Sunday, March 21, 2010

Looks difficult to me

For an application in image processing -- using R for statistical purposes -- I
need to solve the following task:

Given n (e.g. n = 100 or 200) points in the unit square, more or less randomly
distributed. Find a rectangle of maximal area within the square that does not
contain any of these points in its interior.

If a, b are height and width of the rectangel, other constraints may have to be
imposed such as a, b <= 0.5 and/or 0.5 <= a/b <= 2.0 . The rectangle is
allowed to touch the border of the square.

For each new image the points will be identified by the application, like all
stars of a certain brightness on an astronomical picture. So the task will have
to be performed several times.

I assume this problem is computationally hard. I would like to find a solution
that is reasonably fast for n = 100..200 points. Exhaustive search along the
x, y coordinates of the points will not be fast enough.



I doubt there is easy way to solve this. At least it is not obvious to me.


PS: see also http://yetanothermathprogrammingconsultant.blogspot.com/2010/03/looks-difficult-to-me-2.html

7 comments:

  1. I guess the solution rectangle has to be aligned with the unit square?

    This problem reminds me of this one paper of applied topology: "Homological Sensor" networks. In that paper, the dropped sensors on some area. The sensors had a given range, thus defining a neighborhood graph. From this graph, you can construct a higher-dimensional simplicial complex and compute its homology groups. This way, they could detect holes in the area, not covered by points.

    You could interprete your points as sensors and play a little with the range. Of course, the holes found by this method are purely topological, independent of size and shape. It might still be fun to try.

    Link to paper:
    http://www.math.uiuc.edu/~ghrist/preprints/noticesdraft.pdf

    There is already some software available, to compute the homology:
    http://chomp.rutgers.edu/

    In any case, you can learn some new trick :-)

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  2. Well, I would agree that it isn't easy if you are starting from scratch, but this sort of task is fairly straightforward with standard image processing packages.

    Say you have a binary image with your n points as black pixels on a white background. Modify that image by going through each point and blacking out the entire row and entire column of each point. You now have an image with a bunch of lines. These black lines outline a bunch of white rectangles, which are your candidate rectangles.

    Now you can run a standard blob-finding routine to get a list of the rectangles, which you can easily sort by area or apply constraints as you wish.

    I know Intel IPP and Matlab have blob-finding built into their image processing toolboxes. I am not familiar with R . . . you may need to implement your own blob finding routine . . . one place you might look for help is openCV's open source implementation (C++). http://opencv.willowgarage.com/wiki/cvBlobsLib#Algorithm

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  3. I have actually once solved a similar problem (same but with largest circle) with genetic programing. it works very well.

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  4. Hmm, on further review, please disregard my comment. ;)

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  5. The main disadvantage I see with genetic programming is that it is often not clear how good the solutions are. It finds better and better solutions, but how far we are from a global optimum is often just a guess. But it is well known that for some difficult problems it can do extremely well.

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  6. Do the sides of the rectangle have to be parallel with the sides of the unit square? If not this problem is really hard.

    Otherwise I'd go for a largest circle first, then find a large box.

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