Hi Jesper,
without carefully studying all of the code, it seems to me that the
algorithm simply determines the properties of the ellipse having the
same moments (more specifically, 1st and 2nd moments) as the the
particle (using the mask, i.e., the area; the grayscale value is
ignored).
The moment (m,n) is simply
integral (x^m * y^n) dx dy
whereby the integral is executed over the area of the particle.
Some details:
moment (0,0) gives you the area
moment (1,0) / area is the x center (Centroid)
moment (0,1) / area is the y center (Centroid)
2nd-order moments (0,2), (1,1), (2,0) are first converted to central
moments and then used to determine the axes and orientation of the
ellipse.
This method usually results in an ellipse with a slightly different
area than the particle. Therefore, as a last step the ellipse is
scaled to have the same area as the particle.
To me, the algorithm seems like the natural way of doing it, and it
seems that there are many references (but I have not checked any of
them, some may be not really about the subject):
http://dx.doi.org/10.1016/S0191-8141(03)00093-2The refs given there are:
Jain, A.K., 1989. Fundamentals of Digital Image Processing, Random
House, New York.
Jähne, B., 1997. Digital Image Processing: Concepts, Algorithms, and
Scientific Applications, Springer-Verlag, Berlin.
Russ, J.C., 1999. The Image Processing Handbook, CRC Press, Florida.
Hu, M.K., 1962. Visual pattern recognition using moment invariants.
IRE Transactions on Information Theory IT-8, pp. 179–187.
Teague, M.R., 1980. Image analysis via the general theory of moments.
Journal of the Optical Society of America 70, pp. 920–930.
Hope this helps,
Michael
________________________________________________________________
On 20 Oct 2010, at 11:24, JesperSP wrote:
> I'm writing a manuscript for a scientific publication about a new
> plugin that
> I've developed in ImageJ. The plugin is an extension of MTrack2 and
> uses a
> simple ellipse fitting to quantify the swimming rate of nematodes. My
> collaborators say we need to decribe the inner workings of this
> ellipse
> fitting algorithm in detail for the paper. I've located the source-
> code for
> the ellipse fitting at
http://rsb.info.nih.gov/ij/developer/source/ .
> However, deciphering the source isn't as easy as I thought it would
> be.
> Moreover, the source doesn't explicitly mention a scientific
> reference to
> the person that developed the algorithm (but it does mention "Cramer,
> Mathematical Methods of Statistics, Princeton Univ. Press, 945,
> page 283").
> I wanted to contact the original author of the source, Bob Rodieck,
> but from
> what I read it appears that he passed away
>
http://www.ncbi.nlm.nih.gov/pubmed/16212696>
http://www.ncbi.nlm.nih.gov/pubmed/16212696 . If anyone here is
> familiar
> with the math behind the ellipse fitting that can, I'm hoping that
> someone
> can please direct me to more information about this algorithm.
>
> Cheers,
> Jesper
Locked
. If anyone here is familiar with the math behind the ellipse fitting that can, I'm hoping that someone can please direct me to more information about this algorithm.
Cheers,
Jesper
