The Hough Transform I: Finding Lines

I've made reference to the Hough Transform in several previous posts, but I haven't ever done posts on what the transform is in a general sense, and how I've used it (and tried to improve it) in the past. In this series of posts, I'll try to rectify that.

The most basic implementation of the Hough transform is used to find lines in an edge image like the one below.  In this post I'll walk through the basic process of using the Hough transform to find lines in this image.

An image of machinery with edge detection applied. This image would be a good candidate for further processing with the Hough transform. (Source: Wikipedia)

Underlying the Hough transform is the idea that each pixel in an an image tells us something about the chances of there being certain lines in the entire image. If we have a pixel at point p, then it would make sense to look at all the lines that pass through p, and see if any of them happen to pass very close many other points in the image. If a particular line through p passes close to an unusual number of points, then it's a pretty reasonable to assume that this line constitutes an actual feature in the image.

MIT Library Scanning...

...turns out to be a great way to scan figures. It's a ton faster than using my own scanner, and the resolution isn't too much worse. I'm writing a few math-oriented and cs-oriented posts with figures right now, and the book scanner means I can just box figures in my notebooks and use Gimp / Picasa Web Albums to crop things down and clear up the colors. It's a much faster pipeline than what I had before, and I'll hopefully make use of it to put lots of pretty figures in my posts.

Voroni Diagrams and Graphs I: Introduction

Suppose we have a set of points { p₀, p₁, ..., p_n }, each of which lives in R². A Voroni Diagram is a partitioning R² into regions { [p₀], [p₁], ..., [p_n] } so that any point q in [p_i] is at least as close to p_i as any other p_j. Note that, for most points, this should mean that q is strictly closest to p_i, but we're also going to include the boundaries of these regions, so they'll all be closed sets ( If it were to strike your fancy, you could also use open sets. We'll use the closed property a bit later on, but it's mostly personal preference. ).

This is all a perfectly good definition of what Voroni Diagrams are, but it doesn't explain how to find them in practical situations or what they're useful for. In this post, I'll explain a basic approach for finding a Voroni Diagram, and present an (inefficient) algorithm I made up to find what I call the Voroni graph: a graph whose vertices are the { p₀, p₁, ..., p_n }, and whose structure describes the diagram (and provides a quick way to check if q is in a given [p_i], among other things ).