Graphics Reference
In-Depth Information
in the transformed block remain the same from frame to frame, we allow for a scale
and shift:
e
·
I
(
ax
+
by
+
u
,
cx
+
dy
+
v
,
t
+
1
)
+
f
=
I
(
x
,
y
,
t
)
∀
(
x
,
y
)
∈
W
(4.15)
When training images of the target feature under different illuminations are available,
more advanced photometric models can be obtained [
184
].
4.1.2
Harris-Laplace
A major drawback of Harris corners is that they are only extracted for a fixed, user-
defined block size. While setting this block size to a small value (e.g., 7
7 pixels)
enables the extraction of many fine-detail corners, it would also be useful to extract
features that take up a relatively larger portion of the image. That is, we would like to
detect features at different spatial
scales
. Detecting features in
scale space
is a critical
aspect of most modern feature detectors. We first describe howHarris corners can be
extracted at multiple scales, and in the following sections introduce new criteria not
based on the Harris matrix. Lindeberg [
285
,
286
] pioneered the use of scale space for
image feature detection, providingmuchof the theoretical basis for subsequent work.
The key concept of scale space is the convolution of an image with a Gaussian
×
L
(
x
,
y
,
σ
)
=
G
(
x
,
y
,
σ
)
∗
I
(
x
,
y
)
(4.16)
D
D
where
D
takes on a sequence of increasing values, typically a geometrically increas-
ing sequence of scales
σ
0
,
k
2
D
increases, the image gets blurrier,
since the Gaussian acts as a low-pass filter. The idea is similar to the Gaussian pyra-
mid discussed in Section
3.1.2
, except that the output image is not downsampled
after convolution.
Revisiting Section
4.1.1.1
, we can rewrite the Harris matrix evaluated at a point
{
σ
0
,
k
σ
σ
0
,
...
}
.As
σ
(
x
,
y
)
using derivation scale
σ
D
and integration scale
σ
I
as:
∂
L
(
x
,
y
,
σ
D
)
∂
2
∂
L
(
x
,
y
,
σ
D
)
∂
L
(
x
,
y
,
σ
D
)
∂
∂
x
x
y
H
(
x
,
y
,
σ
D
,
σ
I
)
=
G
(
x
,
y
,
σ
I
)
∗
∂
L
(
x
,
y
,
σ
D
)
∂
2
(4.17)
∂
L
(
x
,
y
,
σ
D
)
∂
L
(
x
,
y
,
σ
D
)
∂
x
∂
y
y
Note that
∂
L
(
x
,
y
,
σ
)
=
∂
G
(
x
,
y
,
σ
)
D
D
∗
I
(
x
,
y
)
(4.18)
∂
x
∂
x
since differentiation and convolution are commutative, which implies that we can
take the derivative and smooth the image in either order.
If we compute features at different scales and look at the eigenvalues of the Harris
matrix to decidewhich features are themost significant, we don't want larger features
to outweigh smaller features just because they're computed over a larger domain. We
would like to
scale-normalize
the Harris matrix so that feature quality can be directly
compared across scales. Furthermore, it's sometimes desirable to detect and match
features at different scales — for example, to match a large square of pixels from a
zoomed-in shot with a small square of pixels from a wider angle shot. In this case, we
5
The use of
L
to denote this Gaussian-blurred image is conventional notation and shouldn't
be confused with the Laplacian pyramid images from Chapter
3
.
(
x
,
y
,
σ)
