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GEORGIA INSTITUTE OF TECHNOLOGY
School of Electrical and Computer Engineering
ECE 6258 Digital Image Processing Fall 2003
Problem Set #
Issued: Friday, September 12, 2003 Due (live): Monday, September 22, 2003 Due (video): Monday, October 6, 2003
Problem 3.1 (2-D Recursive Systems): A linear shift-invariant two-dimensional sys- tem is defined by the difference equation
y[n 1 , n 2 ] − 0. 7 y[n 1 + 1, n 2 ] + 0. 2 y[n 1 + 1, n 2 + 1] = x[n 1 , n 2 ]
(a) Determine the system function Hz (z 1 , z 2 ). (b) If the filter is implemented using the recursion equation
y[n 1 , n 2 ] = x[n 1 , n 2 ] + 0. 7 y[n 1 + 1, n 2 ] − 0. 2 y[n 1 + 1, n 2 + 1]
determine the region of support of the impulse response, h[n 1 , n 2 ], i.e. those values of [n 1 , n 2 ] where the impulse response can be nonzero. (c) Specify a sufficient set of boundary values for evaluating y[n 1 , n 2 ] if x[n 1 , n 2 ] is nonzero for 0 ≤ n 1 ≤ N 1 − 1, 0 ≤ n 2 ≤ N 2 − 1. (d) Determine whether or not the filter is stable.
Problem 3.2 (Image Sampling): In a feeble attempt to unify the nation’s sampling grids, Hannibal Hamlin once decreed that henceforth all analog imagery in the coun- try should have the spectral support (bandwidth) indicated below. Fortunately, the decree was struck down by the Supreme Court before it would have become effective.
3 π
− 3 π
− 4 π 4 π
(a) Determine an aliasing matrix U that will permit periodic replication of the spec- trum without aliasing in a way that will minimize the required sampling density. (b) Determine a consistent sampling matrix V that would define an optimal sampling lattice for this spectral support.
(c) What is the sampling density at which all of the country’s images would have been sampled?
Problem 3.3 (Video Sampling): A (somewhat boring) analog video scene consists of a single object o(x, y) moving in front of a black background with constant velocity. The resulting video signal, s(x, y, t) can thus be modeled as
s(x, y, t) = o(x − vxt, y − vyt),
where vx and vy are the horizontal and vertical components of the velocity, measured in mm/s. The two-dimensional Fourier transform of o(x, y) is bandlimited with a circularly-shaped region of support with a radius of W rad/mm.
(a) Determine the (continuous-variable) 3-D Fourier transform of s(x, y, t) in terms of the quantities that have been defined. (b) If the video signal can be represented exactly from its samples, determine a suf- ficient sampling lattice and describe a system that can be used to recover the video signal from its samples. If it cannot be represented exactly, explain why not. (Note: This problem requires some thought.)
Problem 3.4 (The 2-D Discrete Fourier Series (DFS)): Suppose that ˜x[n 1 , n 2 ] is a rectangularly periodic sequence with horizontal period N 1 and vertical period N 2. The sequence ˜x 1 [n] = ˜x[n, n] is then a periodic one-dimensional sequence.
(a) Show that ˜x[n] is a periodic sequence with period N 1 N 2. Show that if N 1 and N 2 have any common integral factors then ˜x[n] will also have a smaller period. (Hint: Draw some pictures.) (b) Assuming that N 1 and N 2 have no common factors, show that the samples of the DFS coefficients X˜ 1 [k] are equal to selected values of X˜[k 1 , k 2 ] and determine the mapping between k and [k 1 , k 2 ].
If the row-column algorithm is used to evaluate the 2-D DFS coefficients, X˜[k 1 , k 2 ] this is an efficient algorithm for computing the one-dimensional DFS X˜ 1 [k] known as the prime factor algorithm. It predates the Cooley-Tukey (FFT) algorithm and is more efficient than that algorithm, but places an inconvenient restriction on the length of the transform.
