ECE544
Statistical Image and Video Processing
Homework #2
Due October 1, 2008
Problem 1 (Computer Experiment on AR Processes)
(a) Compute the mean and the first 4 ×4 covariance samples RX(n),0≤n1,n2≤3 for the five
images (tif format) on the Web site. This will be done by computing spatial averages and
making a stationarity assumption. Discuss the validity of fits of the form RX(n) = Cρ|n1|+|n2|
and RX(n) = Cρknk.
(b) Describe a simple and computationally efficient algorithm for generating realizations of the
following process: x(n) is stationary and Gaussian distributed with mean 100 and covariance
RX(n) = 30 ∗ρ|n1|+|n2|.
(c) Generate a realization of the process in (b), using the following values for ρ: .1, .5 and .95.
Problem 2 (Binary Gibbs Random Fields) Propose an energy function that strongly favors config-
urations made of nearest-neighbor pairs of black pixels surrounded by white pixels, or vice-versa.
Problem 3 (Computer Experiment on Ising Model) Use Gibbs sampling to generate realizations
from the Ising model
P(x) = Z−1e−1
TU(x), U(x) = X
i∼j
xixj
(where each xi=±1, and the sum is over nearest neighbors).
Use a periodic 50 ×50 lattice and equiprobable, i.i.d binary random variables as your initial distri-
bution. Use the following values for T−1: .1, .6, .88, 1.5 and 10. Observe the apparent convergence
of the distribution to a steady state as the number of iterations increases. Explain the aspect of
typical realizations for different values of Tupon convergence.
Problem 4 (Optional). Repeat Problem 3 with the 8-level Potts model.
Problem 5 (Maximum–Likelihood (ML) Parameter Estimation). Consider the GGMRF model
with scale parameter σand exponent p. The energy function takes the form
U(x) = X
i
x2
i+X
i∼j
xi−xj
σ
p
,
where 0 < p ≤2, and the second sum is over all nearest neighbors.
1. Explain why exact ML estimation of σand pis difficult.
2. Develop an approximate ML estimation method for σand pand apply it to the five tif images
on the Web page. Comment.
1