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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
22.058 - lecture 4, Convolution and Fourier Convolution

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 )
This is often given the symbol h(r|r’).
Of course we want the entire output from the whole object function,
¥
òò
-¥
¥
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

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.
IRF(x,y|x0,y0)=PSF(x-x0,y-y0 )
The Point Spread Function (PSF) is a spatially invariant approximation of the IRF.

Space Invariance
Since the Point Spread Function describes the same blurring over the entire sample,
IRF(x,y|x0,y0)ÞPSF(x-x0,y-y0 )
The image may be described as a convolution,
or,
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òò
-¥
E(x,y)=
I(x0,y0)PSF(x-x0,y-y0)dx0dy0
Image(x,y)=Object(x,y)ÄPSF(x,y)+noise

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:
#1. Fold h(x’) about the line x’=0
#2. Displace h(x’) by x
#3. Multiply h(x-x’) * g(x’)
#4. Integrate
-¥

Consider the following two functions:
#1. Fold h(x’) about the line x’=0
#2. Displace h(x’) by x x

Consider the following two functions:

Some Properties of the Convolution
commutative:
associative:
fÄg=gÄf
fÄ(gÄh)=(fÄg)Äh
multiple convolutions can be carried out in any order.
distributive:
fÄ(g+h)=fÄg+fÄh

Convolution Integral
Recall that we defined the convolution integral as,
¥ò
fÄg=f(x)g(x'-x)dx
-¥
One of the most central results of Fourier Theory is the convolution
theorem (also called the Wiener-Khitchine theorem.
where,
Á{fÄg}=F(k)×G(k)
f(x)ÛF(k)
g(x)ÛG(k)

Convolution Theorem
Á{fÄg}=F(k)×G(k)
In other words, convolution in real space is equivalent to
multiplication in reciprocal space.

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
10
5
-3 -2 -1 1 2 3
=
15 ?
10
5
-3 -2 -1 1 2 3

Proof of the Convolution Theorem
¥ò
fÄg=f(x)g(x'-x)dx
The inverse FT of f(x) is,
-¥
¥ò
f(x)=1
F(k)eikxdk
-¥
and the FT of the shifted g(x), that is g(x’-x)
2p
¥ò
g(x'-x)=1
G(k')eik'(x'-x)dk'
-¥
2p

So we can rewrite the convolution integral,
as,
¥ò
fÄg=f(x)g(x'-x)dx
-¥
¥ò
fÄg=1
¥ò
¥ò
dxF(k)eikxdk
4p2
change the order of integration and extract a delta function,
¥ò
¥ò
dkF(k)dk'G(k')eik'x'1
-¥ 14 4 2 4 4 3
-¥
-¥
G(k')eik'(x'-x)dk'
-¥
fÄg=1
2p
eix(k-k')dx
2p
-¥
d(k-k')
¥ò
-¥

or,
¥ò
fÄg=1
¥ò
dkF(k)dk'G(k')eik'x'1
2p
eix(k-k')dx
-¥ 14 4 2 4 4 3
¥ò
¥ò
dkF(k)dk'G(k')eik'x'd
(k-k')
Integration over the delta function selects out the k’=k value.
2p
-¥
d(k-k')
¥ò
-¥
fÄg=1
2p
-¥
-¥
¥ò
fÄg=1
dkF(k)G(k)eikx'
-¥
2p

¥ò
fÄg=1
dkF(k)G(k)eikx'
-¥
2p
This is written as an inverse Fourier transformation. A Fourier
transform of both sides yields the desired result.
Á{fÄg}=F(k)×G(k)

Fourier Convolution

Reciprocal Space
real space reciprocal space

Filtering
We can change the information content in the image by
manipulating the information in reciprocal space.
Weighting function in k-space.

Filtering
We can also emphasis the high frequency components.
Weighting function in k-space.

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}
¥
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

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 ÆAll , Mesh ÆFalse <D admission.edhole.com

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
120
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Optics with lens
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ListPlot @RotateLeft @Abs @ftline D, 64D, 8PlotJoined ÆTrue 15
12.5
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7.5
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ü Fourier transform of vertical line to show modulation
ftline =Fourier @line D;
1000
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600
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2.5
Ö Graphics Ö
ListPlot @RotateLeft @Abs @ftline D, 64D, 8PlotRange ÆAll , PlotJoined ÆTrue <D
20 40 60 80 100120
200
Ö Graphics Ö admission.edhole.com

Optics with lens
120
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2D FT
0 20 40 60 80 100 120
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ftcrop =Fourier @crop D;
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Optics with lens
Projections
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Optics with lens
Projections
xprojection = Fourier @RotateLeft @rotft @@64DD, 64DD;
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Optics with pinhole

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Projections
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Projections
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Deconvolution to determine MTF of Pinhole
20 40 60 80 100 120
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MTF =Abs @pinhole Dê Abs @image D;
ListPlot @MTF , 8PlotRange ÆAll , PlotJoined ÆTrue <10
8
6
4
2
20 40 60 80 100 120
Ö Graphics Ö

FT to determine PSF of Pinhole
ListPlot @MTF , 8PlotRange ÆAll , PlotJoined ÆTrue <D
20 40 60 80 100 120
10
8
6
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2
Ö Graphics Ö PSF =Fourier @RotateLeft @MTF , 64DD;
ListPlot @RotateLeft @Abs @PSF D, 64D, 8PlotRange ÆAll , PlotJoined ÆTrue <D
20 40 60 80 100 120
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Ö Graphics Ö

Filtered FT to determine PSF of Pinhole
ListPlot @MTF , 8PlotRange ÆAll , PlotJoined ÆTrue <D
20 40 60 80 100 120
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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
20 40 60 80 100 120
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Ö Graphics Ö
20 40 60 80 100 120
0.25

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