In theory, the Gaussian function at every point on the image will be non-zero, meaning that the entire image would need to be included in the calculations for each pixel. In practice, when computing a discrete approximation of the Gaussian function, pixels at a distance of more than 3σ are small enough to be considered effectively zero. Thus contributions from pixels outside that range can be ignored. Typically, an image processing program need only calculate a matrix with dimensions ceil(6σ) x ceil(6σ) to ensure a result sufficiently close to that obtained by the entire gaussian distribution.

apart from that, what do you mean by “screen resolution” and why is it 72dpi? that’s pointless to consider. you may consider a formula to cumpute sharpen radius for printed pictures based on resolution, but, when it comes to screen display, resolution means nothing. it’s the zoom level which matters. for 100% zoom (that is, one picture pixel to one screen pixel), my prefered radius is 0.5. from that mark, simply multiply (i.e., for 50% zoom, use half radius). still, choosing radius depending on resolution is not a wise choise. it is better off set according to visual results.

SIGMA in IM == RADIUS in PS/GIMP

sigma, however, is the most important parameter, as it defines the weight of each surrounding pixel. such weight is non zero for ALL pixels, but actually for pixels too distintant from the center, their value means almost nothing and counting them is just a loss of time. that’s what the parameter radous is for. it specifies a boundary to cut off and do not count pixels values. setting this parameter to zero is sesible, as imagemagick will take a value -according to sigma- which is studied to be small enough to reduce processing while obtaining very good (quasi optimal) results.

The unsharp mask filter uses a gaussian blur filter internally. That means the image gets convolved with a gaussian distribution with a standard deviation of sigma. The gaussian distribution has its peak at the center, falling of in an s-curve to boths sides, but never reaching zero. Sigma tells you how broad the distribution is and thus, the amount of blur to be applied. But as the gaussian distribution extends from -infinity to +infinity you cannot calculate the convolution perfectly. This is where the radius parameter comes into play. The radius tells you, how big your kernel for the convolution should be and how much of the gaussian distribution function is actually used. Setting this to something like 2*sigma or higher should give you a reasonably good approximation. Radius is rounded to the next integer, because the kernel used for convolution is made up of whole pixels (as is you digitized image). A kernel size of 0.5 does not make sense.

Now, all that said, the above remarks assume that the image has had correct input sharpening applied to it. If you are not sharpening your images on input (e.g. a RAW file) or you are letting some image display program down sample it for display (without additional sharpening, because downsampling almost always requires additional sharpening), then you may need to increase the sharpening radius to account for the lack of two-pass (input/output) sharpening.

Apologies if I am misinterpreting your use of the ppi formula.