Repairing 3D binary images using the BCC grid with a 4-valued combinatorial coordinate system

Abstract

A 3D binary image I is called well-composed if the set of points in the topological boundary of the cubes in I is a 2-manifold. Repairing a 3D binary image is a process which produces a well composed image (or a polyhedral complex) from the non-well-composed image I.We propose here to repair 3D images by associating the Body-Centered Cubic grid (BCC grid) to the cubical grid. The obtained polyhedral complex is well composed, since two voxels in the BCC grid either share an entire face or are disjoint. We show that the obtained complex is homotopy equivalent to the cubical complex naturally associated with the image I.To efficiently encode and manipulate the BCC grid, we present an integer 4-valued combinatorial coordinate system that addresses cells of all dimensions (voxels, faces, edges and vertices), and allows capturing all the topological incidence and adjacency relations between cells by using only integer operations.We illustrate an application of this coordinate system on two tasks related with the repaired image: boundary reconstruction and computation of the Euler characteristic.