Geometric Analysis of Estimability of Target Object Shape Using Location-Unknown Distance Sensors

We geometrically analyze the problem of estimating parameters related to the shape and size of a 2-D target object on the plane by using randomly distributed distance sensors whose locations are unknown. Based on the analysis using geometric probability, we discuss the estimability of these parameters: which parameters we can estimate and what conditions are required to estimate them. For a convex target object, its size and perimeter length can be estimated, and other parameters cannot be estimated. For a general polygon target object, convexity, in addition to its size and perimeter length, can be estimated. Parameters related to a concave vertex can be estimated when some conditions are satisfied. We also propose a method for estimating the convexity of a target object and the perimeter length of the target object.