## Wednesday, March 18, 2009

### Hough Transform

The Hough transform (pronounced /ˈhʌf/, rhymes with tough) is a feature extraction technique used in image analysis, computer vision, and digital image processing.[1] The purpose of the technique is to find imperfect instances of objects within a certain class of shapes by a voting procedure. This voting procedure is carried out in a parameter space, from which object candidates are obtained as local maxima in a so-called accumulator space that is explicitly constructed by the algorithm for computing the Hough transform.

The classical Hough transform was concerned with the identification of lines in the image, but later the Hough transform has been extended to identifying positions of arbitrary shapes, most commonly circles or ellipses. The Hough transform as it is universally used today was invented by Richard Duda and Peter Hart in 1972, who called it a "generalized Hough transform"[2] after the related 1962 patent of Paul Hough.[3] The transform was popularized in the computer vision community by Dana H. Ballard through a 1981 journal article titled "Generalizing the Hough transform to detect arbitrary shapes".

Theory

In automated analysis of digital images, a subproblem often arises of detecting simple shapes, such as straight lines, circles or ellipses. In many cases an edge detector can 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 as y = mx + b and can be graphically plotted for each pair of image points (x,y). In the Hough transform, a main idea is to consider the characteristics of the straight line not as image points x or y, but in terms of its parameters, here the slope parameter m and the intercept parameter b. Based on that fact, the straight line y = mx + b can be represented as a point (b, m) in the parameter space. However, one faces the problem that vertical lines give rise to unbounded values of the parameters m and b. For computational reasons, it is therefore better to parameterize the lines in the Hough transform with two other parameters, commonly referred to as r and θ (theta). The parameter r represents the distance between the line and the origin, while θ is the angle of the vector from the origin to this closest point (see Coordinates). Using this parametrization, the equation of the line can be written as[4]

$y = \left(-{\cos\theta\over\sin\theta}\right)x + \left({r\over{\sin\theta}}\right)$,

which can be rearranged to r = xcosθ + ysinθ (Shapiro and Stockman, 304).

It is therefore possible to associate to each line of the image, a couple (r,θ) which is unique if $\theta \in [0,\pi]$ and $r \in \mathbf{R}$, or if $\theta \in [0,2\pi]$ and $r \geq 0$. The (r,θ) plane is sometimes referred to as Hough space for the set of straight lines in two dimensions. This representation makes the Hough transform conceptually very close to the two-dimensional Radon transform.

An infinite number of lines can pass through a single point of the plane. If that point has coordinates (x0,y0) in the image plane, all the lines that go through it obey the following equation:-

$r(\theta) = x_0\cdot\cos \theta+y_0\cdot\sin \theta$

This corresponds to a sinusoidal curve in the (r,θ) plane, which is unique to that point. If the curves corresponding to two points are superimposed, the location (in the Hough space) where they cross correspond to lines (in the original image space) that pass through both points. More generally, a set of points that form a straight line will produce sinusoids which cross at the parameters for that line. Thus, the problem of detecting colinear points can be converted to the problem of finding concurrent curves.[5]

Implementation

The Hough transform algorithm uses an array, called accumulator, to detect the existence of a line y = mx + b. The dimension of the accumulator is equal to the number of unknown parameters of the Hough transform problem. For example, the linear Hough transform problem has two unknown parameters: m and b. The two dimensions of the accumulator array would correspond to quantized values for m and b. For each pixel and its neighborhood, the Hough transform algorithm determines if there is enough evidence of an edge at that pixel. If so, it will calculate the parameters of that line, and then look for the accumulator's bin that the parameters fall into, and increase the value of that bin. By finding the bins with the highest values, typically by looking for local maxima in the accumulator space, the most likely lines can be extracted, and their (approximate) geometric definitions read off. (Shapiro and Stockman, 304) The simplest way of finding these peaks is by applying some form of threshold, but different techniques may yield better results in different circumstances - determining which lines are found as well as how many. Since the lines returned do not contain any length information, it is often next necessary to find which parts of the image match up with which lines. Moreover, due to imperfection errors in the edge detection step, there will usually be errors in the accumulator space, which may make it non-trivial to find the appropriate peaks, and thus the appropriate lines.

Example:

#### Figure 4: Lines extracted with the HT.

References
1. ^ Shapiro, Linda and Stockman, George. "Computer Vision," Prentice-Hall, Inc. 2001
2. ^ Duda, R. O. and P. E. Hart, "Use of the Hough Transformation to Detect Lines and Curves in Pictures," Comm. ACM, Vol. 15, pp. 11–15 (January, 1972)
3. ^ P.V.C. Hough, Machine Analysis of Bubble Chamber Pictures, Proc. Int. Conf. High Energy Accelerators and Instrumentation, 1959
4. ^ "Use of the Hough Transformation to Detect Lines and Curves in Pictures".
5. ^ "Hough Transform".
6. ^ Fernandes, L.A.F. and Oliveira, M.M., "Real-time line detection through an improved Hough transform voting scheme," Pattern Recognition, Elsevier, Volume 41, Issue 1, pp. 299–314 (January, 2008).
7. ^ Vosselman, G., Dijkman, S: "3D Building Model Reconstruction from Point Clouds and Ground Plans", International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, vol 34, part 3/W4, October 22-24 2001, Annapolis, MA, USA, pp.37- 44.
8. ^ Tahir Rabbani: "Automatic reconstruction of industrial installations - Using point clouds and images", page 43-44, Publications on Geodesy 62, Delft, 2006. ISBN-13: 978 90 6132 297 9 http://www.ncg.knaw.nl/Publicaties/Geodesy/62Rabbani.html
9. ^ Tahir Rabbani and Frank van den Heuvel, "Efficient hough transform for automatic detection of cylinders in point clouds" in Proceedings of the 11th Annual Conference of the Advanced School for Computing and Imaging (ASCI '05), The Netherlands, June 2005.
10. ^ Image Transforms - Hough Transform
(All data and Pictures obtained from Wiki and www.ddj.com)