Absolute Orientation

Version 1.3.0.0 (2.38 KB) by
Computes the transformation to register two corresponding 3D point sets.
Updated 9 Jun 2010

[s R T error] = absoluteOrientationQuaternion( A, B, doScale)

Computes the orientation and position (and optionally the uniform scale factor) for the transformation between two corresponding 3D point sets Ai and Bi such as they are related by:

Bi = sR*Ai+T

Implementation is based on the paper by Berthold K.P. Horn:
"Closed-from solution of absolute orientation using unit quaternions"
http://people.csail.mit.edu/bkph/papers/Absolute_Orientation.pdf

Authors:
Dr. Christian Wengert, Dr. Gerald Bianchi

ETH Zurich, Computer Vision Laboratory, Switzerland

Parameters:
A 3xN matrix representing the N 3D points
B 3xN matrix representing the N 3D points
doScale Flag indicating whether to estimate the uniform scale factor as well [default=0]

Return:
s The scale factor
R The 3x3 rotation matrix
T The 3x1 translation vector
err Residual error (optional)

Notes: Minimum 3D point number is N > 4

The residual error is being computed as the sum of the residuals:

for i=1:Npts
d = (B(:,i) - (s*R*A(:,i) + T));
err = err + norm(d);
end

Example:

s=0.7;
R = [0.36 0.48 -0.8 ; -0.8 0.6 0 ; 0.48 0.64 0.6];
T= [45 -78 98]';
X = [ 0.272132 0.538001 0.755920 0.582317;
0.728957 0.089360 0.507490 0.100513;
0.578818 0.779569 0.136677 0.785203];
Y = s*R*X+repmat(T,1,4);

%Compute
[s2 R2 T2 error] = absoluteOrientationQuaternion( X, Y, 1);

error = 0;

Noise = [
-0.23 -0.01 0.03 -0.06;
0.07 -0.09 -0.037 -0.08;
0.009 0.09 -0.056 0.012];

Y = Y+Noise;
%Compute
[s2 R2 T2 error] = absoluteOrientationQuaternion( X, Y, 1);

error = 0.33

Cite As

Christian Wengert (2024). Absolute Orientation (https://www.mathworks.com/matlabcentral/fileexchange/22422-absolute-orientation), MATLAB Central File Exchange. Retrieved .

MATLAB Release Compatibility
Created with R13
Compatible with any release
Platform Compatibility
Windows macOS Linux
Categories
Find more on Geometric Transformation and Image Registration in Help Center and MATLAB Answers
Acknowledgements

Inspired: Absolute Orientation - Horn's method

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Version Published Release Notes
1.3.0.0

Based on Bryan Murawski's comments, I reviewed the computation of the residual error. Indeed, it seemed a bit strange, I thus changed the computation a bit so that it reflects the overall error of the transformation.

1.2.0.0