Graphics Reference
In-Depth Information
1
-2
1
Figure 4.7.
(a) Example discrete 9
×
9
Gaussian derivative filters used for com-
puting the Hessian, with
σ
=
1.2. The top
2
L
filter is
∂
(
x
,
y
,
σ)
and the bottom filter is
∂
x
2
2
L
(
x
,
y
,
σ)
∂
∂
y
. Light values are positive, black
values are negative, and gray values are
near zero. (b) Efficient box-filter approx-
imations of the filters at left. Gray values
are 0.
x
∂
1
-1
-1
1
(a)
(b)
4.1.4
Difference-of-Gaussians
Lowe [
306
] made the important observation that the Laplacian-of-Gaussian detector
could be approximated by a
Difference-of-Gaussians
or
DoG
detector. Why is this
the case? From the definition of the Gaussian function in Equation (
4.7
), we can
show that
∂
G
∂σ
2
G
=
σ
∇
(4.26)
If we assume that we generate Gaussian-smoothed images where adjacent scales
differ by a factor of
k
, then we can approximate
∂
G
∂σ
G
(
x
,
y
,
k
σ)
−
G
(
x
,
y
,
σ)
≈
(4.27)
k
σ
−
σ
Equating Equation (
4.26
) and Equation (
4.27
) gives that
2
2
G
(
k
−
1
)σ
∇
≈
G
(
x
,
y
,
k
σ)
−
G
(
x
,
y
,
σ)
(4.28)
That is, the difference of the Gaussians at adjacent scales is a good approximation
to the scale-normalized Laplacian, which we used to construct LoG features in the
previous section. This is highly advantageous, since to compute scale-space features
we had to create these Gaussian-smoothed versions of the image anyway. Figure
4.8
compares a difference of Gaussians with the Laplacian of Gaussian, showing that the
DoG is a good approximation to the LoG.
Therefore, the key quantity for DoG feature detection is the difference of adjacent
Gaussians applied to the original image:
D
(
x
,
y
,
σ)
=
(
G
(
x
,
y
,
k
σ)
−
G
(
x
,
y
,
σ))
∗
I
(
x
,
y
)
(4.29)
=
L
(
x
,
y
,
k
σ)
−
L
(
x
,
y
,
σ)
