Erosion and Dilation on 2-D and 3-D Digital Images: A New Size-Independent Approach

Abstract This paper presents a new approach to achieve ele-mentary neighborhood operations on both 2D and3D binary images by using the Extreme VerticesModel (EVM),a recent orthogonal polyhedra repre-sentation applied to digital images. The operationsdeveloped here are erosion and dilation. In contrastwith previous techniques, this method do not use avoxel-based approach but deal with the inner sec-tions of the object. It allows to build an image size-independent algorithm. The proposed method alsoadmits the use of structuring elements of arbitrarysize and allows to treat 2D and 3D images in iden-tical way using the same algorithm. 1 Introduction A 3D digital image (or volume dataset) can be rep-resented as a map , in such away that every point is assigned a value represent-ing its color. In a binary image the image set is  . In [10] a 3D digital image is defined as anunion of voxels i.e. upright unit cubes whose ver-tices have integer coordinates. Analogously, a 2Ddigital image is defined as a union of pixels. In or-der to generalize this term, we will call the elementsin a n-dimensional digital image n-voxels.The most extensive class of binary images pro-cessing operations is sometimes collectively de-scribed as morphological operations [5]. This in-cludes erosion and dilation which are the base ofmost morphological operations such as opening,closing or hit-or-miss transform. The operations areused in several tasks such as elimination of smallspurious objects, smoothing of object boundaries,