HOUGH ALGORITHM USING EDGE INFORMATION A CLOSER LOOK

HOUGH ALGORITHM USING EDGE INFORMATION A CLOSER LOOK October 5, 1998 1

Hough Algorithm: So far we have seen…. . • Choose an analytic form f(x, y, a 1, a 2, …, an) and choose a range of values for parameters a 1, a 2, a 3, …. , an. • Create accumulator array A(a 1, a 2, …, an) which represents direct match of f(x, y, a 1, a 2, …, an) with binary image. • Local for local maximum which exceeds certain threshold. October 5, 1998 2

Hough Algorithm: Consider this simple modification…. . IDEA ! • Instead of looking at the matching of f(x, y, a 1, a 2, …, an) by incrementing through all (x, y), and, n parameters, suppose we only look at edge pixels in binary image and cast votes for a pixel (x, y) based upon whether f(x, y, a 1, a 2, …, an) passes through the edge point with the same edge orientation. October 5, 1998 3

Hough Algorithm: Revisited An Example…. . f(x, y, a, b, r)=0 f(x, y, a 1, a 2, a 3)=0 (x-a)^2 + (y-b)^2 - r^2 = 0 CIRCLE Edge Orientation from implicit differentiation 2(x-a) + 2(y-b) (dy/dx) = 0 r (a, b) dy/dx = -(x-a)/(y-b) October 5, 1998 4

Hough Algorithm: Revisited Edges Image October 5, 1998 5

Hough Algorithm: Revisited Edges Image October 5, 1998 6

Modified Hough Algorithm (More Efficient) • Note that for p-parameters a 1, a 2, …, ap containing 2 translation parameters that the accumulator array is now (p -2) dimensional. • Also note reduction in the number of pixels that are effectively visited: – Instead of examining all pixels as a possible ‘center’ for the shape we now only examine edge points as possible boundary points for the shape. October 5, 1998 7

GENERALIZED HOUGH ALGORITHM • In general many important shapes do not have an analytic form f(x, y, a 1, a 2, …, an). • How do we incorporate general shape matching into this modified scheme for the Hough Algorithm ? • ANSWER: Use an R-Table to represent a general shape relating edge orientation, f, at a boundary point (Bx, By) to the ‘center’ (Xc, Yc) using vector displacement, r=(Xc-Bx, Yc-By). October 5, 1998 8

GENERALIZED HOUGH ALGORITHM R-TABLE f 1 f 2 (Xc, Yc) f October 5, 1998 f 1 n r 1, r 2, …, , rn 9

Generalized Hough Algorithm Edges October 5, 1998 Image 10