# Geometry of the Hough transforms with applications to synthetic data

## TL;DR

This paper analyzes the geometric properties of Hough transforms, providing bounds for their application in curve detection, and demonstrates robustness on noisy synthetic data with an algebraic approach for exact cases.

## Contribution

It introduces new bounds for the number of Hough transforms needed and explores their robustness under noise, with an algebraic method for precise scenarios.

## Key findings

- Derived bounds for Hough transform application in curve detection.
- Demonstrated robustness of bounds on noisy synthetic data.
- Presented an algebraic approach for exact case analysis.

## Abstract

In the framework of the Hough transform technique to detect curves in images, we provide a bound for the number of Hough transforms to be considered for a successful optimization of the accumulator function in the recognition algorithm. Such a bound is consequence of geometrical arguments. We also show the robustness of the results when applied to synthetic datasets strongly perturbed by noise. An algebraic approach, discussed in the appendix, leads to a better bound of theoretical interest in the exact case.