Is the radial distance really a distance? An analysis of its properties and interest for the matching of polygon features

Abstract

In this paper, we examine the properties of the radial distance which has been used as a tool to compare the shape of simple surfacic objects. We give a rigorous definition of the radial distance and derive its theoretical properties, and in particular under which conditions it satisfies the distance properties. We show how the computation of the radial distance can be implemented in practice and made faster by the use of an analytical formula and a Fast Fourier Transform. Finally, we conduct experiments to measure how the radial distance is impacted by perturbation and generalization and we give abacuses and thresholds to deduce when buildings are likely to be homologous or non-homologous given their radial distance.

Keywords:

• Radial distance
• shape similarity
• polygon data matching
• robustness
• error propagation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data and codes availability statement

The data and code that support the findings of this study are available through the following private link: https://doi.org/10.5281/zenodo.7006944

All the authors participated to the conceptualization, methodology, validation and writing tasks. A.M.O.R produced the state of the art. A.L.G and Y.M. conducted the formal analysis. Codes and scripts were written by I.M.A and Y.M. Eventually, I.M.A. and A.M.O.R. prepared and anotated the data.

Notes

1 Here, by maximal practical value, we may refer to the 95th percentile value, which would directly correspond to the 5% risk rejection threshold for H₀.

2 Always being an integer multiple of 4 ensures that all closed curves have perfectly symmetrical north and south branches, and then that their center of mass is invariably located at the origin even after damping selection by α_n.

3 The upper bound $\sqrt{2}$ is approached with signatures that have arbitrary large values and, thus arbitrary small support, which enables arbitrary small value of the inner product $〈·|·〉$ even with the optimal shift. When we constrain the radial signatures to be bounded by M, their support have a minimum span and an optimal shift can make them overlap and necessarily lowers the upper bound.

Additional information

Funding

This work was supported by the French National Mapping Agency: Institut National de l’Information Géographique et Forestière (IGN) and by the University of Djibouti.

Notes on contributors

Yann Méneroux

Yann Méneroux is a researcher at the French National Mapping Agency. His research interest focuses on GPS trajectories mainly through terrestrial vehicle trajectography with sensor fusion, and usage of collected GPS traces for map construction. His research extends as well to noise analysis and modeling error propagation in applications relying on geographic data.

Ibrahim Maidaneh Abdi

Ibrahim Maidaneh Abdi graduated as a State Engineer in Geoinformation, teaches Geomatics at the University of Djibouti and is currently completing his PhD at the IGN on the” Evaluation of the quality of the OpenStreetMap database with machine learning: Case of the Republic of Djibouti”. His main contribution is the proposal of a general framework allowing to infer the extrinsic quality of a building dataset based only on an intrinsic evaluation, in the context of the absence of reference in most countries in Africa.

Arnaud Le Guilcher

Arnaud Le Guilcher is a researcher in GIS at the LASTIG laboratory. His research interest include data quality, data enrichment, propagation of uncertainties in complex problems involving geographical data, and anonymization, with a specific focus on VGI data.

Ana-Maria Olteanu-Raimond

Ana-Maria Olteanu-Raimond is senior researcher in GIS and co-director of LASTIG laboratory. Her research interests include the integration of heterogeneous spatial data, imperfect information fusion, data matching, and collaborative mechanisms and qualification of VGI for joint use with authoritative spatial data. She is also conducting research on citizen science: data collection, platforms, tasks, motivation and sustainability.