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ECE-S490 Digital Image Processing
Homework 1
1. Let u be a random variable with a cumulative distribution function F. Let v = F(u)
be a new random variable. Show that v is uniformly distributed.
2. Let x(n), n=0,…,N-1 be a discrete time signal. Design a linear interpolator using
the closest 2 neighbors which magnifies the signal by a factor of 2 and 3. Identify
the coefficients of the low pass filter and the upsampling procedure with zero
padding. Apply your results to the following signal.
X(4n) = 1
X(4n+1) = 1
X(4n+2) = 0
X(4n+3) = 0,
for n = 0,1,2,…15
Plot the original signal and the interpolated one using Matlab.
3. Calculate the variance of the original signal in question number 2 and the
variance of the magnified version. Assume that the original signal has an additive
white noise with variance s, i.e.,
Y(n) = x(n) + w(n)
Where w is zero mean with standard deviation s, and x and w are statistically
uncorrelated, what will the noise variance be after magnification process.
4. Let c1 and c2 be two random variables with probability density function p1 and
p2, representing two classes. Let x be an observation that belongs to class1, c1
or class2, c2. Consider the following hypothesis testing method for classification
problem.
H1 : x > T  x belongs to class1
H2 : x =< T  x belongs to class2
For some threshold T. It is known that the threshold T can be chosen so that
total error is minimized where total error is defined as
Total error = probability that x is classified as c1, while it belongs to c2 +
probability that x is classified as c2, while it belongs to c1
i.e.
Total error = Pr { x>T | x belongs to c2} + Pr { x=<T | x belongs to c1}
Show that the total error is minimized when T is placed at the intersection of two
probability density functions. Use graphical arguments for 1D random variables.
5. Let h be a 2D FIR filter and |H| be its magnitude response function. Show that
sum of kernel coefficients is unity iff H has a unity DC gain.
6. Let the point spread function or impulse response of the filter be
2 4 2
16 x h = 0 0 0
-2 - 4 -2
Let the input image be I(m,n) = 16, n, m = 0, …, 3
Compute the filtered image using free boundary conditions.