete histogram.
The thinning of the edge.
While the traditional Canny edge detection implements a good detection result to meet the first two criteria, it does not meet the single response per edge strictly. A mathematical morphology technique to thin the detected edge is developed by Mallat S and Zhong.
Use of Curvelets.
Curvelets have been used in place of the Gaussian filter and gradient estimation to compute a vector field whose directions and magnitudes approximate the direction and strength of edges in the image, to which steps 3 - 5 of the Canny algorithm are then applied. Curvelets decompose signals into separate components of different scales, and dropping the components of finer scales can reduce noise.
Differential geometric formulation of the Canny edge detector.
A more refined approach to obtain edges with sub-pixel accuracy is by using the approach of differential edge detection, where the requirement of non-maximum suppression is formulated in terms of second- and third-order derivatives computed from a scale space representation (Lindeberg 1998) – see the article on edge detection for a detailed description.
Variational formulation of the Haralick–Canny edge detector.
A variational explanation for the main ingredient of the Canny edge detector, that is, 
finding the zero crossings of the 2nd derivative along the gradient direction, was shown 
to be the result of minimizing a Kronrod–Minkowski functional while maximizing the integral 
over the alignment of the edge with the gradient field (Kimmel and Bruckstein 2003). See the article on
regularized Laplacian zero crossings and other optimal edge integrators for a detailed description.
Parameters.
The Canny algorithm contains a number of adjustable parameters, which can affect the computation time and effectiveness of the algorithm.
Conclusion.
The Canny algorithm is adaptable to various environments. Its parameters allow it to be tailored to recognition of edges of differing characteristics depending on the particular requirements of a given implementation. In Canny's original paper, the derivation of the optimal filter led to a Finite Impulse Response filter, which can be slow to compute in the spatial domain if the amount of smoothing required is important (the filter will have a large spatial support in that case). For this reason, it is often suggested to use Rachid Deriche's infinite impulse response form of Canny's filter (the Canny–Deriche detector), which is recursive, and which can be computed in a short, fixed amount of time for any desired amount of smoothing. The second form is suitable for real time implementations in FPGAs or DSPs, or very fast embedded PCs. In this context, however, the regular recursive implementation of the Canny operator does not give a good approximation of rotational symmetry and therefore gives a bias towards horizontal and vertical edges.