EECS 101: HOMEWORK #8

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1. Consider an image irradiance sequence E(x, y, t) in continuous space (x, y) and time (t)
E(x, y, t) = cos(y − t) + 1 0 ≤ x ≤ 2π, 0 ≤ y ≤ 2π, t ≥ 0
a) Determine the components of the optical flow vector u(x, y, t) and v(x, y, t) for this sequence.
b) Write down the optical flow constraint equation.
c) Compute the time and space derivatives of image irradiance that appear in the optical flow
constraint equation for the function given above.
d) Use your answers to the previous parts to show that the optical flow constraint equation holds
for all x, y, t.
e) For the optical flow computed in part a), compute the smoothness integral
es(t) = Z Z (u
2
x + u
2
y + v
2
x + v
2
y
)dxdy
2. Assume the stereo geometry given in class with origin at (0,0,0), a baseline of b=2cm separating the two lens centers, and a distance between each lens and corresponding image plane
of f=0.5cm. Suppose the left and right image planes are each a 1cm x 1cm square with the
origin of each local coordinate system at the center of the square. Assume that we are viewing
a black plane with three bright white spots. The three spots are observed in the left image
at coordinates (x

l
, y′
l
) = {(0.0, 0.2),(−0.3, 0.2),(0.0, 0.1)} and in the right image at coordinates
(x

r
, y′
r
) = {(−0.1, 0.2),(0.1, 0.2),(0.2, 0.1)}
a) Determine the image correspondences for the points.
b) Determine the coordinates (x, y, z) of the three bright points in the scene.
c) Determine the gradient space representation (p, q) for the plane in the scene.
3. Consider a pattern classification problem where we would like to discriminate between two
materials M1 and M2 using a measured color vector (R, G, B) of the unknown material. Assume
that the a priori probability of M1 is 2/5 and that the a priori probability of M2 is 3/5. Suppose
that the probability density for M1 is uniform on the cube 50 ≤ R ≤ 100, 30 ≤ G ≤ 60, 40 ≤
B ≤ 80. Suppose that the probability density for M2 is uniform on the cube 80 ≤ R ≤ 140, 30 ≤
G ≤ 80, 60 ≤ B ≤ 100.
a) What is the conditional pdf p(R, G, B|M1) as a function of (R, G, B)?
b) What is the conditional pdf p(R, G, B|M2) as a function of (R, G, B)?
c) What is the a posteriori probability P(M1|R, G, B) as a function of (R, G, B)?
d) What is the a posteriori probability P(M2|R, G, B) as a function of (R, G, B)?
e) What is the best guess for what material we are looking at as a function of (R, G, B)?
f) What is the probability of error as a function of (R, G, B) if we take the guess in part e)?