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image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
, pixel connectivity is the way in which pixels in 2-dimensional (or
hypervoxel In 3D computer graphics, a voxel represents a value on a regular grid in three-dimensional space. As with pixels in a 2D bitmap, voxels themselves do not typically have their position (i.e. coordinates) explicitly encoded with their values. Ins ...
s in n-dimensional) images relate to their neighbors.


Formulation

In order to specify a set of connectivities, the dimension N and the width of the neighborhood n , must be specified. The dimension of a neighborhood is valid for any dimension n\geq1 . 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. Let M_N^n represent a N-dimensional
hypercubic In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perp ...
neighborhood with size on each dimension of n=2k+1, k\in\mathbb Let \vec represent a discrete vector in the first orthant from the center structuring element to a point on the boundary of M_N^n. This implies that each element q_i \in \ ,\forall i \in \ and that at least one component q_i = k Let S_N^d represent a N-dimensional hypersphere with radius of d=\left \Vert \vec \right \Vert. Define the amount of elements on the hypersphere S_N^d within the neighborhood M_N^n as E. For a given \vec, E will be equal to the amount of permutations of \vec multiplied by the number of orthants. Let n_j represent the amount of elements in vector \vec which take the value j. n_j = \sum_^N (q_i=j) The total number of permutation of \vec can be represented by a multinomial as \frac If any q_i = 0 , then the vector \vec is shared in common between orthants. Because of this, the multiplying factor on the permutation must be adjusted from 2^ to be 2^ Multiplying the number of amount of permutations by the adjusted amount of orthants yields, :E=\frac2^ Let V represent the number of elements inside of the hypersphere S_N^d within the neighborhood M_N^n. V will be equal to the number of elements on the hypersphere plus all of the elements on the inner shells. The shells must be ordered by increasing order of \left \Vert \vec \right \Vert = r. Assume the ordered vectors \vec are assigned a coefficient p representing its place in order. Then an ordered vector \vec, p \in \ if all r are unique. Therefore V can be defined iteratively as :V_ = V_ + E_, V_=0, or :V_ = \sum_^p E_ If some \left \Vert \vec \right \Vert = \left \Vert \vec \right \Vert, then both vectors should be considered as the same p such that
V_ = V_ + E_ + E_, V_=0
Note that each neighborhood will need to have the values from the next smallest neighborhood added. Ex. V_ = V_ + E_ V includes the center hypervoxel, which is not included in the connectivity. Subtracting 1 yields the neighborhood connectivity, G :G=V-1


Table of Selected Connectivities


Example

Consider solving for G, \vec=(0,1,1) In this scenario, N=3 since the vector is 3-dimensional. n_0=1 since there is one q_i=0. Likewise, n_1=2. k=1, n=3 since \max q_i = 1. d=\sqrt=\sqrt. The neighborhood is M^3_3 and the hypersphere is S_3^\sqrt :E=\frac2^=\frac4=12 The basic \vec in the neighborhood N^3_3, \vec = (0,0,0). The Manhattan Distance between our vector and the basic vector is \left \Vert \vec - \vec \right \Vert_1 = 2, so \vec = \vec. Therefore, :G_ = V_ - 1 = E_ + E_ + E_ - 1 = E_ + E_ + E_ :E_ = \frac2^ = \frac1 = 1 :E_ = \frac 2^=\frac 2 = 6 :G = 1 + 6 + 12 - 1 = 18 Which matches the supplied table


Higher values of k & N

The assumption that all \left \Vert \vec \right \Vert = r are unique does not hold for higher values of k & N. Consider N=2, k=5, and the vectors \vec=(0,5), \vec=(3,4). Although \vec is located in M_2^5, the value for r=25, whereas \vec is in the smaller space M_2^4 but has an equivalent value r=25. \vec=(4,4) \in M_^ but has a higher value of r=32 than the minimum vector in M_2^5. For the this assumption to hold, \begin N=2, k \leq 4 \\ N=3, k \leq 2 \\ N=4, k \leq 1 \end At higher values of k & N, Values of d will become ambiguous. This means that specification of a given d could refer to multiple \vec \in M_n^N.


Types of connectivity


2-dimensional


4-connected

4-connected pixels are neighbors to every pixel that touches one of their edges. These pixels are connected horizontally and vertically. In terms of pixel coordinates, every pixel that has the coordinates : \textstyle(x\pm1, y) or \textstyle(x, y\pm1) is connected to the pixel at \textstyle(x, y).


6-connected

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. With one method, in addition to the 4-connected pixels, the two pixels at coordinates \textstyle(x+1,y+1) and \textstyle(x-1,y-1) are connected to the pixel at \textstyle(x,y).


8-connected

8-connected pixels are neighbors to every pixel that touches one of their edges or corners. These pixels are connected horizontally, vertically, and diagonally. In addition to 4-connected pixels, each pixel with coordinates \textstyle(x\pm1,y\pm1) is connected to the pixel at \textstyle(x,y).


3-dimensional


6-connected

6-connected pixels are neighbors to every pixel that touches one of their faces. These pixels are connected along one of the primary axes. Each pixel with coordinates \textstyle(x\pm1, y, z), \textstyle(x, y\pm1, z), or \textstyle(x, y, z\pm1) is connected to the pixel at \textstyle(x, y, z).


18-connected

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. In addition to 6-connected pixels, each pixel with coordinates \textstyle(x\pm1, y\pm1, z), \textstyle(x\pm1, y\mp1, z), \textstyle(x\pm1, y, z\pm1), \textstyle(x\pm1, y, z\mp1), \textstyle(x, y\pm1, z\pm1), or \textstyle(x, y\pm1, z\mp1) is connected to the pixel at \textstyle(x, y, z).


26-connected

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. In addition to 18-connected pixels, each pixel with coordinates \textstyle(x\pm1, y\pm1, z\pm1), \textstyle(x\pm1, y\pm1, z\mp1), \textstyle(x\pm1, y\mp1, z\pm1), or \textstyle(x\mp1, y\pm1, z\pm1) is connected to the pixel at \textstyle(x, y, z).


See also

* Grid cell topology * Moore neighborhood


References

* * *{{Citation , url = http://homepages.inf.ed.ac.uk/rbf/HIPR2/connect.htm , title = Subband Weighting With Pixel Connectivity for 3-D Wavelet Coding , year = 2009 , author = , journal = IEEE Transactions on Image Processing , pages = 52–62 , volume = 18 , issue = 1 , doi = 10.1109/TIP.2008.2007067 , isbn = , accessdate = 2009-02-16 , pmid = 19095518 , last1 = Cheng , first1 = CC , last2 = Peng , first2 = GJ , last3 = Hwang , first3 = WL Digital topology Graph connectivity