Problems, INF 269, Chapter 2
This set of problems is intended for Wednesday September 4, 2002.
It starts with a set of theoretical problems, with the possibility to discuss the various
consequences of the “theorems” shown, and to perform some interactive demonstrations.
Then comes a programming problem, also with possibilities for discussions and demo.
• Theoretical problems
1) Show that if f (x, y) has a Fourier
transform
F (u, v), then |ab|1/2 f (ax, by) has
a Fourier transform |ab|−1/2 F ua , vb .
Discuss the consequences of this.
2) Show that if f (x, y) has a Fourier transform F (u, v), then f (x − a, y − b) has
a Fourier transform F (u, v)e−2πi(au+bv) . Discuss the consequences of this.
3) Use the convolution theorem to obtain an expression for the convolution of two
1D zero-mean Gaussians
√
(x)2
1 − (x)22
1
e− 2a2 ∗ √
e 2b
2πa
2πb
and similarly for the 2D circular symmetric case
1 − x2 +y2 2
1 − x2 +y2 2
2a
e
∗
e 2b
2πa2
2πb2
• Practical programming exercise: (this will not count as a mandatory exercise)
4)
∗ Write a program that creates an N ×N image where the pixel value increases linearly from f (x, y) = ∆ in the lower left hand corner to f (x, y) =
255 − ∆ in the upper right hand corner.
∗ Implement Algorithm 2.3 on page 36 of the textbook to add zero-mean
Gaussian noise to the image f , thus creating a new image g. Do this for a
couple of values of the standard deviation σ. What limitations does ∆ put
on σ ?
∗ Plot the distribution (histogram) of ε = f (x, y) − g(x, y) for the chosen
values of σ.
∗ Compute the actual mean and standard deviation of these distributions/histograms,
and explain if these eventually deviate from a “perfect” zero-mean Gaussian.
∗ How can you use local first order statistics to find whether the noise in an
image is additive or multiplicative?
See you on September 4!
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