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should make sure that the derivatives we compute in creating the Harris matrix are
scale-invariant
— that is, that we compute the same matrix regardless of the image
resolution.
We can determine the correct scale normalization as follows [
127
]. Suppose that
we have two versions of the same image: a high-resolution one
I
(
x
,
y
)
and a low-
resolution one
I
(
x
,
y
)
kx
and
=
. The coordinates of the two images are related by
x
ky
, where
k
is a scale factor greater than 1. If we consider a block centered at
=
y
x
,
y
)
(σ
D
,
σ
I
)
(
with scales
in the low-resolution image, it will correspond to the block
kx
,
ky
)
σ
D
,
k
σ
I
)
centered at
in the high-resolution image. From the
chain rule, we can also compute that the image gradients at corresponding points
satisfy
(
with scales
(
k
I
=
∇
k
∇
I
. Substituting everything into Equation (
4.17
), we have that
1
k
2
H
(
σ
D
,
k
σ
I
)
=
x
,
y
,
σ
D
,
σ
I
)
H
(
x
,
y
,
k
(4.19)
where
H
and
H
are the scale-dependent Harris matrices computed for the high- and
low-resolution images, respectively. This implies that if we compute a scale space
where each derivation scale and integration scale is a multiple of a base scale
σ
, i.e.,
σ
0
,
k
2
{
σ
0
,
k
at
σ
0
,
...
}
, then we should compute the
scale-normalized Harris matrix
as
H
k
2
H
(
x
,
y
,
k
σ
D
,
k
σ
I
)
=
(
x
,
y
,
k
σ
D
,
k
σ
I
)
∂
L
(
x
,
y
,
k
σ
D
)
∂
2
∂
L
(
x
,
y
,
k
σ
D
)
∂
L
(
x
,
y
,
k
σ
D
)
x
∂
x
∂
y
k
2
G
=
(
x
,
y
,
k
σ
)
∗
∂
L
(
x
,
y
,
k
σ
D
)
∂
2
I
∂
L
(
x
,
y
,
k
σ
D
)
∂
L
(
x
,
y
,
k
σ
D
)
∂
x
∂
y
y
(4.20)
That is, in order to directly compare the response from the Harris matrix at different
scales, we must multiply Equation (
4.17
) by the compensation term
k
2
. Now we can
apply the Harris criterion with the same threshold at every scale, that is:
1. Create the scale space of
the image for a fixed set of scales
σ
D
∈
σ
0
,
k
2
{
σ
0
,
k
σ
0
,
...
}
, with
σ
I
=
a
σ
D
. Typical values are
σ
0
=
1.5,
k
∈[
1.2, 1.4
]
,
(see [
325
,
327
]).
2. For each scale, compute the scale-normalizedHarrismatrix in Equation (
4.20
)
andfind all localmaxima of theHarris function inEquation (
4.4
) that are above
a certain threshold.
and
a
∈[
1.0, 2.0
]
Features detected in this way are known as
multi-scale Harris corners
.
This new approach can detect multiple features centered at the same location
(
at different scales. However, we would often rather select a
characteristic scale
that defines the scale at which a given feature is most significant. Lindeberg [
286
]
suggested using the maximum of the
normalized Laplacian
for this purpose:
x
,
y
)
σ
2
2
G
2
G
∂
(
x
,
y
,
σ)
+
∂
(
x
,
y
,
σ)
NL
(
x
,
y
,
σ)
=
∗
I
(
x
,
y
)
(4.21)
∂
x
2
∂
y
2
value specified by the dot in each
image and plot the normalized Laplacian as a function of
Figure
4.4
illustrates the idea; if we fix the
(
x
,
y
)
, we see that the function
assumes amaximumat the same apparent scale in each case (visualized as the radius
σ
