# Reconstruction of r-Regular Objects from Trinary Images

## TL;DR

This paper presents a method for reconstructing r-regular objects from trinary digital images, ensuring the reconstructed shape closely approximates the original in Hausdorff distance and is topologically equivalent.

## Contribution

It introduces a novel reconstruction technique for r-regular objects from trinary images, with guarantees on shape accuracy and topological correctness.

## Key findings

- Reconstructed objects are close to original in Hausdorff norm.
- A homeomorphism exists between reconstructed and original objects.
- The method effectively handles pixel configurations in trinary images.

## Abstract

We study digital images of r-regular objects where a pixel is black if it is completely inside the object, white if it is completely inside the complement of the object, and grey otherwise. We call such images trinary. We discuss possible configurations of pixels in trinary images of r-regular objects at certain resolutions and propose a method for reconstructing objects from such images. We show that the reconstructed object is close to the original object in Hausdorff norm, and that there is a homeomorphism of the plane taking the reconstructed set to the original.