Digital Signal Processing Reference
In-Depth Information
Fig. 1: For image pyramid, there are 4 octaves, and each octave has 5 scales. Initial
image is repeatedly convolved with Gaussians to produce the set of scale space
images. Maxima and minima of the difference-of-Gaussian images are detected by
comparing a pixel (marked with
×
) to its 26 neighbors in 3
×
3 regions at the current
and adjacent scales.
2.2
Accurately Locate Keypoint
Brown's approach uses the Taylor expansion of the scale space function,
,
to fit a 3D quadratic function of the candidate feature point to determine the
interpolated location of the maximum:
Dxy
( ,
,
σ
)
T
2
∂
1
D
D
∂
T
DX D
()
=
( )
+
X
+
X
(4)
X
X
0
∂
X
2
∂
X
T
(, , )
Where
D
and its derivatives are evaluated at the sample point,
X xy
σ
=
is the
T
offset from this point and
is the candidate extrema point. Taking
the derivative of (4) formula with respect to
X
and getting the extreme location.
=
(,
,
)
X XY
σ
0
00 0
2
T
−
1
2
∂
D
∂
D
D
D
∂
D
∂
∂
^
=
+
X
=
0
→= −
(5)
X
2
2
∂
XX
∂
∂
X
∂
X
∂
X
1) Reject low contrast points
The function value at the extremum,
^
()
D
X
, which can be obtained by substituting
^
()
equation (5) into (4). If
D
X
is less than 0.03, it will be discarded, otherwise left.
T
1
∂
D
^
^
DD
()
=
( )
2
+
(6)
X
X
X
0
∂
X
2) Eliminate edge responses
The principal curvatures can be computed through a
22
Hessian matrix,
H
. The
×
maximum eigenvalue is
α
, and the minimum eigenvalue is
β
,
α
=
r
β
,
2
( )
r
+
2
()
H
=+=+
α
β
,
Det H
DD
D
()
=
−
(
)
=
α β
, If
R
<
(
r
T
D
D
r
XX
YY
XX
YY
XY
takes 10), the keypoint will be left, otherwise discarded.
DD
H
DD
XX
XY
=
(7)
YX
YY
2
2
2
r
H
()
(
αβ
αβ
+
)
( )
r
+
T
R
=
=
=
(8)
Det H
()
r
