Re: Describing linear densities and patterns

Hi BC

Sorry for the long delay in replying...

Thanks and good point. The study containers are actual spatial study
areas, or regions on a landscape. These are well defined, but could be
physical (e.g., watershed) or arbitrary boundaries (e.g., geopolitical
boundary). It may be useful to know that the 'lines' are anthropogenic
linear features (e.g., roads, trails, etc.).

Disclaimer: The following is just off the top of my head and without
much thought!

For the density of the curvilinear features, is this not just the
number of such features per unit area? Or perhaps the ratio of the area
occupied by the features to the total area?

For the distribution of the features, I think it depends upon what you
want to do with the information. For example, do you want to be able to
compare two regions, or be able to say something specific about a
particular region etc. One idea might be to parameterise each
curvilinear feature according to some equation (e.g. a polynomial) and
then take the parameters of that equation as a vector. For example, if
you have the polynomial:

y = a0 + a1 x + a2 x^2 + a3 x^3

your vector would be [a0 a1 a2 a3]. That vector can then be considered
to be a point in a 4-D space. The curvilinear features in your region
would then become a set of 4-D vectors, and their distribution might
tell you something useful about the region and may allow you to compare
regions. Note that you may need something other than just a simple
polynomial, and you may need to choose a representation with
appropriate characteristics for the task you have in mind.

Given such a representation, you could then use a density estimation
technique (e.g. a Gaussian Mixture Model or Parzen Windowing) to form
the 'surface' that you mentioned.

In terms of other fields, you might want to look at the medical image
analysis literature, specifically about branching structures such as in
angiography, or the analysis of spiculated (stellate) lesions in x-ray
mammography.

I hope some of that helps.

C

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