This paper describes a new robust regular polygon detector. The regular polygon transform is posed as a mixture of regular polygons in a five dimensional space. Given the edge structure of an image, we derive the a posteriori probability for a mixture of regular polygons, and thus the probability density function for the appearance of a mixture of regular polygons. Likely regular polygons can be isolated quickly by discretising and collapsing the search space into three dimensions. The remaining dimensions may be efficiently recovered subsequently using maximum likelihood at the locations of the most likely polygons in the subspace. This leads to an efficient algorithm. Also the a posteriori formulation facilitates inclusion of additional a priori information leading to real-time application to road sign detection. The use of gradient information also reduces noise compared to existing approaches such as the generalised Hough transform. Results are presented for images with noise to show stability. The detector is also applied to two separate applications: real-time road sign detection for on-line driver assistance; and feature detection, recovering stable features in rectilinear environments.