Canny Edge Edg e Detection Dete ction Tutorial Tutorial This tutorial will teach you how to:
(1) Implement the Canny edge detection algorithm.
INTRODUCTION Edges characterize boundaries and are therefore a problem of fundamental importance in image processing. Edges in images ima ges are areas with strong intensity contrasts – a jump in intensity from one pixel to the next. Edge detecting an image significantly reduces the amount of data and filters out useless information, while preserving the important structural properties in an image. This was also stated in my Sobel and Laplace edge detection tutorial, but I just wanted reemphasize the point of why you would want to detect edges. The Canny edge detection algorithm is known to many as the optimal edge detector. Canny's intentions were to enhance the many edge detectors already out at the time he started his work. He was very successful in achieving his goal and his ideas and methods can be found in his paper, " A Computational Approach to Edge Detection Dete ction". ". In his paper, he followed a list of criteria to improve current methods of edge detection. The first and most obvious is low error rate. It is important that edges occuring in images should not be missed and that there be NO responses to non-edges. The second criterion is that the edge points be well localized. In other words, the distance between the edge pixels as found by the detector and the actual edge is to be at a minimum. A third criterion is to have only one response to a single edge. This was implemented because the first 2 were not substantial enough to completely eliminate the possibility of multiple responses to an edge. Based on these criteria, the canny edge detector first smoothes the image to eliminate and noise. It then finds the image gradient to highlight regions with high spatial derivatives. The algorithm then tracks along these regions and suppresses any pixel that is not at the maximum (nonmaximum suppression). The gradient array is now further reduced by hysteresis. Hysteresis is used to track along the remaining pixels that have not been suppressed. Hysteresis uses two thresholds and if the magnitude is below the first threshold, it is set to zero (made a nonedge). If the magnitude is above the high threshold, it is made an edge. And if the magnitude is between the 2 thresholds, then it is set to zero unless there is a path from this pixel to a pixel with a gradient above T2.
Step 1 In order to implement the canny edge detector algorithm, a series of steps must be followed. The first step is to filter out any noise in the original image before trying to locate and detect any edges. And because the Gaussian filter can be computed using a simple mask, it is used exclusively in the Canny algorithm. Once a suitable mask has been calculated, the Gaussian smoothing can be performed using standard convolution methods. A convolution mask is usually much smaller than the actual image. As a result, the mask is slid over the image, manipulating a square of pixels at a time. The larger the width of the Gaussian mask, the lower is the detector's sensitivity to noise . The localization error in the detected edges also increases slightly as the Gaussian width is increased. The Gaussian mask used in my implementation is shown below.
Step 2 After smoothing the image and eliminating the noise, the next step is to find the edge strength by taking the gradient of the image. The Sobel operator performs a 2-D spatial gradient measurement on an image. Then, the approximate absolute gradient magnitude (edge strength) at each point can be found. The Sobel operator uses a pair of 3x3 convolution masks, one estimating the gradient in the x-direction (columns) and the other estimating the gradient in the y-direction (rows). They are shown below:
The magnitude, or EDGE STRENGTH, of the gradient is then approximated using the formula:
|G| = |Gx| + |Gy| Step 3 Finding the edge direction is trivial once the gradient in the x and y directions are known. However, you will generate an error whenever sumX is equal to zero. So in the code there has to be a restriction set whenever this takes place. When ever the gradient in the x direction is equal to zero, the edge direction has to be equal to 90 degrees or 0 degrees, depending on what the value of the gradient in the y-direction is equal to. If GY has a value of zero, the edge direction will equal 0 degrees. Otherwise the edge direction will equal 90 degrees. The formula for finding the edge direction is just:
theta = invtan (Gy / Gx) Step 4 Once the edge direction is known, the next step is to relate the edge direction to a direction that can be traced in an image. So if the pixels of a 5x5 image are aligned as follows:
x x x x x
x x x x x
x x a x x
x x x x x
x x x x x
Then, it can be seen by looking at pixel "a", there are only four possible directions when describing the surrounding pixels - 0 degrees (in the horizontal direction), 45 degrees (along the positive diagonal), 90 degrees (in the vertical direction), or 135 degrees (along the negative diagonal). So now the edge orientation has to be resolved into one of these four directions depending on which direction it is closest to (e.g. if the orientation angle is found to be 3 degrees, make it zero degrees). Think of this as taking a semicircle and dividing it into 5 regions.
Therefore, any edge direction falling within the yellow range (0 to 22.5 & 157.5 to 180 degrees) is set to 0 degrees. Any edge direction falling in the green range (22.5 to 67.5 degrees) is set to 45 degrees. Any edge direction falling in the blue range (67.5 to 112.5 degrees) is set to 90 degrees. And finally, any edge direction falling within the red range (112.5 to 157.5 degrees) is set to 135 degrees. Step 5 After the edge directions are known, nonmaximum suppression now has to be applied. Nonmaximum suppression is used to trace along the edge in the edge direction and suppress any pixel value (sets it equal to 0) that is not considered to be an edge. This will give a thin line in the output image. Step 6 Finally, hysteresis is used as a means of eliminating streaking. Streaking is the breaking up of an edge contour caused by the operator output fluctuating above and below the threshold. If a single threshold, T1 is applied to an image, and an edge has an average strength equal to T1, then due to noise, there will be instances where the edge dips below the threshold. Equally it will also extend above the threshold making an edge look like a dashed line. To avoid this, hysteresis uses 2 thresholds, a high and a low. Any pixel in the image that has a value greater than T1 is presumed to be an edge pixel, and is marked as such immediately. Then, any pixels that are connected to this edge pixel and that have a value greater than T2 are also selected as edge pixels. If you think of following an edge, you need a gradient of T2 to start but you don't stop till you hit a gradient below T1.
Hough Transform
Common Names: Hough transform
Brief Description The Hough transform is a technique which can be used to isolate features of a particular shape within an image. Because it requires that the d esired features be specified in some parametric form, the classical Hough transform is most commonly used for the detection of regular curves such as lines, circles, ellipses, etc. A generalized Hough transform can be employed in applications where a simple analytic description of a feature(s) is not possible. Due to the computational complexity of the generalized Hou gh algorithm, we restrict the main focus of this discussion to the classical Hough transform. Despite its domain restrictions, the classical Hough transform (hereafter referred to without the classical prefix) retains many applications, as most manufactured parts (and many anatomical parts investigated in medi cal imagery) contain feature boundaries which can be described by regular curves. The main advantage of the Hough transform technique is that it is tolerant of gaps in feature boundar y descriptions and is relatively unaffected by image noise.
How It Works The Hough technique is particularly useful for computing a global description of a feature(s) (where the number of solution classes need not be known a priori), given (possibly noisy) local measurements. The motivating idea behind the Hough technique for line detection is that each input measurement (e.g. coordinate point) indicates its contribution to a globally consistent solution (e.g. the physical line which gave rise to that image point). As a simple example, consider the common problem of fitting a set of line segments to a set of discrete image points (e.g. pixel locations output from an edge detector). Figure 1 shows some
possible solutions to this problem. Here the lack of a priori knowledge about the number of desired line segments (and the ambiguity about wh at constitutes a line segment) render this problem under-constrained.
Figure 1 a) Coordinate points. b) and c) Possible straight line fittings.
We can analytically describe a line segment in a number of forms. However, a convenient equation for describing a set of lines uses parametric or normal notion:
where
is the length of a normal from the origin to this line and
respect to the X-axis. (See Figure 2.) For any point
is the orientation of
on this line,
and
with
are constant.
Figure 2 Parametric description of a straight line.
In an image analysis context, the coordinates of the point(s) of edge segments (i.e.
) in the
image are known and therefore serve as constants in the parametric line equation, while
and
are the unknown variables we seek. If we plot the possible values defined by each , points in cartesian image space map to curves (i.e. sinusoids) in the polar Hough parameter space. This point-to-curve transformation is the Hough transformation for straight lines. When viewed in Hough parameter space, points which are collinear in the cartesian image space become readily apparent as they yield curves which intersect at a common
point.
The transform is implemented by quantizing the Hough parameter space into finite intervals or accumulator cells. As the algorithm runs, each is transformed into a discretized curve and the accumulator cells which lie along this curve are incremented. Resulting peaks in the accumulator array represent strong evidence that a corresponding straight line exists in the image. We can use this same procedure to d etect other features with analytical descriptions. For instance, in the case of circles, the parametric equation is
where and are the coordinates of the center of the circle and is the radius. In this case, the computational complexity of the algorithm begins to increase as we now have three coordinates in the parameter space and a 3-D accumulator. (In general, the computation and the size of the accumulator array increase polynomially with the number o f parameters. Thus, the basic Hough technique described here is only practical for simple curves.)
Guidelines for Use The Hough transform can be used to identify the parameter(s) of a curve which best fits a set of given edge points. This edge description is commonly obtained from a feature detecting op erator such as the Roberts Cross, Sobel or Canny edge detector and may be noisy, i.e. it may contain multiple edge fragments corresponding to a single whole feature. Furthermore, as the output of an edge detector defines only where features are in an image, the work of the Hough transform is to determine both what the features are (i.e. to detect the feature(s) for which it has a parametric (or other) description) and how many of them exist in the image.
In order to illustrate the Hough transform in detail, we be gin with the simple image of two occluding rectangles,
The Canny edge detector can produce a set of boundary descriptions for this part, as shown in
Here we see the overall boundaries in the image, but this result tells us nothing about the identity (and quantity) of feature(s) within this boundary description. In this case, we can use the Hough (line detecting) transform to detect the eight separate straight lines segments of this image and thereby identify the true geometric structure of the subject. If we use these edge/boundary points as input to the Hough transform, a cu rve is generated in polar space for each edge point in cartesian space. The accumulator array, when viewed as an intensity image, looks like
Histogram equalizing the image allows us to see the patterns of information contained in the low intensity pixel values, as shown in
Note that, although
and
are notionally polar coordinates, the accumulator space is plotted
rectangularly with as the abscissa and as the ordinate. Note that the accumulator space wraps around at the vertical edge of the image such that, in fact, there are only 8 real peaks.
Curves generated by collinear points in the gradient image intersect in peaks in the Hough transform space. These intersection points characterize the straight line segments of the original image. There are a number of methods which one might employ to extract these bright points, or local maxima, from the accumulator array. For example, a simple method involves thresholding and then applying some thinning to the isolated clusters of bright spots in the accumulator array image. Here we use a relative threshold to extract the unique points corresponding to each of the straight line edges in the original image. (In other words, we take only those loc al maxima in the accumulator array whose values are equal to or greater than some fixed percentage of the global maximum value.) Mapping back from Hough transform space (i.e. de-Houghing ) into cartesian space yields a set of line descriptions of the image subject. By ov erlaying this image on an inverted version of the original, we can confirm the result that the Hough transform found the 8 true sides of the two rectangles and thus revealed the underlying geometry of the occluded scene
Note that the accuracy of alignment of detected and original image lines, which is obviously not perfect in this simple example, is determined by the quantization of the accumulator array. (Also note that many of the image edges have several detected lines. This arises from having several nearby Hough-space peaks with similar line parameter values. Techniques exist for controlling this effect, but were not used here to illustrate the output of the standard Hough transform.) Note also that the lines generated by the Hough transform are infinite in length. If we wish to identify the actual line segments which generated the transform parameters, further image analysis is required in order to see which portions of these infinitely long lines actually have points on them. To illustrate the Hough technique's robustness to noise, the Canny edge description has been corrupted by 1% salt and pepper noise
before Hough transforming it. The result, plotted in Hough space, is
De-Houghing this result (and overlaying it on the original) yields
(As in the above case, the relative threshold is 40%.) The sensitivity of the Hough transform to gaps in the feature boundary can be investigated by transforming the image
, which has been edited using a paint program. The Hough representation is
and the de-Houghed image (using a relative threshold of 40%) is
In this case, because the accumulator space did not receive as many entries as in previous examples, only 7 peaks were found, but these are all structurally relevant lines. We will now show some examples with natural imagery. In the first case, we have a city scene where the buildings are obstructed in fog,
If we want to find the true edges of the buildings, an edge detector (e.g. Canny) cannot recover this information very well, as shown in
However, the Hough transform can detect some of the straight lines representing building edges within the obstructed region. The histogram equalized accumulator space representation of the original image is shown in
If we set the relative threshold to 70%, we get the following de-Houghed image
Only a few of the long edges are detected here, and there is a lot of duplication where many lines or edge fragments are nearly colinear. Applying a more generous relative threshold, i.e. 50%, yields
yields more of the expected lines, but at the expense of man y spurious lines arising from the many colinear edge fragments. Our final example comes from a remote sensing application. Here we would like to detect the streets in the image
of a reasonably rectangular city sector. We can edge detect the image using the Canny edge detector as shown in
However, street information is not available as output of the edge detector alone. The image
shows that the Hough line detector is able to recover some of this information. Because the contrast in the original image is poor, a limited set of features (i.e. streets) is identified.
Common Variants Generalized Hough Transform The generalized Hough transform is used when the shape of the feature that we wish to isolate does not have a simple analytic equation describing its boundary. In this case, instead of using a parametric equation of the curve, we use a look-up table to define the relationship between the boundary positions and orientations and the Hough parameters. (The look-up table values must be computed during a preliminary phase using a prototype shape.)
For example, suppose that we know the shape and orientation of the desired feature. (See Figure 3.) We can specify an arbitrary reference point which the shape (i.e. the distance reference point
within the feature, with respect to
and angle of normal lines drawn from the boundary to this
) of the feature is defined. Our look-up table (i.e. R-table) will consist of these
distance and direction pairs, indexed by the orientation
of the boundary.
Figure 3 Description of R-table components.
The Hough transform space is now defined in terms of the possible positions of the shape in the image, i.e. the possible ranges of
. In other words, the transformation is defined by:
(The and values are derived from the R-table for particular known orientations .) If the orientation of the desired feature is unknown, this procedure is complicated by the fact that we must extend the accumulator by incorporating an extra parameter to account for changes in orientation.
Interactive Experimentation You can interactively experiment with this operator b y clicking here.
Exercises 1.
Find the Hough line transform of the object s shown in Figure 4.
Figure 4 Features to input to the Hough tr ansform line detector.
2.
Starting from the basic image
create a series of images with which you can investigate the ability of the Hough line detector to extract occluded features. For example, begin using translation and image addition to create an image containing the original image overlapped by a translated copy of that image. Next, use edge detection to obtain a boundary description of your subject. Finally, apply the Hough algorithm to recover th e geometries of the occluded features. 3.
Investigate the robustness of the Hough algorithm to image noise. Starting from an edge detected version of the basic image
try the following: a) Generate a series of boundary descriptions of the image using different levels of Gaussian noise. How noisy (i.e. broken) does the edge description have to be before Hough is unable to detect the original geometric structure of the scene? b) Corrode the boundary descriptions with different levels of salt and pepper noise. At what point does the combination of broken edges and added intensity spikes render the Hough line detector useless? 4.
Try the Hough transform line detector on t he images:
and
Experiment with the Hough circle detector on
and
5.
One way of reducing the computation required to pe rform the Hough transform is to make use of gradient information which is often available as output from an edge detector. In the case of the Hough circle detector, the edge gradient tells us in which direction a circle must lie from a given edge coordinate point. (See Figure 5.)
Figure 5 Hough circle detection with gradient information.
a) Describe how you would modify the 3-D circle detector accumulator array in order to take this information into account. b) To this algorithm we may want to add gradient
magnitude information. Suggest how to introduce weighted incrementing of the accumulator. 6.
The Hough transform can be seen as an e fficient implementation of a generalized matched filter strategy. In other words, if we created a template composed of a circle of 1's (at a fixed ) and 0's everywhere else in the image, then we could convolve it with the gradient image to yield an
7.
accumulator array-like description of all the circles of radius in the image. Show formally that the basic Hough transform (i.e. the algorithm with no use of gradient direction information) is equivalent to template matching. Explain how to use the generalized Hough tr ansform to detect octagons.
References D. Ballard and C. Brown Computer Vision, Prentice-Hall, 1982, Chap. 4. R. Boyle and R. Thomas Computer Vision:A First Course, Blackwell Scientific Publications, 1988, Chap. 5. A. Jain Fundamentals of Digital Image Processing , Prentice-Hall, 1989, Chap. 9. D. Vernon Machine Vision, Prentice-Hall, 1991, Chap. 6.
Local Information Specific information about this operator may be found here. More general advice about the local HIPR installation is available in the Local Information introductory section.
©2003 R. Fisher, S. Perkins, A. Walker and E. Wolfart.
