Vol. 4, No.
4/April 1987/J.
Opt.
Soc. Am.
A 629
Closed-form
solution
of absolute
orientation
using
unit
quaternions
Berthold
K. P. Horn
Department
of Electrical
Engineering, University
of Hawaii
at Manoa, Honolulu,
Hawaii 96720
Received August
6, 1986; accepted November 25, 1986
Finding the
relationship between
two coordinate
systems
using pairs of
measurements
of the coordinates
of a
number
of points in both systems
is a classic photogrammetric
task. It finds applications
in stereophotogrammetry
and in robotics.
I present here
a closed-form solution
to the least-squares
problem
for three or more
points.
Currently
various empirical,
graphical, and
numerical iterative
methods are in
use. Derivation
of the solution
is
simplified
by use
of unit quaternions
to
represent rotation.
I emphasize
a symmetry
property
that a solution
to this
problem ought
to possess. The best translational
offset is the difference between
the centroid of the coordinates
in
one system and
the rotated and scaled
centroid of the
coordinates in
the other system.
The best scale is
equal to the
ratio of the
root-mean-square deviations
of the coordinates in the two
systems from their respective centroids.
These exact
results are to be preferred to
approximate methods based on
measurements of a few selected
points.
The unit quaternion
representing the best
rotation is the eigenvector associated
with the most positive eigenvalue
of
a symmetric 4 X 4 matrix. The
elements of this matrix
coordinates of the points.
1. INTRODUCTION
Suppose that we
are given the
coordinates of a
number of
points as measured in two different Cartesian
coordinate
systems (Fig.
1). The photogrammetric problem
of recover-
ing the transformation between the
two systems from these
measurements is referred
to as that of absolute orientation.'
It occurs
in several contexts, foremost
in relating a stereo
model developed
from pairs of aerial photographs to a geo-
detic coordinate system.
It also is of importance
in robotics,
in which measurements in a camera coordinate
system must
be related
to coordinates in a system attached
to a mechani-
cal manipulator.
Here one speaks of the
determination of
the hand-eye
transform.
2
A. Previous Work
The problem of absolute orientation is usually
treated in an
empirical,
graphical,
or numerical iterative
fashion. 13,
4
Thompson
5
gives a solution to this problem
when three
points are measured.
His method, as well as the simpler one
of Schut,
6
depends on selective
neglect of the extra con-
straints available when all coordinates of three
points are
known. Schut uses unit quaternions
and arrives at a set of
linear equations. I present
a simpler solution to this special
case in Subsection 2.A that
does not require solution of a
system of linear
equations. These methods all suffer from
the defect that
they cannot handle more than three points.
Perhaps
more importantly, they do not even use all
the
information available from
the three points.
Oswal and Balasubramanian
7
developed
a least-squares
method that can handle
more than three points, but their
method does not
enforce the orthonormality of the rotation
matrix.
An iterative method is then used to square up
the
result-bringing it closer to being orthonormal.
The meth-
od for doing this
is iterative, and the result is not the solution
of the original least-squares
problem.
are combinations of sums of products of corresponding
I present a closed-form solution to the least-squares
prob-
lem
in Sections 2 and
4 and show
in Section 5 that
it simpli-
fies
greatly when only three points are
used. This is impor-
tant,
since at times only three
points may be available.
The
solution is different
from those
described
at the beginning
of
this section because it
does not selectively neglect
informa-
tion provided by the
measurements-it uses all of
it.
The groundwork for
the application of quaternions
in
photogrammetry was laid by Schut8
and Thompson.
9
In
robotics,
Salamin'
0
and
Taylor" have
been the
main propa-
gandists.
The use of unit quaternions
to represent rotation
is reviewed in
Section 3 and
the appendixes
(see also Ref.
2).
B. Minimum
Number of Points
The transformation
between
two Cartesian
coordinate
sys-
tems
can be thought of as the result
of a rigid-body motion
and can thus be decomposed into a rotation
and a transla-
tion. In stereophotogrammetry,
in addition, the scale may
not be known.
There are obviously
three degrees of
freedom
to translation.
Rotation
has another three
(direction of
the
axis about
which the rotation takes place plus
the angle of
rotation
about this axis).
Scaling adds one
more degree of
freedom. Three
points known
in both coordinate
systems
provide nine constraints
(three coordinates
each), more
than
enough to
permit determination
of the seven unknowns.
By discarding
two of the constraints,
seven equations
in
seven unknowns
can be developed that allow one to recover
the
parameters. I show
in Subsection 2.A
how to find the
rotation
in a similar fashion,
provided that
the three points
are not collinear. Two points clearly do not provide
enough
constraint.
C. Least
Sum of Squares of Errors
In practice, measurements
are not exact,
and so greater
accuracy in determining the
transformation parameters will
be sought for by
using more than three points. We no longer
0740-3232/87/040629-14$02.00
© 1987 Optical
Society of America
Berthold
K. P. Horn
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