i. In regional processing, it makes sense to link a given set of pixels only if we know that they are part of the boundary of a meaningful region.

ii. Often, we have to work with unstructured environments in which all we have is an edge image and no knowledge about where objects of interest might be.

iii. In such situations, all pixels are candidates for linking and thus have to be accepted or eliminated based on predefined global properties.

iv. Thus, an approach had to be developed based on whether sets of pixels lie on curves of a specified shape. Once detected, these curves form the edges or region boundaries of interest.

v. Given n points in an image, suppose that we want to find subsets of these points that lie on straight lines. One possible solution is to find first all lines determined by every pair of points and then find all subsets of points that are close to particular lines.

vi. This approach involves finding $n(n − 1)/2 ~ n^2$ lines and then performing $(n)(n(n − 1))/2 ~ n^3$ comparisons of every point to all lines. This is a computationally prohibitive task in all but the most trivial applications.

vii. Hough [1962] proposed an alternative approach, commonly referred to as the Hough transform.

viii. Consider a point (x, y) in the xy-plane and the general equation of a straight line in slope-intercept form, $yi = ax + b. $

ix. Infinitely many lines pass through (x, y), but they all satisfy the equation $y = ax + b $for varying values of a and b.

x. However, writing this equation as $b = -xi a + yi$ and considering the ab-plane (also called parameter space) yields the equation of a single line for a fixed pair (x, y).

xi. Furthermore, a second point (x, y) also has a line in parameter space associated with it, and, unless they are parallel, this line intersects the line associated with (x, y) at some point (a', b'), where a' is the slope and b' the intercept of the line containing both (x, y) and (x, y) in the xy-plane. In fact, all the points on this line have lines in parameter space that intersect at(a,b).

xii. Figure 2 illustrates these concepts.

Figure 2

xiii. In principle, the parameter-space lines corresponding to all points (xk, yk) in the xy-plane could be plotted, and the principal lines in that plane could be found by identifying points in parameter space where large numbers of parameter-space lines intersect.

xiv. In simple terms, Hough transform works when a set of discrete pixels are given, and it is to be checked whether these pixels lie on a straight line. if the pixels are found to lie on a straight line,then a line is drawn joining all these points with the help of Hough transform.