Hough transforms can be understood in terms of projections, since each column of the Hough space corresponds to the projection of the binary image along the angle respective to that column. Such a correspondence is clarified by formulating the Hough transform in terms of intervals [1-3], as illustrated in this equation.
Assuming the discrete lines in the spatially quantized space to follow the grid-intersect quantization scheme [2-3], the suitable Hough mapping is given below [2-3], where q, r and w are the normal distance angle, normal distance and the width of the Hough scanning window:
Each of the two points A and B in Figure 1(a) produces a sinuous band in the Hough space in Figure 1(b). Any of the points inside the intersection between the sinuous bands, e.g. the green point P in Figure 1(b), will define a straight band in the original image that comprehends both points A and B.
It is clear that every cell in spatially quantized Hough spaces will correspond to the number of points inside the respective straight band defined in the original image). Thus, each column in the Hough space represents the projection of the objects in the binary image along the orientation respective to that column.
For convex shapes, the original shape can be recovered from its projections as illustrated by the following example:
The Hough envelope is henceforth defined to be the set of two vectors S and I (standing for superior and inferior) defined by the oriented distances from the superior and inferior boundaries of the used region of the discrete Hough space, with respect to a reference baseline B; The baseline B (a sinusoidal in the Hough space) corresponds to the Hough transform of a reference point in the object. Thus, the Hough envelope of a translated shape (along the parameter r) can be straighforwardly obtained in terms of the algebric sum between the original envelope and the translation value.
The Hough envelope provides the extent of the image projections according to the orientations represented in the discrete Hough space (i.e. the accumulator array), with respect to the adopted reference point.
The concept of Hough envelopes is illustrated in the following:
