Editing Hough transform (section)

===Hough transform of curves, and its generalization for analytical and non-analytical shapes===
Although the version of the transform described above applies only to finding straight lines, a similar transform can be used for finding any shape which can be represented by a set of parameters. A circle, for instance, can be transformed into a set of three parameters, representing its center and radius, so that the Hough space becomes three dimensional. Arbitrary ellipses and curves can also be found this way, as can any shape easily expressed as a set of parameters.

The generalization of the Hough transform for detecting analytical shapes in spaces having any dimensionality was proposed by Fernandes and Oliveira.<ref>{{cite journal | last1 = Fernandes | first1 = L.A.F. | last2 = Oliveira | first2 = M.M. | year = 2012 | title = A general framework for subspace detection in unordered multidimensional data | doi = 10.1016/j.patcog.2012.02.033 | journal = Pattern Recognition | volume = 45 | issue = 9| pages = 3566–3579 | bibcode = 2012PatRe..45.3566F }}</ref> In contrast to other Hough transform-based approaches for analytical shapes, Fernandes' technique does not depend on the shape one wants to detect nor on the input data type. The detection can be driven to a type of analytical shape by changing the assumed model of geometry where data have been encoded (e.g., [[euclidean space]], [[projective space]], [[conformal geometry]], and so on), while the proposed formulation remains unchanged. Also, it guarantees that the intended shapes are represented with the smallest possible number of parameters, and it allows the concurrent detection of different kinds of shapes that best fit an input set of entries with different dimensionalities and different geometric definitions (e.g., the concurrent detection of planes and spheres that best fit a set of points, straight lines and circles).

For more complicated shapes in the plane (i.e., shapes that cannot be represented analytically in some 2D space), the [[Generalised Hough transform]]<ref>{{cite journal | last1 = Ballard | first1 = D.H. | year = 1981 | title = Generalizing the Hough transform to detect arbitrary shapes | journal = Pattern Recognition | volume = 13 | issue = 2| pages = 111–122 | doi = 10.1016/0031-3203(81)90009-1 | bibcode = 1981PatRe..13..111B | hdl = 1802/13802 | hdl-access = free }}</ref> is used, which allows a [[Feature (computer vision)|feature]] to vote for a particular position, orientation and/or scaling of the shape using a predefined look-up table.The Hough transform accumulates contributions from all pixels in the detected edge.