In automated analysis of digital images, a subproblem often arises of detecting simple shapes, such as straight lines, circles or ellipses. In many cases anedge detectorcan be used as a pre-processing stage to obtain image points or image pixels that are on the desired curve in the image space. Due to imperfections in either the image data or the edge detector, however, there may be missing points or pixels on the desired curves as well as spatial deviations between the ideal line/circle/ellipse and the noisy edge points as they are obtained from the edge detector. For these reasons, it is often non-trivial to group the extracted edge features to an appropriate set of lines, circles or ellipses. The purpose of the Hough transform is to address this problem by making it possible to perform groupings of edge points into object candidates by performing an explicit voting procedure over a set of parameterized image objects (Shapiro and Stockman, 304).
The simplest case of Hough transform is the linear transform for detecting straight lines. In the image space, the straight line can be described asy=mx+bwhere the parametermis the slope of the line, andbis the intercept (y-intercept). This is called the slope-intercept model of a straight line. In the Hough transform, a main idea is to consider the characteristics of the straight line not as discrete image points (x1,y1), (x2,y2), etc., but instead, in terms of its parameters according to the slope-intercept model, i.e., the slope parametermand the intercept parameterb. In general, the straight liney=mx+bcan be represented as a point (b,m) in the parameter space. However, vertical lines pose a problem. They are more naturally described asx=aand would give rise to unbounded values of the slope parameter m. Thus, for computational reasons, Duda and Hart proposed the use of a different pair of parameters, denoted
and
(theta), for the lines in the Hough transform. These two values, taken in conjunction, define apolar coordinate.
The parameter
represents the algebraic distance between the line and theorigin, while
is the angle of the vector orthogonal to the line and pointing toward the half upper plane (seeCoordinates). If the line is located above the origin,
is simply the angle of the vector from the origin to this closest point. Using this parameterization, the equation of the line can be written as
which can be rearranged to
(Shapiro and Stockman, 304).
It is therefore possible to associate with each line of the image a pair (r,θ) which is unique ifand
, or ifand
. The (r,θ) plane is sometimes referred to asHough spacefor the set of straight lines in two dimensions. This representation makes the Hough transform conceptually very close to the two-dimensionalRadon transform. (They can be seen as different ways of looking at the same transform.)
For an arbitrary point on the image plane with coordinates, e.g., (x0,y0), the lines that go through it are the pairs (r,θ) with
,
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