Image Processing Reference
In-Depth Information
However, most important, Eq. (12.72) shows that the condition
k
t
=0will be best
fulfilled when the difference between the two directions is maximum, (
θ
2
−
θ
1
)=
π/
2. Ideally this happens when the 2D image has perfectly balanced directions, i.e.,
both eigenvalues of the 2D structure tensor of
g
are large and are equal to each other.
Ill-conditioned numerical computations can be avoided by assuring that: (i) there is
a nonzero motion, i.e.,
∂f
∂t
>
0; (ii) the (2D spatial) gradient in the 2D image is
nonzero, i.e.,
>
0; (iii) the 2D image
g
lacks linear symmetry such that there
are at least two distinct directions in it, i.e., the most significant eigenvalue of the
2D structure tensor of
g
has the multiplicity 2. We summarize these findings in the
following lemma.
Lemma 12.6 (Spatial directions constraint).
The lack of linear symmetry is a suffi-
cient and necessary condition for an image
g
(
x, y
)
to satisfy for a translation of it be
computable from the corresponding spatiotemporal image
f
(
x, y, t
)
. The minimum
number of directions that must be contained in
g
to allow computation of an unam-
biguous translational motion is 2, provided that they are sufficiently distinct.
∇
g
12.7 Discrete Structure Tensor by Tensor Sampling in
N
D
In discretizing the
N
D tensor, we will follow an analogous approach to that devel-
oped in Sect. 10.11; for this reeason we only state the results without derivations or
proofs. The computation and discretization in the
r
-domain is done by utilizing the
Parseval-Plancherell theorem and by assuming that the interpolation function and
the multiplicative window defining a local image, when need be, are two Gaussians
with
σ
p
and
σ
w
, respectively.
The structure tensor for
E
3
The structure tensor for the 3D Euclidean space was defined in the spectral domain
and for continuous images via Eq. (12.13). In numerous situations, these quantities
need to be estimated on a discrete (Cartesian) 3D grid in the spatial domain, (the
r
-domain). Furthermore, this should, in many applications, be done quite often, e.g.,
for local images around every point.
Lemma 12.7.
The structure tensor is estimated in the TLS error sense, by averaged
tensor (outer) products
1
4
π
2
f
l
)
T
m
l
.
S
=
(
∇
f
l
)(
∇
(12.74)
l
where
f
l
is the gradient of
f
(
r
)
at the point
r
l
, which belongs to a regular discrete
grid, and
m
l
is an averaging kernel that consists of a discrete Gaussian. The discrete
gradients are estimated by filtering the original image
f
l
with a kernel consisting of
a discrete gradient of a Gaussian.
∇
