ECE544
Statistical Image and Video Processing
Homework #3
Due October 15, 2008
Problem 1 (Hidden Markov Model). Consider the following HMM. The hidden sequence
S0, S1, S2, S3is a homogeneous Markov chain with alphabet Λ = {0,1}and transition prob-
ability matrix Q=0.9 0.1
0.1 0.9. The observed variables are X={X0,·· · , X7} ∈ Λ8. The
conditional pmf for Xgiven Sis given by
p(x|s) =
3
Y
i=0
Q(x2i|si)Q(x2i+1|si).
Give a computationally efficient algorithm for evaluating p(x7|s0).
Problem 2 (Simplified graphical model for hand silhouettes). Say the global orientation of
the hand is represented by a variable O, all fingers have prespecified lengths, and the angle
subtended by fingers 1 and 5 (the pinkie and the thumb) is denoted by A. The orientation
of the five fingers is represented by angular variables F1,· · · , F5. In this simplified model, the
hand is parameterized by the 7 variables F1,·· · , F5,O, A.
These variables are modeled as random with the following joint pdf:
p(f1,· · · , f5, o, a) = 1
Zψ15(f1, f5|o, a)ψ12(f1, f2)ψ23 (f2, f3)ψ34(f3, f4)ψ45(f4, f5)p(o)p(a).
1. Draw a graphical representation of this model, and give a computationally efficient algo-
rithm for evaluating p(f3).
2. Does this model appear physiologically reasonable?
Hint: to think about this question, lay one hand flat on a table, hold the pinkie and the
thumb firm with the other hand, and try to move the three free fingers one at a time.
3. Assume now that O=o∗and A= 10 (degrees) with probability 1, and that
ψi,i+1(fi, fi+1) = exp{fi−fi+1 }1{fi<fi+1}:i= 1,2,3
= exp{1
4(fi−fi+1)}1{fi<fi+1}:i= 4
ψ15(f1, f5|o∗,10) = exp{106|f1+ 5|+ 106|f5−5|}.
Write a computer program and evaluate the expected value of F3.
Hint: you should be able to make a rough guess of that value before writing the program.
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