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Point set alignment
Closed-form solution of absolute orientation using unit quaternions
Berthold K. P. Horn
Department of Electrical Engineering, University of Hawaii at Manoa
Presented by Ashley Fernandes
Abstract
• Finding relationship between coordinate systems (absolute orientation)
• Closed-form
• Use of quaternions for rotation
• Use of centroid for translation
• Use of root-mean-square deviations for scale
Disadvantages of previous methods
• Cannot handle more than three points
• Do not use information from all three points
• Iterative instead of least squares
Introduction - TransformationTransformation between two
Cartesian coordinate systems
Translation Rotation Scaling
Introduction - Method
• Minimize error
• Closed-form solution
• Use of quaternions
• Symmetry of solution
Selective discarding constraints
X axis
Y axis
Z axis
l
Maps points from left hand to right hand coordinate system
Rotation
Finding the translation
Measured coordinates in left and right hand systems
Form of translation
Residual error
To be minimized
Scale factorTranslational offset
Rotated vector from left coordinate system
Centroids of sets of measurements
Centroids
New coordinates
Error term
Sum of squares of errors
where
Centroids of sets of measurements
Translation, when r’o = 0
Error term, when r’o = 0
Total error term to be minimized
Symmetry in scaleSuppose we tried to find
, or so we hope.
But, or
Instead, we use
Total error becomes
To minimize w.r.t. scale s, first term should be zero, or
Scale
Why unit quaternions
• Easier to enforce unit magnitude constraint on quaternion than orthogonal constraint on matrix
• Closely allied to geometrically intuitive concept of rotation by an angle about an axis
Quaternions
Dot product
Square of magnitude
Conjugate
Product of quaternion and its conjugate
Inverse
Unit quaternions and rotation
if
We use the composite product which is purely imaginary.
Note that this is similar to
Also, note that
Finding the best rotationandwhere
Introducing the matrix3x3
that contains all the information required to solve the least-squares problem forrotation.
where and so on.
Then,
Eigenvector maximizes matrix product
Unit quaternion that maximizes
is eigenvector corresponding to most positive eigenvalue of N.
Eigenvalues are solutions of quartic in that we obtain from
After selecting the largest positive eigenvalue we find the eigenvector
by solving
Nature of the closed-form solution
• Find centroids rl and rr of the two sets of measurements
• Subtract them from all measurements
• For each pair of coordinates, compute x’lx’r, x’ly’r, … z’lz’r of the components of the two vectors.
• These are added up to obtain Sxx, Sxy, …Szz.
Nature of the closed-form solution
• Compute the 10 independent elements of the 4x4 symmetric matrix N
• From these elements, calculate the coefficients of the quartic that must be solved to get the eigenvalues of N
• Pick the most positive root and use it to solve the four linear homogeneous equations to get the eigenvector. The quaternion representing the rotation is a unit vector in the same direction.
Nature of the closed-form solution
• Compute the scale from the symmetrical form formula, i.e. the ratio of the root-mean-square deviations of the measurements from their centroids.
• Compute the translation as the difference between the centroid of the right measurements and the scaled and rotated centroid of the left measurement.