Segmentation From 2D to 3D

Using a 2D image, the image is represented initially as a MxN matrix (which is then converted to a (MxN)x(MxN) Laplacian matrix). Each node is connected to the node above it, below it, to the right of it, and to the left of it. These connections are provided using weighted edges. The ultimate representation looks like a grid.

Taking this to the 3D realm, we simple represent the initial image using a MxNxL matrix (which is then converted to a (MxNxL)x(MxNxL) Laplacian matrix).  Each slice of the image would be 1 MxN slice and the entire 3D image would have L slices, again, each consisting of MxN nodes. Each node would then be connected to the node above it, below it, to the right of it, to the left of it, and now, to the front of it and behind it. These connections, again, would be provided using weighted edges and the the would be weighted using the same formula as the 2D image. 

Using this concept, converting a Random Walker algorithm to accomodate 3D images is pretty easy since all that would have to be done is adding MxN matrices to the originally MxNx1 matrix to eventually have an MxNxL matrix; and also adding more edges to each node. 

Again, all the same formulas for the 2D image segmentation would still be applicable.