Finding Polygons

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Finding Polygons with K edges in an image
written by Gidi Sidis
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Motivation

• This is one step further in Perceptual
  Organization (after lines circles).
• Automated machines can use this function in order
  to scan images and identify pieces
• Analyzing satellite or airplane images (searching
  for buildings and etc)

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How can it be done ?

• Let the following image be our subject

And lets say we want to find all polygons with 4
edges.
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• The main idea is to find all the lines in the
  image by doing edge detection and using Hough
  Transform on the result.

edge detection
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hough transform result
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Lines found - marked on image
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• After all lines have been found we search for
  each line the appropriate segment(s) in the
  image. When done, we create a matrix of size
  num_segments num_segments

neighborhood matrix of size 55
Segments found - marked on image
Comment There are 2 matrixes do to the fact that
a point is represented by (x,y)
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• For each segment i and segment j that intersect,
  we update the matrix(i,j) to be the intersecting
  point.

neighborhood matrix after updating
X
Segment 5
Segment 3
Y
(62,94)
Segment 2
Segment 1
Segment 4
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• Basically, we found a neighborhood matrix of all
  segments which represents a graph where each
  segment is a vertex, and each intersecting point
  is an edge.

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• All that remains to be done is to scan the graph
  for all circles of size K. Each circle is a
  polygon of K edges in the image.

neighborhood matrix after updating
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Some problems in implementation

• The algorithm is highly sensitive to inaccuracy
  (which is a side product of transforming from
  continuous to discrete and vice versa), plus
  there is numeric inaccuracy done by the cpu which
  have a finite memory space.
• The algorithm is sensitive to noise in the image.
  
• The problem of finding all circles in a graph is
  a NP Hard problem.

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Some solutions to the problems

• For the first and second problem as stated
  previously, the simplest solution is to use an e
  environment for all numeric calculations that
  deals with binary comparison (as in the case of 2
  segments intersection and etc).

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• For the third problem, there is no simple
  solution (if there was I would be a very rich
  man), however, we are not scanning the graph
  for all circles but only the circle of size K
  thus we can block the running time to NK where N
  is the number of segments found in the image
  (notice that the number of segments is much
  smaller than the number of edge point which by
  them the Hough Transform runs).

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Some Conclusions points

• The problem of finding polygons in a complex
  image, especially a noisy one, is fairly hard and
  will not be treated well by this algorithm /
  program, however this algorithm could be useful
  and will give pretty good results for simple
  images.
• You can check out the implementation pseudo code
  and more information in my final report at
  www.cs.bgu.ac.il/sidisg/vision_project_report.doc
  
• You can check out the MatLab program which
  implements the algorithm at www.cs.bgu.ac.il/sid
  isg/find_polygons_program.zip

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more examples
The result of searching for all 4 edges polygons
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more examples cont.
The result of searching for all 3 edges polygons
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more examples cont.
The result of searching for all 5 edges polygons
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more examples cont.
The result of searching for all 4 edges polygons