Question

I have a matrix A (Mx2), and matrix B (Nx2), with M>=N.
I have to compute the perpendicular euclidean distance to test similarity between A and B. Refering to the image, a single euclidean distance is the segment between each pi and p’i, once computed every single perpendicular distance, I can compute the mean or the sum to get a single parameter. Do you know a matlab routine or such a script that do this?

Answer 1

If you know that p1 .. p4 lie between p0 and p5, then you can use the area of a triangle to calculate the perpendicular distance easily:

d(p0p5, p1) = 2 * area(p0p1p5) / norm(p5-p0)

The area of a triangle is found by

area(p0p1p5) = sqrt(s * (s-norm(p0-p1)) * (s-norm(p1-p5)) * (s-norm(p5-p0)))

where

s = (norm(p0-p1) + norm(p1-p5) + norm(p5-p0)) / 2

I think you can gain a performance improvement using the dot product for projection and pythagorean theorem:

d(p0p5, p1) = sqrt(norm(p1-p0)^2 - dot(p1-p0,p5-p0)^2/norm(p5-p0)^2)

In either case, if your cost function involves mean-squared error, you can eliminate the sqrt call by just evaluating the squared-error instead. Norm-squares are also cheaper to calculate than calling norm and then squaring (use a dot product instead).