ECE544
Statistical Image and Video Processing
Homework #1
Due TUEDAY September 23, 2008
Problem 1 (Object Randomly Thrown on the Floor). Let ψ(t) be a deterministic signal
defined over the square [0,1]2, and extended outside the boundaries of the square using periodic
extensions. Consider the random process x(t) = ψ(t−d), where dis a random location
parameter, uniformly distributed over the unit square.
(a) Give expressions for the mean and covariance of the process x(t). Is x(t) WSS?
(b) Is x(t) ergodic in the mean? ergodic in correlation?
(c) Is x(t) strictly stationary?
(d) Repeat (a)–(c) when the distribution of the location parameter dis nonuniform.
(e) Repeat (d) when ψ(t) is a strictly stationary periodic random process (with period equal
to the unit square), and dis independent of ψ(t).
Problem 2. Let x(t) be a stationary random process in the plane, and define y(t) = x(Rθt)
where Rθis a 2 ×2 rotation matrix with rotation angle θuniformly distributed over [0,2π).
(a) (8 pts) Prove that y(t) is stationary and isotropic.
(b) (7 pts) Prove that z(t) = T[y(t)] is stationary and isotropic, where T[·] is an arbitrary
point operation.
Problem 3. Formulate a statistical model for the 9 images in the Lecture Notes. A detailed
statistical model is not requested here, but you are asked to indicate whether the random
process that has generated those images is stationary, isotropic, and/or ergodic.
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