The algorithm consists of blurring the measured image with
a regularizing smoothing function . There are several possible
choices of the function , however this project isn't concerned
with finding an optimal ; we will use the simple unitary Gaussian
function in three dimensions2
Now we have the problem that Equation 10 becomes unreliable when becomes small. The sophisticated way to avoid this problem is to use an averaging filter. A suitable averaging term is for an arbitrarily suitable weighting term. is the sum of the neighbourhood in , where usually is taken as the 4-element neighbourhood of and . The averaging procedure is then
In order to simplify things somewhat, we used the observation that the most common pattern under which is for thin, two dimensional spherical shells3. Thus, the flood-filling technique is excessive and would rarely come into play in normal images. Our solution will sacrifice some image quality in order to decrease the complexity of this stage of the algorithm.
We adopt basically the same strategy as above, except that we define Equation 11 to be valid for , for some suitably close to 0. We then process the image iteratively, not recursively, and if a position with is encountered we simply continue by assigning and continuing as in Equation 11.