johnchau123
Member level 1
Hi, everyone. When I did a question on Wiener filter, I do not know how to deal with it.
The question is as follows.
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It is known that a certain object g(x,y) of size 256X256 is filtered by h(x,y) and then is corrupted by white additive Gaussian noise n(x,y) to become i(x,y). We observe i(x,y), and our goal is to recover g(x,y). We intend to use Wiener filter.
It is defined that Φg(fx,fy) is power spectral density of g(x,y) and Φn(fx,fy) is the power spectral density of the noise n(x,y) and assume that Φn(fx,fy)÷Φg(fx,fy) ≈ 0.1 at all frequencies.
In this particular problem, we have some prior measurement for h(x,y) and found that it can be represented as h(x,y) = k1(exp[-k2((x²+y²)]w(x,y), w(x,y) = 1 for -3 ≤ x ≤ 3 and equals 0 for other x and y.
In other word, h(x,y) is a 7X7 filter. Furthermore, the parameters k1 and k2 are real positive values and complete specify h(x,y). We require that Σ[x = -∞ to ∞]Σ[y = -∞ to ∞] h(x,y) = 1.
(a) Sketch the shape of W(fx, 0) as a function of fx for a very small value of k2 (≈ 0).
(b) Sketch the shape of W(fx, 0) as a function of fx for a very large value of k2 (≈ ∞).
(c) Suppose now that the imaging system has been changed: n(x,y) is also filtered by h(x,y) before corrupting the observation. Derive the Wiener filter (should be accurate for all values of k2). How is it compared to the inverse filter of the original imaging system?
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Here's what have I thought or done.
In part (a) and part (b),
I think k1 is the scaling factor only since it helps to satisfy Σ[x = -∞ to ∞]Σ[y = -∞ to ∞] h(x,y) = 1. So, if k2 is very small, the effect of x and y will be larger than the case in which k2 is very large from the h(x,y) equation in which h(x,y) is directly proportion to exp[-f(x,y)]. So, for small value of k2, it behaves as a low pass filter and most high frequency components will not be filtered, a good filter. When k2 is very small, it behaves as a high pass filter in which the Wiener filter cannot function well.
While I have totally no idea in part (c).
Thank you very much for reading such as long question. I am sorry about that. But I hope you can help me, thanks.
Thanks all!!
The question is as follows.
_____________________________
It is known that a certain object g(x,y) of size 256X256 is filtered by h(x,y) and then is corrupted by white additive Gaussian noise n(x,y) to become i(x,y). We observe i(x,y), and our goal is to recover g(x,y). We intend to use Wiener filter.
It is defined that Φg(fx,fy) is power spectral density of g(x,y) and Φn(fx,fy) is the power spectral density of the noise n(x,y) and assume that Φn(fx,fy)÷Φg(fx,fy) ≈ 0.1 at all frequencies.
In this particular problem, we have some prior measurement for h(x,y) and found that it can be represented as h(x,y) = k1(exp[-k2((x²+y²)]w(x,y), w(x,y) = 1 for -3 ≤ x ≤ 3 and equals 0 for other x and y.
In other word, h(x,y) is a 7X7 filter. Furthermore, the parameters k1 and k2 are real positive values and complete specify h(x,y). We require that Σ[x = -∞ to ∞]Σ[y = -∞ to ∞] h(x,y) = 1.
(a) Sketch the shape of W(fx, 0) as a function of fx for a very small value of k2 (≈ 0).
(b) Sketch the shape of W(fx, 0) as a function of fx for a very large value of k2 (≈ ∞).
(c) Suppose now that the imaging system has been changed: n(x,y) is also filtered by h(x,y) before corrupting the observation. Derive the Wiener filter (should be accurate for all values of k2). How is it compared to the inverse filter of the original imaging system?
_____________________________
Here's what have I thought or done.
In part (a) and part (b),
I think k1 is the scaling factor only since it helps to satisfy Σ[x = -∞ to ∞]Σ[y = -∞ to ∞] h(x,y) = 1. So, if k2 is very small, the effect of x and y will be larger than the case in which k2 is very large from the h(x,y) equation in which h(x,y) is directly proportion to exp[-f(x,y)]. So, for small value of k2, it behaves as a low pass filter and most high frequency components will not be filtered, a good filter. When k2 is very small, it behaves as a high pass filter in which the Wiener filter cannot function well.
While I have totally no idea in part (c).
Thank you very much for reading such as long question. I am sorry about that. But I hope you can help me, thanks.
Thanks all!!