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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)

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point set alignment

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
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
Disadvantages of previous methods
  • Cannot handle more than three points
  • Do not use information from all three points
  • Iterative instead of least squares
introduction transformation
Introduction - Transformation

Transformation between two

Cartesian coordinate systems

Translation

Rotation

Scaling

introduction method
Introduction - Method
  • Minimize error
  • Closed-form solution
  • Use of quaternions
  • Symmetry of solution
selective discarding constraints
Selective discarding constraints

X axis

Y axis

Z axis

l

Maps points from left hand to right hand coordinate system

Rotation

finding the translation
Finding the translation

Measured coordinates in left and right hand systems

Form of translation

Scale factor

Translational offset

Rotated vector from left coordinate system

Residual error

To be minimized

centroids of sets of measurements
Centroids of sets of measurements

Centroids

New coordinates

Error term

where

Sum of squares of errors

centroids of sets of measurements10
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
Finding the scale

Total error term, since

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

symmetry in scale
Symmetry in scale

Suppose 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
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
Quaternions

Representation

If

Multiplication

quaternions15
Quaternions

Multiplication expressed as product of

orthogonal matrix4x4 and vector4

quaternions16
Quaternions

Dot product

Square of magnitude

Conjugate

Product of quaternion and its conjugate

Inverse

unit quaternions and rotation
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
Relationship to other notations

If angle is Θ

and axis is unit vector

composition of rotations
Composition of rotations

First rotation

Second rotation

Since

Combined rotation

finding the best rotation
Finding the best rotation

We must find the quaternion

that maximizes

Let

and

finding the best rotation21
Finding the best rotation

What we have to maximize

finding the best rotation22
Finding the best rotation

where

and

Introducing the matrix3x3

that contains all the information required to solve the least-squares problem for

rotation.

where

and so on.

Then,

eigenvector maximizes matrix product
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
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 solution25
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 solution26
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

Thank you.

The end.