Mahalanobis distance between a point and a line

2014 4 24
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hi all,
I am wondering if fellow reader can help me out to find a proper paper to calculate the Mahalanobis distance between a line and a point. I have found paper discussing "whitening transform" used to compute the Mahalanobis distance between a plane and a point, but could not find anything to for line/point.
mainly working in 3D.
thanks,

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José-Luis
José-Luis 2014-4-24
How is this a Matlab question?
Elvis Chen
Elvis Chen 2014-4-24
if you wish:
Does anyone have a matlab implementation of Mahalanobis distance between a point and a line in 3D?
thanks,

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Adam
Adam 2014-5-3

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I think this paper might has what you're looking for: Robust 3D Line Extraction from Stereo Point Clouds . I'm currently working on implementing this algorithm myself.

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Elvis Chen
Elvis Chen 2014-5-6
thanks, I'll look into this paper and report back (time allowing).
Elvis Chen
Elvis Chen 2014-5-6
The paper you cited has exactly what I want. Upon closer inspection, however, it contains a critical error. What equation 7 describes is actually the Eigen decomposition (noted the R and R'), and not SVD as the paper transcribes. Please make a note of it if you are implementing it.
That is, R can be obtained using eig(sigma), not svd(sigma).
of course, please correct me if I am wrong.
thanks,
Elvis Chen
Elvis Chen 2014-5-7
编辑：Elvis Chen 2014-5-12
Equation 10) is also wrong. It calculates the adjacent of the triangle (figure 3a) but what we want is the opposite of the triangle.
can't believe this is an IEEE conference paper, with these 2 critical errors.
(edit:) SVD will work though it gives a different solution (but correct) than eigen decomposition. SVD also works since the covariance matrix is symmetric.
Kirk
Kirk 2015-1-24
That paper breaks the transformation down into rotation and scale, but you can just use the whitening transformation (which does not separate the two).
Matlab example:
white = sqrt(Sigma); % Inverse of whitening transformation
l0w = white\(l0-Mean); % Demean, then whiten the point on the line
l1w = white\l1; % Whiten the direction vector
l1w = l1w/norm(l1w); % Normalize the direction vector
d = l0(2)*l1(1)-l0(1)*l1(2); % Distance from (0,0) to line
Where Mean and Sigma are the moments of your 2D Gaussian, white is the whitening transformation, l0 is a point on the line, l1 is the direction vector of your line, and d is the Mahalanobis distance. This appeared to work for me.

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José-Luis
José-Luis 2014-4-24

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doc pdist
doc mahal
And by distance between a point and a line do you mean the shortest distance? Also, the mahalanobis distances requires you know the covariances of the distribution (that you are assuming to be normal) you are calculating the distance to. How do you expect to find the covariance of a single point? Or of a line?
Elvis Chen
Elvis Chen 2014-4-24

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sorry, I realized that I was not clear in my original question.
For my application, the said covariance is actually the uncertainty associated with the point and it is given. And yes, I am looking for the shortest distance between a point and a line, given the uncertainties of this particular point.
That is, give a line defined by AB, and a point P_3x1, and the associated uncertainty C(P)_3x3, I like to find a closed-form solution for a point C_3x1 on AB that has the shortest Mahalanobis distance from P.
I apologize for not making my question clear. Thank you for your time,
Image Analyst
Image Analyst 2014-5-3

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Not sure what the noise/uncertainty has to do with it, but for a given point, and a given line, why can't you just use the formula to find the distance: http://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html

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Elvis Chen
Elvis Chen 2014-5-6
thanks, but what you describes is to find the Euclidean distance between a point and a line. I am looking for Mahalanobis distance, not Euclidean distance.

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