This section provides a general overview of the distance measures or metrics that are used to measure transformations in SE(n). It is not exhaustive, but instead focuses on those that are frequently used in geometric sensor data alignment and this work in particular. The treatment is divided into translation and rotation measures as before, with the predominant focus being on rotation measures. For the group of translations, d : Rn ×Rn → R+ is a distance measure that maps two vectors in Rn to a non-
negative real number. Similarly, for the group of rotations, d:SO(n)×SO(n)→R+
is a distance measure that maps two elements of the rotation group in SO(n) to a non-negative real number.
In this section, Lp-norm functions are written with their appropriate subscripts.
However, whenever a norm subscript is not notated in this document, the norm is to be understood as anL2 norm unless otherwise specified. In addition, the subscript notationLp is used instead of the more common superscript notationLp for consistency
with the computer vision literature. 3.2.1 Euclidean Distance
The natural distance measure for translations is the Euclidean distance dE, not least because the translation group is isomorphic to Euclidean space. For the group of translations, the Euclidean distance between two n-vectors x and yin Rn is given by
theL2-norm of the vector difference, that is
dE(x,y) =kx−yk2 =q(x−y)|(x−y) = v u u t n X i=1 (xi−yi)2 (3.17)
where xi and yi are the components of x and y respectively. This metric is suitable
for both 2D and 3D translations. When the only relevant information is the relative ordering of translations from a fixed translation, the squared Euclidean distance may be used instead, which is more efficient to calculate.
3.2.2 Angular Distance
The natural distance measure for rotations is the angular distanced∠that takes values in the range [0, π]. The angular distance between a rotationRand the identity rotation Iis given by the angle of rotation∠(R) of the matrixR, chosen such that 06 ∠(R)6 π. In 2D, this corresponds to the (unsigned) rotation angle about the origin, reversing the direction of rotation if necessary. In 3D, this corresponds to the rotation angle about an axis, reversing the direction of the axis if necessary. In both cases, the rotation angle is the angle between v⊥ andRv⊥ wherev⊥ is a vector perpendicular to the axis
of rotation. For two rotation matrices R1 and R2, the angular distanced∠(R1,R2) is defined as the angle of rotation∠(R|1R2) of the matrixR1|R2, or equivalentlyR1R|2, R2|R1 orR2R|1, again chosen to be in the range [0, π] [Hartley and Kahl, 2009].
For R∈SO(2) or R∈SO(3), the angular distance is given by d∠(R1,R2) =d∠(R|1R2,I) = √1
wherekAkF denotes the Frobenius norm given by q P P |aij|2. The factor of ( √ 2)−1
arises from the skew-symmetry of log(R) and the observation that the rotation angle is given bykrk2, for [r]×= log(R). Taking the Frobenius norm of both sides gives the
expression klog(R)kF =k[r]×kF =
√
2krk2.
In 2D, the angular distance can be expressed explicitly as d∠(R1,R2) = arccos trace(R | 1R2) 2 ! . (3.19)
It may also be calculated using min(|θ1−θ2|,2π− |θ1−θ2|) whereθi=∠(Ri). In 3D,
the angular distance can be written as
d∠(R1,R2) = arccos trace(R | 1R2)−1 2 ! . (3.20)
These trace expressions make use of the properties of the eigenvalues of two- and three- dimensional rotation matrices. An alternative approach for 3D rotations is to compute the angle from the quaternion representation of the rotation matrices. If ˆq1 and ˆq2 are unit quaternions representing the same rotations asR1 and R2 respectively, then
d∠(R1,R2) = 2 arccos(|w|) = 2 arccos|hˆq1,qˆ2i| (3.21)
where (w,v) = ˆq−1
2 qˆ1using the Hamilton product,h·,·iis the quaternion inner product
and the positive absolute value is taken to ensure that the angle lies in the range [0, π] as required [Hartley et al., 2013].
3.2.3 Chordal Distance
Another rotation distance measure is found by calculating the Euclidean distance be- tween two rotation matrices in their embedding space Rn×n. Designated the chordal
distance dC between two matricesR1 and R2, it is given by
dC(R1,R2) =kR1−R2kF (3.22) where the Frobenius norm k · kF is an extension of the Euclidean norm to matrices. An advantage of this measure is that is computationally inexpensive, without matrix multiplications or trigonometric functions.
For rotations in bothSO(2) andSO(3), Hartley et al. [2013] show that the angular distance is related to the chordal distance by
dC(R1,R2) = 2√2 sin
1
2d∠(R1,R2)
3.2.4 Quaternion Distance
Similar to the definition of the chordal distance, the quaternion distance is found by calculating the Euclidean distance between two unit quaternions in their embedding spaceR4. However, since antipodal points of the unit quaternion sphere are identified,
ˆ
q and −qˆ represent the same rotation. The problem of choosing the correct sign is
resolved by defining the quaternion distance between two unit quaternions ˆq1 and ˆq2, the quaternion representations of rotation matricesR1 and R2 respectively, as
dQ(R1,R2) = min{kqˆ1−qˆ2k2,kˆq1+ ˆq2k2} (3.24) where both the positive and negative branches of ˆq2are considered. Unlike the angular and chordal distances, the quaternion distance does not exist for 2D rotations.
As obtained in Hartley et al. [2013], the angular distance is related to the quaternion distance by dQ(R1,R2) = 2 sin 1 4d∠(R1,R2) . (3.25) 3.2.5 Angle-Axis Distance
The angle-axis distance is also defined using the Euclidean distance, in this case between two angle-axis vectors in R3. However, if the angle-axis vectors are restricted to lie
within the ball Bπ3 of radius π, as previously defined, then this measure would have discontinuities. For example, rotations close to π radians about opposite axes are close by the angular distance measure but not close by the Euclidean distance measure in Bπ3, since they are almost antipodal. To remove these discontinuities, the angle- axis distance dAA between two rotation matrices R1 and R2 in SO(3) considers all equivalent angle-axis vectorsr1 andr2, including those outside Bπ3, and is defined as
dAA(R1,R2) = minr
1,r2kr1−r2k2 (3.26)
where exp([r1]×) =R1 and exp([r2]×) =R2. Like the quaternion distance, the angle-
axis distance does not exist for 2D rotations.
As observed in Hartley et al. [2013], the angle-axis distance is not bi-invariant. Therefore, there is no equality relationship between the angular distance and the angle- axis distance. However, the useful inequalities
d∠(R1,R2)6dAA(R1,R2)6 π2d∠(R1,R2) (3.27) can be shown for this distance measure [Li and Hartley, 2007; Hartley and Kahl, 2009].