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In-Depth Information
G(
σ
0
)
G(k
σ
0
)
−
G(
σ
0
)
G(k
σ
0
)
G(k
2
σ
0
)
−
G(k
σ
0
)
G(k
2
σ
0
)
G(k
3
σ
0
)
−
G(k
2
σ
0
)
G(k
3
σ
0
)
=
G(2
σ
0
)
G(
σ
0
)
Figure 4.9.
Lowe's octave structure for computing the scale-space DoG representation of an
image. In this example,
s
3. The white images represent Gaussians with the same sequence
of scales applied to octaves of images of different resolutions. The images at each octave are
half the size of the ones above. The gray images represent the differences of adjacent Gaussian-
filtered images. Features are detected as extrema in both the spatial and scale dimensions of
the DoG function. For example, the response at the black pixel must be larger or smaller than all
of its white neighbors.
=
The specific DoG-based detection algorithm proposed by Lowe includes several
additional refinements:
1. Instead of finding a single characteristic scale for each feature, local extrema
(i.e., both maxima and minima) of the DoG function in Equation (
4.29
) with
respect to both space and scale are computed. That is, the DoG value must
be larger or smaller than all twenty-six of its neighbors (see Figure
4.9
). To
compute the extrema at the “ends” of each octave of equally sized images,
we compute an extra pair of images such that the first and last images
of adjacent octaves represent the same scale. This way we don't have to
directly compare images of different sizes. (These extra images aren't shown in
Figure
4.9
.)
2. After detecting the DoG extrema, we further localize each keypoint's position
in space and scale by fitting a quadratic function to the twenty-seven val-
ues of
D
(
x
,
y
,
σ)
at and around the detected point
(
x
i
,
y
i
,
σ
i
)
. This function is
given by:
+
x
y
σ
x
y
σ
x
y
σ
1
2
g
Q
(
x
,
y
,
σ)
=
D
(
x
i
,
y
i
,
σ
)
+
(4.30)
i
where
g
is the 3
×
1 gradient and
is the 3
×
3 Hessian of the function
D
with respect to
using the usual finite-
difference approximations. The updated location and scale canbe shown tobe
(
x
,
y
,
σ)
, evaluated at
(
x
i
,
y
i
,
σ
i
)
