Re: finding irregular areas

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg1041] Re: finding irregular areas
• From: rubin at msu.edu (Paul A. Rubin)
• Date: Wed, 10 May 1995 07:35:33 -0400
• Organization: Michigan State University

```In article <3ocht3\$ovh at news0.cybernetics.net>,
steve at christensen.cybernetics.net (Steve Christensen) wrote:
->Greetings:
->
->I have a colleague who is looking for a full or partial Mathematica
->solution to the following situation.
->
->He has some aerial photographs of agricultural fields.  These fields are
->broken up by fences.  The fields are not of any nice shape being defined
->by rivers, roads, etc.  He was to be able to scan the photos into
->his computer and then find the areas of the various fenced areas.
->
->He will need to be able to define the fenced areas with a mouse or
->similar device and then feed the numbers to a program to find
->the area.
->
->Any ideas on how to handle all or part of this kind of problem?
->
->[I did suggest that he just cut the photos into the various areas
->and weight each and compare to a known area's weight, but that
->was not technical enough.  :-)  ]
->
->Thanks.
->
->Steve Christensen

I'll assume that he can live with a piecewise linear approximation to the
boundaries (he can always jack up the number of pieces), and that he can
find a software package (paint program, maybe?) that will take a scanned
image, let him indicate a specific spot in the image with a pointing
device, and record the Cartesian coordinates of that spot.  He uses this to
make lists of the "corners" of the regions (without having to trace the
full boundary).  Feed those lists to Mma and use
DiscreteMath`ComputationalGeometry`DelaunayTriangulation (pause for breath)
to break the region into nonoverlapping triangles.  That function (I
*refuse* to type the name again) returns lists of indices of vertices,
which he can convert back to lists of coordinates of the vertices.  He
passes the coordinate lists to a function that computes the area of a
triangle (left to the reader as an exercise, I've got places to go) and
sums the triangle areas to get the area of the region.

So the tough part would seem to be getting from the photos to the
coordinates of the boundary corners.  I'm sure a Geographic Information
System could do this easily if he (a) had access to one and (b) knew how to
use it.  We have one here, but I (a') don't and (b') don't (and darned
proud of it!).

Frankly, I thought your suggestion was considerably more practical.  If
he's a professor (with access to student "volunteers"), he could try a
Monte Carlo approach:  post several identical copies of the photo along a
wall; issue darts to the students and have them throw at the photos; use
the ratio of hits in a particular region to total hits on that photo, times
the total area captured in the photo, to estimate the area of that region
(averaging across photos).

Paul

**************************************************************************
* Paul A. Rubin                                  Phone: (517) 432-3509   *
* Department of Management                       Fax:   (517) 432-1111   *
* Eli Broad Graduate School of Management        Net:   RUBIN at MSU.EDU    *
* Michigan State University                                              *
* East Lansing, MI  48824-1122  (USA)                                    *
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Mathematicians are like Frenchmen:  whenever you say something to them,
they translate it into their own language, and at once it is something
entirely different.                                    J. W. v. GOETHE

```

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