What Is the Fourier Transform of the Unblurred Image Intensity?

[Hart]

Homework Statement

I(x) is the intensity of an image after passing through a material which
blurs each point according to a point spread function given by:

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

The Fourier transform of I(x) is given by:

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

Where A is a constant.

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

(ii) Hence find the original unblurred image intensity I_{0}x

Homework Equations

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

The Attempt at a Solution

F[I_{0}] = \frac{1}{\sqrt{2 \pi}}\left( \frac{F<i>}{F<s>} \right)</s></i>

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

Calculation of F:

..after some calculations..

F<s> = \sqrt{\frac{2}{\pi}}\left(\frac{a}{a^{2}+x^{2}} \right) </s>

Calculation of F:

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

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

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

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

Therefore can now combine these expressions to get the answer:

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

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

 

[gabbagabbahey]

Calculation of F:

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

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

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

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

I'm not sure what you are doing here. You are given I(k)=F[I(x)], so why are you trying to (inverse?) Fourier transform it and then call the result F?

 

[Hart]

To clarify, for the last part of my attempt I used:

F[I_{0}] = \frac{1}{\sqrt{2 \pi}}\left( \frac{F<i>}{F<s>} \right)</s></i>

.. confused now where to go with this, not really sure how much of my attempt is correct or vaguely right or indeed just completely useless!?

 

[gabbagabbahey]

Again, you are already given the Fourier transform of I(x)

I(k)\equiv F

 

[Hart]

I(k)\equiv F

.. where does this come from? I don't recall it being stated before within this thread.


anyways, so:

I(k) = F<i> = \left[ \frac{2Ae^{-ix}}{ix(ab)^{2}}\right] </i>

??

..which is the Fourier transform of I(x).

 

[gabbagabbahey]

.. where does this come from? I don't recall it being stated before within this thread.

Did you not read my first response?

... You are given I(k)=F[I(x)]...

It certainly looks to me like I said I(k)=F[I(x)]

anyways, so:

I(k) = F<i> = \left[ \frac{2Ae^{-ix}}{ix(ab)^{2}}\right] </i>

??

..which is the Fourier transform of I(x).

I'm not sure how to make this any clearer. You are told what I(k) is in your problem statement

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

I(k) and F[I(x)] are two different ways of writing the exact same thing; both represent the Fourier transform of I(x).

F<i>=I(k)=\frac{A}{\left( a^{2}+k^{2} \right) \left( b^{2}+k^{2} \right)}</i>

Does this not make sense to you?

 

[Hart]

...

The Fourier transform of I(x) is given by:

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

Which is equivalent to stating:

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

Ok.. I do get that now.

So now I need to find Fourier transform I_{0}.. Assume I don't just set k=0 ?

 

[gabbagabbahey]

So now I need to find Fourier transform I_{0}.. Assume I don't just set k=0 ?

Right, so now you use the convolution theorem:

F[I_{0}] = \frac{1}{\sqrt{2 \pi}}\left( \frac{F<i>}{F<s>} \right)</s></i>

You now know what F is, so if you calculate F, you can use the above convolution theorem. There was nothing wrong with this part of your attempt, however, both your original F and your F were incorrect.

Calculation of F:

..after some calculations..

F<s> = \sqrt{\frac{2}{\pi}}\left(\frac{a}{a^{2}+x^{2}} \right) </s>

You do realize that F is supposed to represent the Fourier transform of S(x), and hence your result should be a function of k, not x, right?...Clearly, you've done something very wrong in your calculation.

Since you are trying to find the Fourier transform of S(x), a good place to start would be to find S(x)...So,, if S(x&#039;-x)=e^{-a|x&#039;-x|}, what is S(x)?.

 

[Hart]

Calculation of F:

f(x) = e^{-\alpha|x|}

\tilda{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-ikx}dx = \sqrt{\frac{2}{\pi}}\left(\frac{\alpha^{2}}{\alpha^{2}+k^{2}}\right)

.. it was actually meant to be with k not x in the final expression, just noticed it when I looked through my calculations again. Hopefully this is actually the correct method and answer.

Then using the Convolution Formula:

F[I_{0}] = \frac{1}{\sqrt{2 \pi}}\left( \frac{F<i>}{F<s>} \right) = \frac{1}{\sqrt{2 \pi}}\left( \frac{\frac{A}{\left( a^{2}+k^{2} \right) \left( b^{2}+k^{2} \right)}}{\sqrt{\frac{2}{\pi}}\left(\frac{\alpha^{2}}{\alpha^{2}+k^{2}}\right)}\right)</s></i>

F[I_{0}] = \frac{1}{\sqrt{2 \pi}}\left({\frac{A\sqrt{\frac{2}{\pi}}\left(\frac{\alpha^{2}}{\alpha^{2}+k^{2}}\right)}{\left( a^{2}+k^{2} \right) \left( b^{2}+k^{2} \right)}\right)

.. which just needs to be simplified.

Any good?!?

 

[gabbagabbahey]

Calculation of F:

f(x) = e^{-\alpha|x|}

Shouldn't this be

S(x) = e^{-a|x|}

...where are \alpha and f(x) coming from? F is the Fruier transform of S(x), not f(x).