Pixel connectivity

In image processing, pixel connectivity is the way in which pixels in 2-dimensional (or hypervoxels in n-dimensional) images relate to their neighbors.In order to specify a set of connectivities, the dimension N and the width of the neighborhood n, must be specified.A common width is 3, which means along each dimension, the central cell will be adjacent to 1 cell on either side for all dimensions.represent a N-dimensional hypercubic neighborhood with size on each dimension ofrepresent a discrete vector in the first orthant from the center structuring element to a point on the boundary ofrepresent a N-dimensional hypersphere with radius ofDefine the amount of elements on the hypersphere, E will be equal to the amount of permutations ofmultiplied by the number of orthants.represent the amount of elements in vectorThe total number of permutation ofis shared in common between orthants.Because of this, the multiplying factor on the permutation must be adjusted fromMultiplying the number of amount of permutations by the adjusted amount of orthants yields, Let V represent the number of elements inside of the hypersphereV will be equal to the number of elements on the hypersphere plus all of the elements on the inner shells.Assume the ordered vectorsare assigned a coefficient p representing its place in order.Subtracting 1 yields the neighborhood connectivity, GTherefore, Which matches the supplied table The assumption that allare unique does not hold for higher values of k & N. ConsiderThis means that specification of a given d could refer to multipleThese pixels are connected horizontally and vertically.6-connected pixels are neighbors to every pixel that touches one of their corners (which includes pixels that touch one of their edges) in a hexagonal grid or stretcher bond rectangular grid.There are several ways to map hexagonal tiles to integer pixel coordinates.These pixels are connected horizontally, vertically, and diagonally.These pixels are connected along one of the primary axes.18-connected pixels are neighbors to every pixel that touches one of their faces or edges.These pixels are connected along either one or two of the primary axes.26-connected pixels are neighbors to every pixel that touches one of their faces, edges, or corners.These pixels are connected along either one, two, or all three of the primary axes.