Reconstruction of r-Regular Objects from Trinary Images
This addresses a specific problem in digital image processing and computational geometry for reconstructing objects from imperfect data, but it appears incremental as it builds on known concepts of r-regularity and trinary images.
The paper tackles the problem of reconstructing r-regular objects from trinary images, where pixels are classified as black, white, or grey based on their position relative to the object, and shows that the reconstructed object is close to the original in Hausdorff norm and homeomorphic to it.
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.