Absolute Orientation - Horn's method

Solves weighted absolute orientation problem using Horn's quaternion-based method.

ABSOR is a tool for least squares estimation of the rotation -- and optionally also the

scaling and translation -- that maps one collection of point coordinates to

another. It is based on Horn's quaternion-based algorithm. The function works for both 2D and 3D coordinates, and also gives the option of weighting the coordinates non-uniformly. The code avoids for-loops so as to maximize speed.

DESCRIPTION:

As input data, one has

A: a 2xN or 3xN matrix whos columns are the coordinates of N source points.

B: a 2xN or 3xN matrix whos columns are the coordinates of N target points.

The basic syntax

[regParams,Bfit,ErrorStats]=absor(A,B)

solves the unweighted/unscaled registration problem

min. sum_i ||R*A(:,i) + t - B(:,i)||^2

for unknown rotation matrix R and unknown translation vector t.

ABSOR can also solve the more general problem

min. sum_i w(i)*||s*R*A(:,i) + t - B(:,i)||^2

where s>=0 is an unknown global scale factor to be estimated, along with R and t,

and w is a user-supplied N-vector of weights. One can include/exclude any

combination of s, w, and translation t in the problem formulation. Which

parameters participate is controlled using the syntax,

[regParams,Bfit,ErrorStats]=absor(A,B,'param1',value1,'param2',value2,...)

with parameter/value pair options,

'doScale' - Boolean flag. If TRUE, the global scale factor, s, is included.

Otherwise, it is assumed that s=1. Default=FALSE.

'doTrans' - Boolean flag. If TRUE, the translation, t, is included. Otherwise,

zero translation is assumed. Default=TRUE.

'weights' - The length N-vector of weights, w. Default, no weighting.

OUTPUTS:

regParams: structure output with estimated registration parameters,

regParams.R: The estimated rotation matrix, R

regParams.t: The estimated translation vector, t

regParams.s: The estimated scale factor.

regParams.M: Homogenous coordinate transform matrix [s*R,t;[0 0 ... 1]].

For 3D problems, the structure includes

regParams.q: A unit quaternion [q0 qx qy qz] corresponding to R and

signed to satisfy max(q)=max(abs(q))>0

For 2D problems, it includes

regParams.theta: the counter-clockwise rotation angle about the

2D origin

Bfit: The rotation, translation, and scaling (as applicable) of A that

best matches B.

ErrorStats: structure output with error statistics. In particular,

defining err(i)=sqrt(w(i))*norm( Bfit(:,i)-B(:,i) ),

it contains

ErrorStats.errlsq = norm(err)

ErrorStats.errmax = max(err)