Game Development Reference
In-Depth 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 two-dimensional 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 Rayleigh-Ritz
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
×