Graphics Reference
In-Depth Information
intensities have been normalized as described in Section
4.2.1
. The dimension of the
descriptor is the number of intensity bins times the number of rings. Since there are
no angular subdivisions of the rings, the descriptor is rotation-invariant.
4.2.4
Invariant-Based Descriptors
The first step in SIFT and the similar approaches in the previous section is the rota-
tion of a patch around the estimated feature location so that the dominant gradient
orientation points in a consistent direction. An alternative is to design a descriptor
that's invariant to the rotation of the patch in the first place, bypassing this estimation
and explicit rotation. Such approaches are generally based on
invariant functions
of the patch pixels with respect to a class of geometric transformations — typically
rotations or affine transformations. For example, the spin image discussed in the
previous section is a crude rotation-invariant descriptor, but substantial discrimi-
native information may be lost as the intensities from increasingly larger rings are
aggregated into histograms.
Schmid and Mohr [
430
] popularized the idea of using
differential invariants
for
constructing rotation-invariant descriptors. That is, the descriptor is constructed
using combinations of
increasingly higher-order derivatives of the Gaussian-
smoothed image
L
(
x
,
y
)
given by Equation (
4.16
). For example, the total intensity
∂
L
(
x
,
y
)
∂
2
∂
L
(
x
,
y
)
∂
2
(
x
,
y
)
, sum of squared gradient magnitude
(
x
,
y
)
L
(
x
,
y
)
+
, and
x
y
sum of Laplacians
(
x
,
y
)
∂
2
L
2
L
(
x
,
y
)
+
∂
(
x
,
y
)
y
2
are all invariant to rotation as long as the
sums are taken over equivalent circular regions (such as the ones we obtain at the
end of the affine adaptation process, Figure
4.12
). A vector of these three quanti-
ties could be used as a descriptor. Since an image can be uniquely defined by its
derivatives (e.g., consider a Taylor series), the more differential invariants we use,
the more uniquely we describe the region around the feature location. For exam-
ple, there are five differential invariants that use combinations of up to second
derivatives, and nine differential invariants that use combinations of up to third
derivatives. On the other hand, the higher-order derivatives we need, the more diffi-
cult they are to accurately estimate from an image patch, especially in the presence
of noise.
Another approach is to use
moment invariants
, which are computed using both
image intensities and spatial coordinates. The
x
2
∂
∂
th
moment of a function defined
(
m
,
n
)
over a region is the average value of
x
m
y
n
f
. The (0,0) moment is thus the average
value of the function, and the (1,0) and (0,1) moments give the center of gravity
with respect to the function. Higher-order moments represent moments of inertia
and skewness. Flusser [
147
] derived combinations of moments that were invariant
to rotations based on earlier work by Hu [
205
]. Van Gool et al. [
510
] enumerated the
affine-invariant moments up to
m
(
x
,
y
)
2, which can be used to construct affine-
invariant feature descriptors. Mikolajczyk and Schmid [
328
] suggested that moment
invariants could be applied to
x
and
y
gradient images as well.
Schaffalitzky and Zisserman [
423
] proposed to use a bank of complex filters whose
magnitude responses are invariant to rotation, indexed by two positive integers and
given by the coefficients
+
n
≤
m
n
G
K
mn
(
)
=
(
+
)
(
−
)
(
)
x
,
y
x
iy
x
iy
x
,
y
(4.40)
