Direction in 2D - Vision with Direction - page 164

10.5 Linear Symmetry Tensor: Directional Dominance

In this section we will mean the complex structure tensor when we refer to the struc-
ture tensor. An “ideal” linear symmetry is present in the image, when λ max >> 0
and λ min =0. Such images have a directional dominance in that there is a single
and well-identified direction of isocurves. One way to quantitate this property is by
measuring λ max −

λ min and k max , which are jointly given by the complex scalar

I 20 [28]. When λ max − λ min increases, so does the evidence for the image being lin-

early symmetric, and hence we have a crisp direction in the image. For Hermitian 3
positive semi definite matrices, which includes the structure tensor Z , the dimension
of the eigenvector space is equal to the multiplicity of the corresponding eigenvalue,
which is the number of times the latter is repeated. The eigenvectors are orthogonal
if they belong to two different eigenvalues. Because there are at most two different
eigenvalues in 2D, there is no need to encode both eigenvectors. The linear symmetry
quality of a 2D image has also been called the “line” property [225], and the “stick”
property [161]. The linear symmetry tensor is a special type of structure tensor such
that

|

Z L = 1
2

I 20 |− iI 20

(10.42)

iI 20

| I 20 |

The tensor is fully equivalent to the scalar quantity I 20 , which determines Z L
uniquely, which is, in turn, a spatial average of ( f ):

10.6 Balanced Direction Tensor: Directional Equilibrium

Certain images lack direction, i.e., when a direction is attempted to be fit to the
power spectrum there is not one optimal axis but there are an infinite (uncountable)
number of them, e.g., the image of sand in Fig. 10.13 or the image in Fig. 10.10.
This property is captured by the structure tensor via the condition λ max = λ min , i.e.,
the smallest (or the largest) eigenvalue is repeated twice making its multiplicity 2.
The condition actually does not describe the presence of a property but the lack of
it. It describes the lack of linear symmetry. An image with a structure tensor having
λ max = λ min , has previously been called “perfectly balanced”, in analogy with the
terminology used in mechanics [28]. The term expresses that there is a directional
equilibrium in that no single direction dominates over the others. Such images lack a
single direction 4 that is more significant than other directions, a property that justifies
the use of the term “balanced directions” or “balancedness”, both referring to an
equilibrium of directions. Balancedness is quantitated by λ min because, for a fixed
λ max , the larger λ min , the closer it gets to λ max . When λ min reaches its upper bound,
which is λ max , then the structure tensor has one eigenvalue which has a multiplicity
2. The least eigenvalue λ min can be used to signal the presence of a balanced image,
3 We recall that a Hermitian matrix Z fulfills Z = Z H .
4 The pattern may still have a group direction, although it may lack a single direction domi-
nating others, e.g., see Fig. 10.10.