Massachusetts
Institute
of
Technology
Department
of
Computer
Science
and
Electrical
Engineering
6.801/6.866
Machine
Vision
Handed
out:
2004
Nov
4th
Due
on:
2004
Nov
12th
Problem
1:
In
the
continuous
version
of
the
optical
ﬂow
problem,
we
ﬁnd
the
functions
u(x,
y)
and
v(x,
y)
that
minimize
2
2
2
(uE
x
+
vE
y
+
E
t
)
2
+
λ(u
2
+
u
y
+
v
+
v
y
)dx
dy
x
x
D
The
Euler
equations
for
this
problem
yield
λ±u
=
(uE
x
+
vE
y
+
E
t
)E
x
λ±v
=
(uE
x
+
vE
y
+
E
t
)E
y
where
E
x
,
E
y
,
and
E
t
are
the
derivatives
of
image
brightness.
What
are
the
natural
boundary
conditions?
Problem
2:
The
formulation
of
the
optical
ﬂow
problem
as
above
uses
as
a
measure
of
“unsmoothness’’
the
sum
of
squares
of
ﬁrst
partial
derivatives
of
the
components
of
the
optical
ﬂow.
Consider
a
sphere
centered
at
X
=
Y
=
0
,of
radius
R
,
rotating
about
an
axis
parallel
to
the
y
axis
with
angular
velocity
ω
.
Suppose
the
sphere
is
far
away
from
the
camera
(in
relation
to
its
radius)
and
we
approximate
perspective
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 Spring '04
 BertholdHorn
 Derivative, Partial differential equation, Frame rate, dx dy, optical ﬂow problem

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