# Topological alterations of 3D digital images under rigid transformations

Published in HAL (research report), 2014

Recommended citation: Pluta K., Kenmochi Y., Passat N., Talbot H., Romon P. (2016) Topological alterations of 3D digital images under rigid transformations. HAL, https://hal.archives-ouvertes.fr/hal-01333586

Author(s): Pluta K., Kenmochi Y., Passat N., Talbot H. and Romon P.

Abstract: Rigid transformations in $\mathbb{R}^n$ are known to preserve the shape, and are often applied to digital images. However, digitized rigid transformations, defined as digital functions from $\mathbb{Z}^n$ to $\mathbb{Z}^n$ do not preserve shapes in general; indeed, they are almost never bijective and thus alter the topology. In order to understand the causes of such topological alterations, we first study the possible loss of voxel information and modification of voxel adjacencies induced by applications of digitized rigid transformations to 3D digital images. We then show that even very simple structured images such as digital half-spaces may not preserve their topology under these transformations. This signifies that a simple extension of the two-dimensional solution for topology preservation cannot be made in three dimensions.

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