A combination of RANSAC and DBSCAN methods for solving the multiple geometrical object detection problem

In this paper we consider the multiple geometrical object detection problem. On the basis of the set A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} containing data points coming from and scattered among a number of geometrical objects not known in advance, we should reconstruct or detect those geometrical objects. A new efficient method for solving this problem based on the popular RANSAC method using parameters from the DBSCAN method is proposed. Thereby, instead of using classical indexes for recognizing the most appropriate partition, we use parameters from the DBSCAN method which define the necessary conditions proven to be far more efficient. Especially, the method is applied to solving multiple circle detection problem. In this case, we give both the conditions for the existence of the best circle as a representative of the data set and the explicit formulas for the parameters of the best circle. In the illustrative example, we consider the multiple circle detection problem for the data point set A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} coming from 5 intersected circles not known in advance. The method is tested on numerous artificial data sets and it has shown high efficiency. The comparison of the proposed method with other well-known methods of circle detection in real-world images also indicates a significant advantage of our method.