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2. 22.058 - lecture 4, Convolution and
Fourier Convolution
Convolution
Fourier Convolution
Outline
· Review linear imaging model
· Instrument response function vs Point spread function
· Convolution integrals
• Fourier Convolution
• Reciprocal space and the Modulation transfer function
· Optical transfer function
· Examples of convolutions
· Fourier filtering
· Deconvolution
· Example from imaging lab
• Optimal inverse filters and noise
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3. Instrument Response Function
The Instrument Response Function is a conditional mapping, the form of the map
depends on the point that is being mapped.
{ }
IRF(x,y|x0,y0)=Sd(x-x0 )
22.058 - lecture 4, Convolution and
Fourier Convolution
This is often given the symbol h(r|r’).
Of course we want the entire output from the whole object function,
¥
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-¥
¥
òò
-¥
and so we need to know the IRF at all points.
d(y-y0 )
E(x,y)=
¥
òò
-¥
I(x,y)S{d(x-x0)d(y-y0)}dxdydx0dy0
E(x,y)=
¥
òò
-¥
I(x,y)IRF(x,y|x0,y0)dxdydx0dy0
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4. IRF(x,y|x0,y0)=PSF(x-x0,y-y0 )
22.058 - lecture 4, Convolution and
Fourier Convolution
Space Invariance
Now in addition to every point being mapped independently onto the detector,
imaging that the form of the mapping does not vary over space (is independent of
r0). Such a mapping is called isoplantic. For this case the instrument response
function is not conditional.
The Point Spread Function (PSF) is a spatially invariant approximation of the IRF.
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5. 22.058 - lecture 4, Convolution and
Fourier Convolution
Space Invariance
Since the Point Spread Function describes the same blurring over the entire sample,
The image may be described as a convolution,
or,
IRF(x,y|x0,y0)ÞPSF(x-x0,y-y0 )
¥
òò
-¥
E(x,y)=
I(x0,y0)PSF(x-x0,y-y0)dx0dy0
Image(x,y)=Object(x,y)ÄPSF(x,y)+noise
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6. Convolution Integrals
Let’s look at some examples of convolution integrals,
¥ òg(x')h(x-x')dx'
f(x)=g(x)Äh(x)=
So there are four steps in calculating a convolution integral:
22.058 - lecture 4, Convolution and
Fourier Convolution
#1. Fold h(x’) about the line x’=0
#2. Displace h(x’) by x
#3. Multiply h(x-x’) * g(x’)
#4. Integrate
-¥
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7. Convolution Integrals
22.058 - lecture 4, Convolution and
Fourier Convolution
Consider the following two functions:
#1. Fold h(x’) about the line x’=0
#2. Displace h(x’) by x x
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8. Convolution Integrals
22.058 - lecture 4, Convolution and
Fourier Convolution
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9. Convolution Integrals
22.058 - lecture 4, Convolution and
Fourier Convolution
Consider the following two functions:
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10. Convolution Integrals
22.058 - lecture 4, Convolution and
Fourier Convolution
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11. Some Properties of the Convolution
22.058 - lecture 4, Convolution and
Fourier Convolution
commutative:
associative:
multiple convolutions can be carried out in any order.
distributive:
fÄg=gÄf
fÄ(gÄh)=(fÄg)Äh
fÄ(g+h)=fÄg+fÄh
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12. Recall that we defined the convolution integral as,
¥ò
One of the most central results of Fourier Theory is the convolution
theorem (also called the Wiener-Khitchine theorem.
22.058 - lecture 4, Convolution and
Fourier Convolution
where,
Convolution Integral
fÄg=f(x)g(x'-x)dx
-¥
Á{fÄg}=F(k)×G(k)
f(x)ÛF(k)
g(x)ÛG(k)
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13. Convolution Theorem
In other words, convolution in real space is equivalent to
multiplication in reciprocal space.
22.058 - lecture 4, Convolution and
Fourier Convolution
Á{fÄg}=F(k)×G(k)
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14. Convolution Integral Example
We saw previously that the convolution of two top-hat functions
(with the same widths) is a triangle function. Given this, what is
the Fourier transform of the triangle function?
15 ?
22.058 - lecture 4, Convolution and
Fourier Convolution
15
10
5
-3 -2 -1 1 2 3
=
10
5
-3 -2 -1 1 2 3
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15. Proof of the Convolution Theorem
¥ò
fÄg=f(x)g(x'-x)dx
¥ò
F(k)eikxdk
-¥
¥ò
G(k')eik'(x'-x)dk'
-¥
22.058 - lecture 4, Convolution and
Fourier Convolution
The inverse FT of f(x) is,
-¥
f(x)=1
2p
and the FT of the shifted g(x), that is g(x’-x)
g(x'-x)=1
2p
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16. Proof of the Convolution Theorem
So we can rewrite the convolution integral,
¥ò
fÄg=f(x)g(x'-x)dx
¥ò
G(k')eik'(x'-x)dk'
-¥
¥ò
¥ò
dkF(k)dk'G(k')eik'x'1
eix(k-k')dx
-¥ 14 4 2 4 4 3
22.058 - lecture 4, Convolution and
Fourier Convolution
as,
-¥
¥ò
fÄg=1
¥ò
dxF(k)eikxdk
4p2
-¥
-¥
change the order of integration and extract a delta function,
fÄg=1
2p
2p
-¥
d(k-k')
¥ò
-¥
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17. Proof of the Convolution Theorem
¥ò
¥ò
¥ò
dkF(k)dk'G(k')eik'x'1
eix(k-k')dx
2p
-¥ 14 -¥
4 2 4 4 3
d(k-k')
¥ò
¥ò
dkF(k)dk'G(k')eik'x'd
(k-k')
¥ò
dkF(k)G(k)eikx'
-¥
22.058 - lecture 4, Convolution and
Fourier Convolution
or,
fÄg=1
2p
-¥
fÄg=1
2p
-¥
-¥
Integration over the delta function selects out the k’=k value.
fÄg=1
2p
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18. Proof of the Convolution Theorem
¥ò
dkF(k)G(k)eikx'
-¥
This is written as an inverse Fourier transformation. A Fourier
transform of both sides yields the desired result.
22.058 - lecture 4, Convolution and
Fourier Convolution
fÄg=1
2p
Á{fÄg}=F(k)×G(k)
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19. Fourier Convolution
22.058 - lecture 4, Convolution and
Fourier Convolution
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20. 22.058 - lecture 4, Convolution and
Fourier Convolution
Reciprocal Space
real space reciprocal space

21. 22.058 - lecture 4, Convolution and
Fourier Convolution
Filtering
We can change the information content in the image by
manipulating the information in reciprocal space.
Weighting function in k-space.
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22. 22.058 - lecture 4, Convolution and
Fourier Convolution
Filtering
We can also emphasis the high frequency components.
Weighting function in k-space.
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23. Modulation transfer function
i(x,y) = o(x,y)ÄPSF(x,y) + noise
c c c c
I(kx,ky)=O(kx,ky)×MTF(kx,ky)+Á{noise}
22.058 - lecture 4, Convolution and
Fourier Convolution
¥
òò
-¥
E(x,y)=
¥
òò
-¥
I(x,y)S{d(x-x0)d(y-y0)}dxdydx0dy0
¥
òò
-¥
E(x,y)=
¥
òò
-¥
I(x,y)IRF(x,y|x0,y0)dxdydx0dy0
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24. 22.058 - lecture 4, Convolution and
Fourier Convolution
Optics with lens
‡ Input bit mapped image
sharp =Import @" sharp . bmp" D;
Shallow @InputForm @sharp DD
Graphics@Raster@<<4>>D,Rule@<<2>>DD
s =sharp @@1, 1DD;
Dimensions @s D
8480,640<
ListDensityPlot @s , 8PlotRange school.edhole.com ÆAll , Mesh ÆFalse <D

25. 22.058 - lecture 4, Convolution and
Fourier Convolution
Optics with lens
crop =Take @s , 8220 , 347 <, 864, 572 , 4<D;
‡ look at artifact in vertical dimension
rot =Transpose @crop D;
line =rot @@20DD;
ListPlot @line , 8PlotRange ÆAll , PlotJoined ÆTrue <D
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26. 22.058 - lecture 4, Convolution and
Fourier Convolution
Optics with lens
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ü Fourier transform of vertical line to show modulation
ftline =Fourier @line D;
ListPlot @RotateLeft @Abs @ftline D, 64D, 8PlotJoined ÆTrue 2.5
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Ö Graphics Ö
ListPlot @RotateLeft @Abs @ftline D, 64D, 8PlotRange ÆAll , PlotJoined ÆTrue <D
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27. 22.058 - lecture 4, Convolution and
Fourier Convolution
Optics with lens
2D FT
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ftcrop =Fourier @crop D;
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28. 22.058 - lecture 4, Convolution and
Fourier Convolution
Optics with lens
Projections
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29. 22.058 - lecture 4, Convolution and
Fourier Convolution
Optics with lens
Projections
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xprojection = Fourier @RotateLeft @rotft @@64DD, 64DD;
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30. Optics with pinhole
22.058 - lecture 4, Convolution and
Fourier Convolution
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31. 22.058 - lecture 4, Convolution and
Fourier Convolution
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32. 22.058 - lecture 4, Convolution and
Fourier Convolution
2D FT
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33. Optics with pinhole
22.058 - lecture 4, Convolution and
Fourier Convolution
Projections
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34. 22.058 - lecture 4, Convolution and
Fourier Convolution
Projections
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35. Deconvolution to determine MTF of Pinhole
22.058 - lecture 4, Convolution and
Fourier Convolution
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MTF =Abs @pinhole Dê Abs @image D;
ListPlot @MTF , 8PlotRange ÆAll , PlotJoined ÆTrue <10
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Ö Graphics Ö
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36. FT to determine PSF of Pinhole
22.058 - lecture 4, Convolution and
Fourier Convolution
MTF =Abs @pinhole Dê Abs @image D;
ListPlot @MTF , 8PlotRange ÆAll , PlotJoined ÆTrue <D
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Ö Graphics Ö PSF =Fourier @RotateLeft @MTF , 64DD;
ListPlot @RotateLeft @Abs @PSF D, 64D, 8PlotRange ÆAll , PlotJoined ÆTrue <D
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37. Filtered FT to determine PSF of Pinhole
22.058 - lecture 4, Convolution and
Fourier Convolution
MTF =Abs @pinhole Dê Abs @image D;
ListPlot @MTF , 8PlotRange ÆAll , PlotJoined ÆTrue <D
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FÖi lGterraphi=csT Öable @Exp @- Abs @x - 64Dê 15D, 8x, 0, 128 <D ê êN;
ListPlot @Filter , 8PlotRange ÆAll , PlotJoined ÆTrue <D
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