Talk:Image segmentation
It seems to me that connectedness for a digital image is rather distinct from just general "connected." I'd be willing to write up something about this, but I'm not sure if it would be seen as a little too specialized? Is this definition of connectedness too esoteric to add to the "connected" page? Wegsjac 04:23, 5 Apr 2005 (UTC)
Can you explain the variables please
If someone is reading this to learn (the only reason to read it that I know of) it might be a good idea to explain what each symbol means in the formula. I think a trend might exist to use the same name across image processing literature, but we uninitiated may not know what mu, Gamma, f, and g represent. Is v for variance? is mu some sort of mean? ar f and g from the fundamental form? or what?
Mumford-Shah
I did some searching and Mumford-Shah appears in a lot of journal papers but I didn't find much reference to it on the web itself, which implies to me that it's a more advanced technique that isn't all that relevant to a general article on image processing segmentation. It also doesn't explain its variables (as mentioned above), and it seems to do more to confuse than to educate. So I'm removing it to here. If someone has any idea what it is, perhaps they could create an article for it, rather than mushing into this article? -- Zawersh 16:18, 15 May 2006 (UTC)
It is true that the Mumford Shah functional may be a little sophisticated for a general audience. Those variables are not, to my knowledge, used in image analysis across peer-reviewed papers. They are specific to this functional.
Here is some discussion of the functional as defined: f is a piecewise smooth approximation to the data, g. That is, f is a smoothed version of g that also preserves the main edges in g. The set of "segmenting curves", Gamma, typically run along the edges. Those curves are the only locations across which large gradients can develop (are not penalized). The middle term is the "smoothness constraint" that penalizes gradients in f except at edges. The notation R-Gamma means the integral of the gradients of the whole image (R) except where those gradients cross the set of curves that segment the image (Gamma). The mu and nu values are just constants that adjust the relative weights in the functional. These are usually chosen empirically.
So, as approaches infinity, the piecewise constant image, f, approaches the data, g, and the segmenting curves become noise or inconsequential. As nu approaches 0, the functional will put segmenting curves everywhere and again f will approach the data g (because in that case smoothness enforced nowhere); in this case the functional value goes to zero. Without the middle term, the piecewise constant image, f, decouples from the segmenting curves, Gamma, so f goes to g, Gamma goes to the empty set, and the functional value goes to 0. None of these are particularly interesting. However, when these parameters are chosen judiciously, segmentations that "look right" can be found.
A discussion of the mathematics and the generality of the Mumford Shah functional can be found in "Variational methods in image segmentation" by J.M.Morel and S.Solimini. Another discussion of its history from the 1985 paper up until 2000 can be found in "Energy Formulations of Medical Image Segmentations" by J. Kaufhold.
Removed from article:
An example of a global segmentation criterion is the famous Mumford-Shah functional. This functional measures the degree of match between an image and its segmentation. A segmentation consists of a set of non-overlapping connected regions (the union of which is the image), each of which is smooth and each of which has a piecewise smooth boundary. The functional penalizes deviations from the original image, deviations from smoothness within in each region and the total length of the boundaries of all the regions. Mathematically,
