Discrete topology on N-dimensional square tessellated grids
Topology preservation and detection is a well known concept in the processing of 2- and 3-dimensional binary images. These images can be considered as sets that are mapped on square tessellated (hyper cubic) grids. This paper describes how the formalisms derived for 2D and 3D images can be expanded to N-dimensional images, i.e. binary point sets that are aligned on an N -dimensional square tessellated grid. Topology preserving thinning of objects in an image, informally referred to as skeletonization, is based on the successive erosion of the boundary of an object until locally a primitive shape is detected, e.g. in 3D a surface or a curve. The detection of primitive shapes is done using shape primitives. The generation of shape primitive detectors is based on the possibility to describe the primitives for intrinsic or object dimensions N  = N ? 1 by quadratic equations of the form x N = ? ( a n x x + b n x n 2 ) . From this, primitives for lower object dimensions can be derived. A formula is derived that predicts the number of unique shape primitives in each dimension. Their application in measurements on shapes, in conditions for topology detection as used in topology preserving thinning, as well as the determination of standing wave patterns in topological kernels, is described. Finally as on each element of an image at least one of the primitives matches, this can be used to measure the total content of length, area, volume, etc from the objects in the image.
mathematical morphology digital topology n-dimensional shape n-dimensional topological kernels n-dimensional standing waves