[d,w,rx,cv,wx] = best_dir( x, [a , sx ] )
Some points x, are observed and one assumes that they belong to
parallel planes. There is an unknown direction d s.t. for each
point x(i,:), one has :
x(i,:)*d == w(j(i)) + noise
where j is known(given by the matrix a ), but w is unknown.
Under the assumption that the error on x are i.i.d. gaussian,
best_dir() returns the maximum likelihood estimate of d and w.
This function is slower when cv is returned.
INPUT :
-------
x : D x P P points. Each one is the sum of a point that belongs
to a plane and a noise term.
a : P x W 0-1 matrix describing association of points (rows of
x) to planes :
a(p,i) == 1 iff point x(p,:) belongs to the i'th plane.
Default is ones(P,1)
sx : P x 1 Covariance of x(i,:) is sx(i)*eye(D).
Default is ones(P,1)
OUTPUT :
--------
d : D x 1 All the planes have the same normal, d. d has unit
norm.
w : W x 1 The i'th plane is { y | y*d = w(i) }.
rx : P x 1 Residuals of projection of points to corresponding plane.
Assuming that the covariance of x (i.e. sx) was known
only up to a scale factor, an estimate of the
covariance of x and [w;d] are
sx * mean(rx.^2)/mean(sx) and
cv * mean(rx.^2)/mean(sx), respectively.
cv : (D+W)x(D+W)
Covariance of the estimator at [d,w] ( assuming that
diag(covariance(vec(x))) == sx ).
wx : (D+W)x(D*P)
Derivatives of [w;d] wrt to x.
Author : Etienne Grossmann
Created : March 2000
Package: vrml
