Direction in 2D - Vision with Direction - page 166

as the image must look like a ball, or a junction. The images in Fig. 10.10 as well
as in Fig. 10.13 represent textures in which at every point there is a “ball” tensor
of approximately the same magnitude. We will discuss the use of balanced direction
tensor as a corner detector in Sect. 10.9.

10.7 Decomposing the Complex Structure Tensor

In general, an image is neither perfectly linearly symmetric, e.g., Figs. 10.2-10.6, nor
does it totally lack it, e.g., Fig. 10.13. Instead it has the qualities of both types. The
amount of evidence for the respective type can be obtained from the structure tensor.
The structure tensor decomposition can always be achieved into its linear symmetry
and balanced direction components easily using its complex form:

I 11 − = Z L + Z B

Z = 1
2

iI 20

(10.44)

iI 20

I 11

where

| , I 11 −| .

Z L = 1
2

I 20 |− iI 20

Z B = 1
2

I 20 | 0

and (10.45)

iI 20

| I 20 | 0 I 11 −| I 20 |

Conversely, we also wish to study what happens when joining regions having dif-
ferent structure tensors. Without loss of generality, we consider a composition of
a region consisting of two subregions, each having a different structure tensor, Z
and Z , respectively. This is a realistic scenario since two neighboring regions in an
image might differ in their local structure tensors, and the local structure tensor at
a border point between the two regions is needed. Because the components of the
structure tensor are integrals, they can be computed as the sum of two integrals, each
taken over the respective regions. Accordingly, the structure tensor, Z , of a boundary
point is obtained by the addition

Z = p Z + q Z

(10.46)

where p , q are two real positive scalars, with p + q =1, that are proportional to the
areas of the two constituent regions. Following the definition of Z , we obtain

I 20 = pI 20 + qI 20

(10.47)

I 11 = pI 11 + qI 11

(10.48)

where I ·· , I

, and I ·· are the structure tensor parameters of the first, the second and

··

the joint patches.

Example 10.6. In Fig. 10.11 we have two regions labelled A , and B . There are four
localimages,calledimageshere,andthesearemarkedas1 , ··· , 4 withtheirborders
shown as (color) circles. Let images 2 and 4 have the (complex) structure tensors Z
and Z , respectively. The corresponding tensor components are therefore