Boundary and shape of binary images

TL;DR

This paper investigates the geometric properties of binary images with known boundary length and area, providing bounds on the largest connected component and largest inscribed ball, enhancing understanding of shape constraints.

Contribution

It introduces sharp bounds for the largest connected component and inscribed ball size in binary images based on boundary and area measurements.

Findings

Sharp lower bounds for the largest connected component size.

Results on the size of the largest inscribed ball in binary images.

Analysis applicable to both simply connected and general components.

Abstract

In this paper we will consider an unknown binary image, of which the length of the boundary and the area of the image are given. These two values together contain some information about the general shape of the image. We will study two properties of the shape in particular. Firstly, we will prove sharp lower bounds for the size of the largest connected component. Secondly, we will derive some results about the size of the largest ball containing only ones, both in the case that the connected components are all simply connected and in the general case.