A Hough Transform Approach to Solving Linear Min-Max Problems

We begin with a quick overview of a natural extension to the Hough Transform Hough (1962) to 3D. We note that the following theorems, although proved for 3D, hold equally well in 2D; the proofs are identical if we rename the z coordinate into y. Given a point p = (a, b, c) in R, we define its dual plane as H (p) = ` (x, y) = ax + by − c, and given a plane π (x, y) = ax + by + c, we define its dual point as H (π) = (a, b,−c). The usefulness of these definitions is highlighted in the following lemmas.