Direction in 2D - Vision with Direction - page 171

in a specific basis. A representation of the same tensor in a different basis can only
be obtained by a similarity transformation using the basis transformation matrix.
For first-order tensors and vectors, a similar subtle difference exists. The first-order
tensor is represented as a vector in a specific basis. Another representation of it can be
obtained by a linear transformation corresponding to a basis change. The zero-order
tensors represent physical quantities that are scalars. Their numerical representations
do not change with basis transformations.
As illustrated by Fig. 10.9, the error function e ( k ) employed by the total least
square (TLS) error represents the spectral power weighted by its shortest (orthog-
onal) distance to the estimated axis k . This makes e a zero-order tensor and k a
first-order tensor. If we apply a basis change e.g., rotate the coordinates of the power
spectrum, e ( k ) will not change at all and only the numerical representation of k will
change. The new direction vector, k , will be coupled to the old k linearly using the
inverse of the matrix that caused the basis change.
To appreciate the TLS error in this context we compare it to the mean square
(MS) error which is extensively used in applications where one has a black box
controlled by known inputs resulting in a measurable output. In such applications
there is thus a response measurement , y , that may contain measurement errors and
that is to be explained by means of another set of known (error-free) variables, called
explanatory variables X , via a linear model

y = Xk (10.55)

Here k is the unknown regression parameter, which will be estimated by minimizing
the following residual:

2

min
k

y − Xk (10.56)

Adapted to our 2D direction estimation problem, the MS error yields:

e ( γ )=

2 2 d

min
γ

ω y − γω x | F ( ω x ,ω y ) | ω (10.57)

This is the classical regression problem . Here, the direction coefficient γ is unknown
and will be estimated from the data F ( ω x ,ω y ). The unknown γ is related to the di-
rection vector k =(cos θ, sin θ ) T

cos θ

as γ = −

sin θ . Notice that the integrand measures

the distance between the data point and a point on the k -axis to be fitted. This

distance is in general not the shortest, distance as illustrated by the vector d in Fig.
10.14. The MS error would accordingly depend on the coordinate axis directions to
the effect that after a basis change, the new error using the same data will be dif-
ferent. Likewise, the new direction k will not be given by multiplying the inverse
of the basis transformation matrix with k , the estimated direction before the basis
change. In consequence, neither the MS error nor γ are tensors. One can associate
k and kk T

ω

cos( θ )
sin( θ )

. These quantites are not tensors, although they are
conventional vectors and matrices. A more detailed discussion of the TLS error can
be found in [115] and [59].
Through Taylor expansion a spatial interpretation of e ( k ), as an alternative to its
original interpretation, the spectral inertia, can be obtained. In this view, the structure

to every γ = −