It had to be thought of. The authors describe transformations in Hough space that are analogous to the shifts, translations, and scale changes in Fourier transform and Laplace transform spaces and add an inverse transform. These seem to make it easy to detect an arbitrary curve even when the parametric form of that curve is unknown.

Curves are represented by straight-line segments. For example, a circle is represented by the set of tangents; since the curves in question are represented on a discrete raster and have binary values, the set of tangents is finite. Every pair of points on the curve will transform to one point of weight 2 in Hough space; the points that do not lie on the Hough locus for the original curve can then be eliminated by a threshold operation. By translation, rotation, and scaling in the thresholded Hough space, the Hough locus can then be brought to standard form. This form is of course dependent on the shape of the original curve and can be used to find the parametric form of the original curve, where that is unknown. This is valid even for curves not given by analytical expressions. The transformations in Hough space are designed to bring the unknown object into a symmetrical position with respect to the origin.

For example, circles are brought to center at the origin; the transformations in Hough space depend on the radius. Ellipses are also brought to center at the origin, with the major axis rotated to lie parallel to one orthogonal axis.

An inverse Hough transform, which takes into account the transformations that have previously been executed, is defined to find the position of the original curve. For example, the centers of circles of now known radii are found, and the centers of ellipses of now known orientation and lengths of major and minor axes are found.

According to the authors, “The successive Hough space transformations [can] be used to determine and parametrize lines and curves in the input scene.”