Frugal random exploration strategy for shape recognition using statistical geometry

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Frugal random exploration strategy for shape recognition using statistical geometry

Samuel Hidalgo-Caballero, Alvaro Cassinelli, Emmanuel Fort, and Matthieu Labousse

Phys. Rev. Research 6, 023103 – Published 26 April 2024

Abstract

Very distinct strategies can be deployed to recognize and characterize an unknown environment or a shape. A recent and promising approach, especially in robotics, is to reduce the complexity of the exploratory units to a minimum. Here, we show that this frugal strategy can be taken to the extreme by exploiting the power of statistical geometry and introducing different invariant features. We show that an elementary robot devoid of any orientation or location system, exploring randomly, can access global information about an environment such as the values of the explored area and perimeter. The explored shapes are of arbitrary geometry and may even nonconnected. From a dictionary, this most simple robot can thus identify various shapes such as famous monuments and even read a text.

Received 9 January 2024
Accepted 20 March 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.023103

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Ballistic transport Search strategy

Statistical Physics & Thermodynamics

Authors & Affiliations

Samuel Hidalgo-Caballero ^1,2, Alvaro Cassinelli³, Emmanuel Fort², and Matthieu Labousse ^1,*

¹Gulliver, CNRS, ESPCI Paris, Université PSL, 75005 Paris, France, European Union
²Institut Langevin, ESPCI Paris, Université PSL, CNRS, 75005 Paris, France, European Union
³School of Creative Media City, City University of Hong Kong,18 Tat Hong Ave, Kowloon Tong, Hong Kong

^*Corresponding author: matthieu.Labousse@espci.fr

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Vol. 6, Iss. 2 — April - June 2024

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Images

Figure 1
(a) Simple and (b) complex nonconvex domain Ω intersected by a nonoriented line $L$ . Two points $P_{1}$ with curvilinear abscissa $l_{1}$ and $P_{2}$ with abscissa $l_{2}$ inside the domain belonging to a line $L$ are plotted. This line has (a) two entries in ( $i_{1}, i_{2}$ ) and two exits ( $o_{1}, o_{2}$ ) or (b) $N$ entries in ( $i_{1}, ..., i_{N})$ and $N$ exits in ( $o_{1}, ..., o_{N})$ .
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Figure 2
Principle and implementation of robotic shape recognition by random exploration. (a) Photo and structure of the autonomous Sphero toy robot. (b)–(d) Exploration of a projected Eiffel Tower shape (b) by random straight-line motion acquiring only the chord lengths. Using statistical geometry to compute invariant quantities and shape parameters such as the perimeter $P$ and the area $A$ of the explored shape enables recognition from a dictionary. Snapshots of accumulated experimental light blinks emitted by the robot during its exploration of the Eiffel Tower for $N = 40, 200,$ and 1000 lines (b) and for a triangle (c), the Chichen Itzá pyramid (d), and the Statue of Liberty for $N = 1000$ lines (e). (f) Chord length probability density function (PDF) for the Eiffel Tower shape with the mean chord length invariant ${〈 ℓ 〉}_{chords}$ and $A^{1 / 2} = {(\frac{π}{3} \frac{{〈 L_{3} 〉}_{lines}}{{〈 L_{1} 〉}_{lines}})}^{1 / 2}$ for comparison. (g) Trajectories followed by the estimated parameters ( $P, A)$ as the number of lines $N$ increases (increasing line contrast) in the perimeter-area representation space for various explored shapes. 75%, 95%, and 99% statistical confidence limits associated to each shape estimated by Monte Carlo simulations for $N = 1000$ (ellipses with increasing contrast and decreasing size).
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Figure 3
Convergence properties. Common color code: Circular shape (pink), Chichén Itzá pyramid (green), triangle (brown), Statue of Liberty (blue), Eiffel Tower (orange). (a) Probability density function of chords for the four remaining shapes. The normalized mean chord length invariant ${〈 ℓ 〉}_{chords}$ and ${\tilde{A}}^{1 / 2} = {(\frac{π}{3} \frac{{〈 {\tilde{L}}_{3} 〉}_{lines}}{{〈 {\tilde{L}}_{1} 〉}_{lines}})}^{1 / 2}$ for comparison. Lengths are normalized by the maximum chord length for each shape and are indicated by a tilde sign. (b),(c) Evolution of the area and perimeter estimated autonomously by the robot during its exploration. Thick lines correspond to an experimental realization, thinner lines correspond to Monte Carlo simulations and give an indication of the reduction of the fluctuations over different realizations. (d),(e) Evolution of the standard deviation of the surface and perimeter computed from 1000 independent Monte Carlo simulations as the function the number of lines and follows a 1/2 power law convergence $σ_{A} ≃ σ_{A, 0} / \sqrt{N}$ and $σ_{P} ≃ σ_{P, 0} / \sqrt{N}$ . (f) Evolution of the prefactor $σ_{A, 0}$ or $σ_{P, 0}$ with the shape entropy of the domain defined as the Kullback-Leibler divergence (KL) of the chord distribution. The dashed grey line is a guide for the eye. The chord distribution of a circular shape is taken as a reference for the KL calculation. (g),(i) Likelihood landscape in the representation space for $N = 200$ (g) and $N = 1000$ (i). The borders indicate a 95% of likelihood of recognition and strongly depend on the dictionary entries. (h),(j) Relative number of lines contributing to a 95% likelihood of recognition for $N = 200$ (h) and $N = 1000$ (i).
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Figure 4
Global versus local exploration strategies applied to reading. (a) Likelihood landscape in the {Perimeter-Area} representation for the projected capital letters of the alphabet with a likelihood larger than 95%. (b) Specifies the number of lines required to distinguish a given letter without ambiguity, in this case with more than 95% of probability. (c),(d) Global strategy. The dictionary is composed of words in the {Perimeter-Area} representation space. (c) Words sampled from the preamble of the United Nations charter as a dictionary [33] with the associated 95% likelihood domains. The words can be distinguished if there is an absence of anagrams. (d) Number of lines, normalized by the of letter in the word, required to distinguish a given letter with a probability larger than 95%. (e),(f) Comparison of both strategies applied to the word “FREEDOM” with successive identification of each letter (e) and the global estimate (f) of the word perimeter and area. (g),(h) Area estimation of the word FREEDOM and its associated error, with a local (g) or a global (h) strategy. With this moderately small pool of words, the two strategies are of about the same precision.
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