[Insight-users] diameter of segmented object

[Miller, James V (Research)]

There may be some tricks in the Machine Vision literature for this. But I
would probably just brute force it by computing the eigenvalues and
eigenvectors of the scatter matrix of the indices in your binary object.

Let I be an index that is set "on". We'll assume I is a column vector.

Let M be the sample mean of the indices set "on"

M = Sum_i I_i / N

Let C be the covariance matrix of the indices set "on"

C = Sum_i (I_i - M)*(I_i - M)' / (N-1)

(C and M can be computed in one pass using something like C = (S - M*M'/N) /
N-1 where S = Sum_i I_i * I_i)

Compute the eigenvalue and eigenvectors of C.

The eigenvector associated with the largest eigenvalue is the direction of
the data that has the maximum "spread".  You now project your data onto this
"axis" and keep track of the min and max point once projected on this axis.
The difference between the min and max point will be the diameter.

I think an eigenvector problem is order d^2 (where d is the dimension of the
problem, i.e 2d image, 3d image, etc.). So the overall time 
for this approach is something like 2N + d^2.  Assuming N >> d, this will be
a time savings.



-----Original Message-----
From: Benjamin King [mailto:king . benjamin at mh-hannover . de]
Sent: Tuesday, November 18, 2003 7:18 AM
To: ITK
Subject: [Insight-users] diameter of segmented object


Hi all,

I want to compute the diameter of a segmented voxel object. As a first 
approach, I took the binarized image and subtracted an eroded version to 
get the boundary points. Then I compute the distance of each pair of voxels 
and take the maximum.
I need I much faster way (i.e. not O(N^2)) to do this.

Any suggestion is much appreciated,
  Benjamin

-- 
Benjamin King
Institut fur Medizinische Informatik
Medizinische Hochschule Hannover
Tel.: +49  511  532-2663
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