⃗
that satisfies the Optical Flow
1. Smallest Optical Flow (4 pts): What velocity
Constraint Equation
+
⃗
+ = 0 has the smallest magnitude | |? Hint: This can be
solved geometrically as was outlined in class by considering the OFCE in , space.
v
(0,−
(− ,0)
u
2. Moving Gaussian Blob (6 (pts): , , A) Gaussian=1 blob is observed over2 time to have brightness
−2 2( 2−2( 1+ 2) +( 1+ 2) )
a. What are , , and ? Hint: You should find that these derivatives have a simple form.
b. The Optical Flow Constraint Equation is + + = 0. Write this out using the results of Part a. and simplify it as much as possible. For example, you should be able to cancel terms that occur in each of , , and .
( , , ) =
+ 1
[( −
)2
+(− )2
]
3. Quadratic Optical Flow (8 pts : Suppose the image brightness is given by
0
2
1
2
a. What are Ix, Iy, and It? Hint: You should find that these derivatives have a simple form.
b. Express the Optical Flow Constraint Equation + + = 0 in the simplest terms possible for this image sequence.
c. The equation from b. must hold for all x, y, and t. Find a constant solution for u and v that makes this true, that is, such that u and v do not depend on x, y, and t.
flow, where at each iteration, the optical flow ( , ), ( , ) is updated according to
4. Iterative Optical Flow(8 pts): We saw in class an iterative method for computing optical
( , )
new
2
+ 4
−1
∑
old( ) −
= [
∈neighbors( , )
old
[ ( , )]
2+4]
∑
( ) −
∈neighbors( , )
a. Show that this is equivalent to
2
+ 4
−
∑
old( ) −
( , )
new
1
∈neighbors( , )
[ ( , )]
=
4
2 + 4 2
+ 16
[−
2
+ 4]
∑
old
( ) −
∈neighbors( , )
( , )
= ̅
−
(
̅
+
̅
+ )
b. Show that this is equivalent to update equations
new
old
4
old
old
2
+ 2
+
( , ) = ̅
−
(
̅
+
̅
+ )
new
old
4
old
old
2
+ 2
+
where ̅
old
old
, ̅
are the averages of the 4 neighbors of ( , ), ( , ). Hint: You only
need to show this for new because
new follows an identical derivation.
c. In the case that
= 0, what do the update equations reduce to?
