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

Coordinate systems

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

Finding the scale

Total error term, since

To minimize w.r.t. scale s, first term should be zero, or

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

QuaternionsRepresentation

If

Multiplication

Quaternions

Multiplication expressed as product oforthogonal matrix4x4 and vector4

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

Relationship to other notations

If angle is Θ

and axis is unit vector

Composition of rotations

First rotation

Second rotation

Since

Combined rotation

Finding the best rotationWe must find the quaternion that maximizes

Let and

Finding the best rotation

What we have to maximize

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.

Thank you.

The end.