Absolute Orientation

Computes the transformation to register two corresponding 3D point sets.

Description

[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"
The paper can be downloaded here:
http://people.csail.mit.edu/bkph/papers/Absolute_Orientation.pdf

Authors:
Dr. Christian Wengert, Dr. Gerald Bianchi

Copyright:
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;

%Add noise
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

• Acknowledgements: This submission has inspired the following:; Absolute Orientation - Horn's method
• MATLAB release: MATLAB 6.5 (R13)

[Thomas Pieper]

Very good implementation of Horn's paper, but the functions crossprodquaternion and crossprodquaternion2 are still missing.

[Christian Wengert]

I am sorry for the inconvenience caused by the missing crossprod function files. I uploaded the new version and it should be online soon.
Cheers

[Dave Ligthart]

Great implementation!

[Bryan Murawski]

Great work, but I believe that there is a mistake in your error metric computation. If you want to compute the sum of the squared error like you're doing (I assume for performance reasons) you should divide by Na to compute the average squared error and then sqrt that quantity (ie. sqrt(err/Na)).

[Dirk-Jan Kroon]

What is the advantage of this absolute orientation method?

If I want the least-squares transformation matrix, with your coordinates it is simple :
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);

>> X(4,:)=1; Y(4,:)=1;
>> Y/X

ans =

    0.2520 0.3360 -0.5600 45.0000
   -0.5600 0.4200 0.0000 -78.0000
    0.3360 0.4480 0.4200 98.0000
         0 0 0 1.0000

[Christian Wengert]

Here is an interesting paper to the topic:

A Comparison of Four Algorithm s for Estimating tD Rigid Transformations

http://www.homepages.inf.ed.ac.uk/rbf/MY_DAI_OLD_FTP/rp765.pdf

Actually the Horn approach (unit quaternions) and the above mentioned (SVD?) are pretty much equivalent

[Matt J]

@Dirk-Jan, the method that you've shown is not a constrained least squares estimation. The transformation matrix that it produces is therefore not gauranteed to be of the form
[sR,T;zeros(1,3), 1].

If you add noise to X and Y, you will see that your method does not produce the same results as the absolute orientation solver, nor will it be in the desired family of transformations.

[Peter]

In your result, you compute the residual error as:
err =0;
for i=1:Na
   d = (B(:,i) - (s*R*A(:,i) + T));
   err = err + norm(d);
end

Wouldn't it be more appropriate to compute the sum of squared errors, because this is what you actually minimize? (so just use norm(d).^2 instead)

A nice addition would be to add the symmetric scale computation as mentioned later in the paper as a third option.

[Georg Stillfried]

Just what I was looking for