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 desired 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.
The Hough transform has many applications, as most manufactured parts (and many anatomical parts investigated in medical imagery) contain feature boundaries which can be described by regular curves or straight lines.
The main advantage of the Hough transform is that it is tolerant of gaps in feature boundary descriptions and is relatively unaffected by image noise.
The Hough technique is 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). The diagram below shows some possible solutions to this problem.
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 
is the orientation of
with respect to the X-axis. (See Figure 2.) For any point 
on this line,
and
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. (i.e. a multidimensional
array). As the algorithm runs, each
is transformed into a discretized
curve and the accumulator cells which lie along this curve are incremented.
Peaks in the accumulator array represent strong evidence that a corresponding straight line exists in the image.
We can use this same procedure to detect 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 of parameters.
Thus, the basic Hough technique described here is only practical for simple
curves.) 
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 by using an edge detector such as the zero crossings of the laplacian. The edge image may be noisy, i.e. it may contain multiple edge fragments corresponding to a single whole feature.
Since 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 description) and how many of them exist in the image.
In order to illustrate the Hough transform in detail, we begin with the simple image of two occluding rectangles,
An 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 curve is generated in polar
space for each edge point in cartesian space. The accumulator array, when
viewed as an intensity image, looks like
Enhancing the contrast of this image allows us to see the patterns of information contained in the low intensity pixel values.
Although
and
are notionally polar coordinates, the accumulator space is plotted rectangularly.
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 local 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 overlaying 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.
The accuracy of alignment of detected and original image lines is not perfect due to the quantization of the accumulator array.
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 edge description is been corrupted by noise.
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 following image
The Hough representation is
The de-Houghed image 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 images. 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 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 accumulator space representation of the original image is shown here:
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
This yields more of the expected lines, but at the expense of many spurious lines arising from the many colinear edge fragments.
Another 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 as follows:
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.
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.
For example, suppose that we know the shape and orientation
of the desired feature. (See Figure 3.) We can specify an arbitrary reference
point 
within the feature, with respect to which the shape (i.e. the distance
and
angle of normal lines drawn from the boundary to this reference point 
)
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.
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.
