From Hough transform to integral geometry [image processing]

The Hough transform (HT) is a classical tool for multiple alignment detection in image processing, based on the property that Aligned Points are transformed into Intersecting Curves (APIC). Among the alternative transforms which possess the APIC property, one of the most interesting is polar transform (PT) which exchanges a point (a, b) and the straight line with equation ax+by=1. This transform represents classical duality between a pole and its polar line with respect to the unit circle. Another property common to both HT and PT is the correspondence between the shapes´ Boundary Length and connected Areas (BLA property), either directly or compensated by a weight function, allowing an efficient measure of these lengths on a digital screen. PT is shown to be connected to projective and integral (or stochastic) geometry with an important role given to a weight function 1/d³.