Converting point data into a weighted raster

Question

I am completely re-writing my question as i think it was quite confusing before:

I have a number of different points, each point has a -z value. I need to create some sort of raster or a series of polygons which buffers these points.. but i have a few complicated rules!

1. If the adjacent points are of the same z value then the buffer must merge into one polygon.
2. if the adjacent point is a different z value - then the buffers must be weighted in accordance to the depth of the z values, i.e. if one z-value is -10, and the other is -5, then the buffer of the -10 point will take up 75% of the distance between the 2 points. But because there are many points, and they are all related, this buffering will be different on different sides of the point, see diagram!
3. The idea of the weighted voronoi diagram has been suggested to me and that would be perfect if it weren't for all of the points being related to all of the points around its!
4. I know how i would do it on paper - i would divide the space between all points by 10, and then based on the difference between the two points choose where the buffer would start (if there is no difference between the points then the buffer starts in the middle).
5. Although i cannot work out the formula for determining where the buffer starts.. i know it must be derived from the DIFFERENCE between the two points - but i do not have the mathmatical capacity to work out how i would use this to work out the distance!

I am using Arcmap 9.3.1 and am open to python scripting! If you need me to try and explain anything further i will try! i am very confused myself!

Answer 1

I'm unclear on the drawing portion of your question, so I hope I'm not totally off base here, but it seems to me that you could apply a buffer based on the z-value field around each of your points. Take the resulting polygon and use the polygon to raster tool to create your final raster (if you have ArcInfo). If you don't have ArcInfo I believe gdal_rasterize should do the trick.

Note: If your buffer distance around each point isn't directly correlated to your z-value, just add an extra buffer distance field.

ESRI link: http://webhelp.esri.com/arcgiSDEsktop/9.3/index.cfm?TopicName=Polygon_to_Raster_(conversion)

gdal_rasterize link: http://www.gdal.org/gdal_rasterize.html

Answer 2

If I understand your question, an algorithm that may work (i.e. it does in my head, but I've not tried it), and would require some scripting goes something like this:

1. Generate the Voronoi/Theissen polygons of your points.
2. Take the inverse of those polygons, that is there will now be edges connected to your original points.
3. For each edge e connected to point p, calculate the relative distance along e such that it is a proportion of the end points' weights. I.e. w / (w' + w), where w is the weight of p and w' is the weight of p', the other end of e. (If you need the inverse, that is -10 is "heavier" than -2, just subtract this value from 1).
4. Scale e in proportion to the value obtained in 3), and its new end point will mark the weighted boundary between the two points. Associate this boundary point with p. Do this for every edge, remembering to duplicate the new boundary point for p' so you won't need to calculate it a second time.
5. You will now have a ring of points associated with each of your original points. Join these up into polygons which have a weight attribute that is the same as the point that they enclose.
6. Merge all adjacent polygons with the same weight attribute.

You may need to check that each point p has more than two edges radiating from it to avoid degenerate polygons which may happen with corner cases, but I think that should work.

Answer 3

I attempted to solve a similar problem recently - I needed to divide clusters of weighted points into regions. Here is the method that ended up working for me:

1. Compute a kernel density raster to distribute the effect of the various points over space. You'll need to play with the bandwidth to make sure that larger points flow into smaller points in way that suits you.

2. Using the raster calculator, find the inverse of the kernel density raster.

3. Using the method mentioned here by @whuber calculate watersheds.

4. Convert the watersheds raster to polygons, join to your original points, and dissolve polygons with the same z-value.

If your bandwidth is appropriate you should end up with polygons that follow the distribution of the point weights. They won't be as mathematically determined as you mention, but they should be close.

Alternate #2: Why not use weighted voronoi and then dissolve regions with the same z-value? Would that not solve the interaction problem between adjacent points?

Alternate #3: If you truly need mathematical precision, you may need a kernel density calculation with a variable bandwidth related to the z-value. You'd have to write this out yourself, but it would simply be a matter of calculating a kernel around each point, adding it to a temporary raster, moving to the next point, adding the next point's kernel and so on. It would be difficult, but it would give you precisely the shape you need.