Graphics Reference
In-Depth Information
Figure 4.5.
Harris-Laplace features detected in a pair of images of the same scene. The radius of
each circle indicates the characteristic scale of the feature located at that circle's center. (Fairly
aggressive non-maximal suppression was used so that the features don't overwhelm the image.
In practice, a much larger number of features is detected.)
(Note that it's still possible to generate multiple detections at the same
)
location with different characteristic scales, but the scales will be somewhat
separated.)
(
x
,
y
Figure
4.5
illustrates a pair of images and a subset of their detected Harris-Laplace
features, using circles to indicate each feature's scale. We can see that the features
center on distinctive regions of the image and that the detected scales are natural.
More important, many of the same scene locations are detected at the same apparent
scales, indicating thepromiseofHarris-Laplace features for automaticmatching. This
critical property is called
scale covariance
.
7
4.1.3
Laplacian-of-Gaussian and Hessian-Laplace
In addition to proposing the Laplacian for scale selection, Lindeberg [
286
] also pro-
posed a method for feature detection using the
scale-normalized Hessianmatrix
of
second derivatives:
2
L
2
L
∂
(
x
,
y
,
σ
D
)
∂
(
x
,
y
,
σ
D
)
∂
x
∂
y
∂
x
2
S
2
D
(
x
,
y
,
σ
)
=
σ
(4.23)
D
2
L
2
L
∂
(
x
,
y
,
σ
D
)
∂
(
x
,
y
,
σ
D
)
∂
x
∂
y
∂
y
2
As before,
L
(
x
,
y
,
σ
)
is the Gaussian-filtered image at the specified scale. Note that
D
2
L
2
L
∂
(
x
,
y
,
σ
D
)
+
∂
(
x
,
y
,
σ
D
)
trace
S
2
D
(
x
,
y
,
σ
)
=
σ
(4.24)
D
x
2
y
2
∂
∂
∂
2
2
L
2
L
2
L
(
x
,
y
,
σ
D
)
∂
(
x
,
y
,
σ
D
)
∂
(
x
,
y
,
σ
D
)
det
S
4
D
(
x
,
y
,
σ
)
=
σ
−
(4.25)
D
∂
x
2
∂
y
2
∂
x
∂
y
In particular, we can see that the absolute value of the Hessian's trace is the same as
the normalized Laplacian in Equation (
4.21
).
7
As seen earlier, this property is often colloquially called
scale invariance
.
