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ECE468: HOMEWORK 7
due 03/
(20pts) Problem 7.15 from the textbook.
(60pts) Given a linear system
g(x, y) = h(x, y) ⋆ f (x, y) + η(x, y) ,
where f (x, y) and g(x, y) are input and output, h(x, y) is the system’s impulse response, and η(x, y) is additive noise that is known to be spatially invariant and uncorrelated with the input. Let F (u, v), G(u, v), H(u, v), and N(u, v) denote the DFT of the corresponding 2D space-domain functions f (x, y), g(x, y), h(x, y), and η(x, y). Also, let Sf (u, v) and Sη(u, v) be the mean power spectral densities of input f (x, y) and noise η(x, y) defined by
Sf (u, v) = E
|F (u, v)|^2
, Sη(u, v) = E
|N(u, v)|^2
Derive the expression of Wiener filter HW (u, v)
HW (u, v) =
H∗(u, v)Sf (u, v) |H(u, v)|^2 Sf (u, v) + Sη(u, v)
that is used to reconstruct f (x, y) as
F^ ˆ (u, v) = HW (u, v)G(u, v) ,
so that the estimate Fˆ (u, v) minimizes the mean squared error ǫ
ǫ(u, v) = E
|F (u, v) − Fˆ (u, v)|^2
Hint: To minimize ǫ(u, v) you need to solve dǫ(u, v) dHW (u, v)
where ǫ(u, v) = E
|F (u, v) − HW (u, v)[H(u, v)F (u, v) + N(u, v)]|^2
Also, since f (x, y) and η(x, y) are uncorrelated, we have
E {F (u, v)N∗(u, v)} = E {F (u, v)∗N(u, v)} = 0.
- (20pts) (a) Write a MATLAB code for generating a degraded image g(x, y) from the input f (x, y), where the degradation is caused by a linear motion blur and additive Gaussian noise as
g(x, y) = h(x, y) ⋆ f (x, y) + η(x, y). It is known that the linear motion blur comes from a horizontal translation of the image by 7 pixels and a rotation by 45 ◦. Also, the Gaussian noise has zero mean and variance of 0.01, for pixel values defined in the interval [0, 1]. As input use the image provided on the course website.
(b) Write a MATLAB code for reconstructing f (x, y) as fˆ (x, y) = hW (x, y) ⋆ g(x, y) by using the Wiener filter, hW (x, y), where the parameters of the image degradation are known and equal to those given in (a).
Submit the listing of your MATLAB code, Figure 1 showing the degraded image from (a), and Figure 2 showing the reconstructed image from (b).
Hint: Use the following commands: ‘fspecial(’motion’,...)’, ‘imfilter()’, ‘imnoise(’gaussian’,...)’, ‘deconvwnr()’
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