Game Development Reference
InDepth Information
(a)
(b)
Figure 5.5.
For smooth images with strong dominant local orientation, such as the one
shown in (a), local orientation may be estimated using the gradient. However, if the
image is corrupted by noise, as shown in (b), the gradients are noisy and require further
processing. Notice that gradients in the neighborhood of extrema have opposite signs
and would cancel each other out if processed with a smoothing filter.
For the spatial weight, a twodimensional Gaussian function,
exp
,
2
2
ρ
2
1
2
πρ
2
−
x
G
ρ
(
x
)=
with standard deviation
ρ
is used. The set
(
x
) refers to a local neighborhood
of
x
with reasonable cutoff (e.g., with radius 3
ρ
), and
N
=
y
∈N
(
x
)
G
ρ
(
y

G
ρ

−
x
)
denotes the corresponding normalization term.
Equation (5.1) cannot be solved directly, but rewriting the scalar product
in matrix form as
g
(
y
)
T
v
and utilizing the symmetry of the scalar prod
uct, yields
g
(
y
)
,v
1
x
)
v
T
g
(
y
)
g
(
y
)
T
v.
v(
x
) = arg max
v
=1
G
ρ

·
G
ρ
(
y
−

y
∈N
(
x
)
Moreover, since
v
does not depend on
y
, it follows by linearity that
v
T
J
ρ
(
x
)
v,
v(
x
) = arg max
v
=1
(5.2)
where
1
x
)
g
(
y
)
g
(
y
)
T
.
J
ρ
(
x
)=
G
ρ

·
G
ρ
(
y
−
(5.3)

y
∈N
(
x
)
The outer product
g
(
x
)
g
(
x
)
T
is called the
structure tensor
at
x
,and
J
ρ
(
x
)is
called the
smoothed structure tensor
at
x
. Since it is a sum of outer products
weighted by nonnegative coecients, the smoothed structure tensor is a sym
metric positive semidefinite 2
2 matrix. Therefore, from the RayleighRitz
theorem [Horn and Johnson 85, Section 4.2], it follows that the solution of Equa
tion (5.2) is given by the eigenvector associated with the major eigenvalue of
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