(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[tex]I(x)[/tex] is the intensity of an image after passing through a material which

blurs each point according to a point spread function given by:

[tex]S\left(x'-x\right)=e^{-a\left|x'-x\right|}[/tex]

The Fourier transform of [tex]I(x)[/tex] is given by:

[tex]I(k) = \frac{A}{\left( a^{2}+k^{2} \right) \left( b^{2}+k^{2} \right)}[/tex]

Where A is a constant.

(i) Find the Fourier transform [tex]I_{0}^{~}[/tex] of the unblurred image intensity.

(ii) Hence find the original unblurred image intensity [tex]I_{0}x[/tex]

2. Relevant equations

[tex]I(x') = \int_{-\infty}^{\infty}I_{0}(x)S\left(x'-x\right)dx = \left(I_{0}*S \right)(x')[/tex]

3. The attempt at a solution

[tex]F[I_{0}] = \frac{1}{\sqrt{2 \pi}}\left( \frac{F}{F} \right)[/tex]

(Then can inverse Fourier transform this to get the undistorted image intensity [tex][I_{0}k][/tex])

Calculation of F:

..after some calculations..

[tex] F = \sqrt{\frac{2}{\pi}}\left(\frac{a}{a^{2}+x^{2}} \right) [/tex]

Calculation of F:

[tex]F= A\int_{-\infty}^{\infty}\frac{e^{-ixk}}{\left(a^{2}+k^{2}\right)+\left(b^{2}+k^{2}\right)} dk[/tex]

[tex]F= A\left[\frac{e^{-ixk}}{(-ix)(a^{2}+k^{2})(b^{2}+k^{2})}\right]\right|^{\infty}_{-\infty}[/tex]

[tex]F= A\left[ \frac{2e^{-ix}}{ix(ab)^{2}}\right][/tex]

[tex]F= \left[ \frac{2Ae^{-ix}}{ix(ab)^{2}}\right][/tex]

Therefore can now combine these expressions to get the answer:

[tex]F[I_{0}] = \left( \frac{A\left(iax(ab)^{2}\right)e^{-ix}}{a^{2}+x^{2}} \right)[/tex]

But this looks rather messy, so I assume I've done something wrong somewhere!?!

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# Homework Help: Convolution - Image Processing

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