Graphics Reference
In-Depth Information
U
2
λ
(5.31)
v
0
=
−
+
Ω
1
.
U
3
λ
+
U
1
λ
x
u
x
=
λ
2
.
(5.32)
u
y
=
Ω
3
+
U
1
λ
y
(5.33)
λ
2
.
Ω
3
+
U
2
λ
x
λ
2
v
x
=
−
(5.34)
+
U
2
λ
y
U
3
λ
(5.35)
v
y
=
λ
2
.
The system of equations is underconstrained as there are fewer number of
equations than there are number of unknowns.
λ
determines the structure of
the scene.
An image feature or shape will undergo a transformation for the image
velocity field. The transformation from a shape at time
t
to the deformed
shape at time
t
+
δt
can be approximated by an ane transformation. We can
write as the first order approximation
u
v
=
u
0
v
0
+
u
x
x
y
+
O
x
2
,xy,y
2
)
.
u
y
(5.36)
v
x
v
y
Cipolla and Blake neglected the non-linear term
O
x
2
,xy,y
2
) in their anal-
ysis. One can decompose the velocity gradient term into three components
with each term having a simple geometric significance, invariant under the
transformation of the image coordinate system.
u
x
=
0
+
div
v
2
10
01
+
u
y
1
10
−
curl
v
2
v
x
v
y
cos
μ
10
0
cos
μ
−
sin
μ
sin
μ
def
v
2
sin
μ
cos
μ
−
1
−
sin
μ
cos
μ
0
+
div
v
2
10
01
(5.37)
1
10
−
curl
v
2
=
cos 2
μ
,
sin 2
μ
+
def
v
2
sin 2
μ
−
cos 2
μ
where
div
v
=(
u
x
+
v
y
)
curl
v
−
(
u
y
−
v
x
)
=
(5.38)
(
def
v
)cos2
μ
=(
u
x
−
v
y
)
(
def
v
)sin2
μ
=(
u
y
+
v
x
)
.
The curl, divergence, and magnitude of deformation are scalar invariants and
do not depend on a particular choice of coordinate system. The axes of max-
imum extension and compression rotate with rotations of the image plane
Search WWH ::
Custom Search