Ensemble computation approach to the Hough transform

TL;DR

This paper presents an ensemble computation approach to analyze the complexity of the classical Hough transform, demonstrating it can be optimized similarly to the fast Hough transform using divide-and-conquer techniques.

Contribution

It introduces a constructive ensemble computation method that provides complexity bounds for the classical Hough transform and relates it to the fast Hough transform, suggesting potential circuit size improvements.

Findings

Classical Hough transform has at most $ ext{O}(n^3 / ext{log} n)$ complexity.

The ensemble approach yields the same asymptotics as the fast Hough transform for certain patterns.

Potential for smaller circuit sizes than previously thought for the classical Hough transform.

Abstract

It is demonstrated that the classical Hough transform with shift-elevation parametrization of digital straight lines has additive complexity of at most $\mathcal{O}(n^3 / \log n)$ on a $n\times n$ image. The proof is constructive and uses ensemble computation approach to build summation circuits. The proposed method has similarities with the fast Hough transform (FHT) and may be considered a form of the "divide and conquer" technique. It is based on the fact that lines with close slopes can be decomposed into common components, allowing generalization for other pattern families. When applied to FHT patterns, the algorithm yields exactly the $\Theta(n^2\log n)$ FHT asymptotics which might suggest that the actual classical Hough transform circuits could smaller size than $\Theta(n^3/ \log n)$.