Image Processing Reference
In-Depth Information
12
Direction in
N
D, Motion as Direction
In this chapter we extend our discussion of the direction estimation problem from a
2D setting in Cartesion and curvilinear coordinates to an estimation problem in
N
dimensions (
N
D). We start the discussion in 3D. To the extent that we can recognize
the solutions from treatment of 2D, the solutions of 3D problems related to direction
estimation will be inherited from their 2D analogues. This will provide an oppor-
tunity to focus on specific issues of direction estimation not present in 2D. In turn,
the experience from 3D will help us draw conclusions on the problem of direction
estimation in
N
D.
12.1 The Direction of Hyperplanes and the Inertia Tensor
Whether or not a 3D image has a direction can be studied in the Fourier transform
domain
1
in analogy with Chap. 10, although the computations will be carried out in
the spatial domain. Let
f
be a positive real function defined on
E
3
with
F
being its
3D Fourier transform. As before, we will sometimes call
f
an image and its values as
gray values, although the values of
f
can represent a variety of physical properties in
applications. They can, for example, represent the light intensities as observed on a
video camera sensor to form image sequences,
f
(
x, y, t
), or the absorption of X-ray
in an organic tissue to form X-ray tomography images,
f
(
x, y, z
). The function
f
will still be called an image, even if it represents a local image.
Intuitively, if
f
has a “direction” then this has to be related to the locus of
f
's
isogray values, points which have the same function values. In 3D the loci of isogray
values are more complex to describe than in 2D because
f
's being constant can be
achieved along a 1D curve, the simplest of which is a line, or along a 2D surface, the
simplest of which is a plane. First, we will discuss the
simple manifolds
, namely the
1
An equivalent formulation in the spatial domain utilizing the Lie operators [31] as is done
in Sect. 11.2, is possible in 3D Cartesian coordinates. We prefer the frequency-domain
derivation for its straightforward geometric interpretation in terms of the habitual line and
plane fitting.
