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 ) = S{δ (x − x0 )δ (y − y0 )}
This is often given the symbol h(r|r’).
Of course we want the entire output from the whole object function,
∞ ∞
E( x, y) =
∫∫ ∫∫ I(x, y)S{δ ( x − x0 )δ (y − y0 )}dxdydx0 dy0
−∞ −∞
∞ ∞
E( x, y) =
∫∫ ∫∫ I(x, y)IRF(x, y | x0 , y0 )dxdydx0 dy0
−∞ −∞
and so we need to know the IRF at all points.
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.
IRF(x, y | x0 , y0 ) = PSF(x − x0 , y − y0 )
The Point Spread Function (PSF) is a spatially invariant approximation of the IRF.
22.058 - lecture 4, Convolution and Fourier Convolution
Space Invariance
Since the Point Spread Function describes the same blurring over the entire sample,
IRF(x, y | x0 , y0 ) ⇒ PSF(x − x0 , y − y0 )
The image may be described as a convolution,
∞
E( x, y) =
∫∫ I (x0 , y0 )PSF(x − x0 , y − y0 )dx0 dy0
−∞
or,
Image( x, y) = Object ( x, y ) ⊗ PSF( x, y ) + noise
22.058 - lecture 4, Convolution and Fourier Convolution
Convolution Integrals
Let’s look at some examples of convolution integrals,
∞
f (x) = g( x) ⊗ h( x) =
∫ g(x')h( x − x' )dx'
−∞
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
22.058 - lecture 4, Convolution and Fourier Convolution
Convolution Integrals
Consider the following two functions:
#1. Fold h(x’) about the line x’=0
#2. Displace h(x’) by x
22.058 - lecture 4, Convolution and Fourier Convolution
x
Convolution Integrals
22.058 - lecture 4, Convolution and Fourier Convolution
Convolution Integrals
Consider the following two functions:
22.058 - lecture 4, Convolution and Fourier Convolution
Convolution Integrals
22.058 - lecture 4, Convolution and Fourier Convolution
Some Properties of the Convolution
commutative:
f ⊗ g = g ⊗ f
associative:
f ⊗ (g ⊗ h) = ( f ⊗ g) ⊗ h
multiple convolutions can be carried out in any order.
distributive:
f ⊗ (g + h) = f ⊗ g + f ⊗ h
22.058 - lecture 4, Convolution and Fourier Convolution
Convolution Integral
Recall that we defined the convolution integral as,
f ⊗g=
∞
∫ 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.
ℑ{ f ⊗ g} = F(k)⋅ G(k)
where,
f (x) ⇔ F(k)
g(x) ⇔ G(k)
22.058 - lecture 4, Convolution and Fourier Convolution
Convolution Theorem
ℑ{ f ⊗ g} = F(k)⋅ G(k) In other words, convolution in real space is equivalent to
multiplication in reciprocal space.
22.058 - lecture 4, Convolution and Fourier Convolution
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?
=
-3
-2
-1
15
15
10
10
5
5
1
2
3
-3
22.058 - lecture 4, Convolution and Fourier Convolution
-2
-1
?
1
2
3
Proof of the Convolution Theorem
f ⊗g=
∞
∫ f (x)g(x'−x)dx
−∞
The inverse FT of f(x) is,
1 ∞
f (x) =
∫ F(k )eikx dk
2 π −∞
and the FT of the shifted g(x), that is g(x’-x)
1 ∞
g(x'−x) =
∫ G(k' )eik'(x'−x )dk'
2π −∞
22.058 - lecture 4, Convolution and Fourier Convolution
Proof of the Convolution Theorem
So we can rewrite the convolution integral,
f ⊗g=
∞
∫ f (x)g(x'−x)dx
−∞
as,
f ⊗g=
1 ∞
∞
∞
4π 2 −∞
−∞
−∞
∫ dx ∫ F(k)eikx dk ∫ G(k' )eik'( x'− x )dk'
change the order of integration and extract a delta function,
∞
∞
1 ∞
ik'x' 1
f ⊗g=
∫ dkF(k ) ∫ dk'G(k')e
∫ eix(k −k')dx
2π −∞
2π −∞
−∞
1442443
δ (k−k')
22.058 - lecture 4, Convolution and Fourier Convolution
Proof of the Convolution Theorem
∞
∞
1
1 ∞
f ⊗g=
∫ dkF(k ) ∫ dk'G(k')eik'x' ∫ eix(k −k')dx
2π −∞
2π −∞
−∞
1442443
δ (k−k')
or,
∞
1 ∞
f ⊗g=
∫ dkF(k ) ∫ dk'G(k')eik'x'δ (k − k')
2π −∞
−∞
Integration over the delta function selects out the k’=k value.
1 ∞
f ⊗g=
∫ dkF(k)G(k)eikx'
2π −∞
22.058 - lecture 4, Convolution and Fourier Convolution
Proof of the Convolution Theorem 1 ∞
f ⊗g=
∫ dkF(k)G(k)eikx'
2π −∞
This is written as an inverse Fourier transformation. A Fourier
transform of both sides yields the desired result.
ℑ{ f ⊗ g} = F(k)⋅ G(k)
22.058 - lecture 4, Convolution and Fourier Convolution
Fourier Convolution
22.058 - lecture 4, Convolution and Fourier Convolution
Reciprocal Space
real space
22.058 - lecture 4, Convolution and Fourier Convolution
reciprocal space
Filtering
We can change the information content in the image by
manipulating the information in reciprocal space.
Weighting function in k-space.
22.058 - lecture 4, Convolution and Fourier Convolution
Filtering
We can also emphasis the high frequency components.
Weighting function in k-space.
22.058 - lecture 4, Convolution and Fourier Convolution
Modulation transfer function
i(x, y) = o( x, y) ⊗ PSF( x, y) + noise
c
c
c
c
I (k x , k y ) = O(k x , k y ) ⋅ MTF(kx , k y )+ ℑ{noise}
∞ ∞
E( x, y) =
∫∫ ∫∫ I ( x, y)S{δ ( x − x0 )δ (y − y0 )}dxdydx0 dy0
−∞ −∞
∞ ∞
E( x, y) =
∫∫ ∫∫ I ( x, y)IRF(x, y | x0 , y0 )dxdydx0 dy0
−∞ −∞
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 , 1 DD;
Dimensions
@s D
8 480, 640<
ListDensityPlot
@s ,
8PlotRange
ÆAll ,
Mesh ÆFalse
22.058 - lecture 4, Convolution and Fourier Convolution
<D
Optics with lens
crop
= Take @s , 8220 , 347 <, 864 , 572 , 4 <D;
‡ look at artifact in vertical dimension
rot
120
line
100
= Transpose
@crop D;
= rot @@20 DD;
ListPlot
@line
, 8PlotRange
ÆAll ,
PlotJoined
120
80
100
60
80
40
60
40
20
20
0
0
20
40
60
80
100
120
Ö Graphics Ö
22.058 - lecture 4, Convolution and Fourier Convolution
40
60
80
100
120
ÆTrue <D
Optics with lens
120
ListPlot
100
15
80
12.5
@RotateLeft
@Abs @ftline
D, 64 D, 8PlotJoined
10
60
7.5
40
5
20
40
60
80
100
2.5
120
ü Fourier transform of vertical line to show modulation
ftline
ListPlot
= Fourier
@RotateLeft
@line
D;
@Abs @ftline
D, 64 D,
800
600
400
200
Ö Graphics Ö
40
60
80
100
40
60
Ö Graphics Ö
8PlotRange
ÆAll , PlotJoined
1000
20
20
120
22.058 - lecture 4, Convolution and Fourier Convolution
80
100
120
ÆTrue <D
Æ
Optics with lens
2D FT
ftcrop
120
@crop D;
120
100
100
80
80
60
60
40
40
20
0
0
= Fourier
20
20
40
60
80
100
120
22.058 - lecture 4, Convolution and Fourier Convolution
0
0
20
40
60
80
100
120
Optics with lens
Projections
1000
800
600
120
400
100
200
80
20
40
60
80
100
120
20
40
60
80
100
120
60
40
1000
800
20
600
0
0
20
40
60
80
100
120
400
200
22.058 - lecture 4, Convolution and Fourier Convolution
Optics with lens
Projections
xprojection
= Fourier
@RotateLeft
@rotft
@@64 DD, 64 DD;
300
120
250
200
100
150
80
100
50
60
20
40
60
80
100
120
200
20
0
0
40
150
20
40
60
80
100
120
100
50
20
22.058 - lecture 4, Convolution and Fourier Convolution
40
60
80
100
120
Optics with pinhole
22.058 - lecture 4, Convolution and Fourier Convolution
120
120
100
100
80
80
60
60
40
40
20
20
0
0
20
40
60
80
100
120
22.058 - lecture 4, Convolution and Fourier Convolution
0
0
20
40
60
80
100
120
2D FT
120
120
100
100
80
80
60
60
40
40
20
0
0
20
20
40
60
80
100
120
22.058 - lecture 4, Convolution and Fourier Convolution
0
0
20
40
60
80
100
120
Optics with pinhole
Projections
1200
1000
800
120
600
100
400
200
80
20
40
60
80
100
120
20
40
60
80
100
120
60
1400
40
1200
1000
20
800
0
0
600
20
40
60
80
100
120
400
200
22.058 - lecture 4, Convolution and Fourier Convolution
Projections
300
250
250
200
200
150
150
100
100
50
20
20
40
60
80
100
40
60
80
100
120
120
200
200
150
150
100
100
50
20
20
40
60
80
100
120
22.058 - lecture 4, Convolution and Fourier Convolution
40
60
80
100
120
Deconvolution to determine MTF of Pinhole
1200
1000
800
MTF = Abs @pinhole
600
ListPlot
400
Dê Abs @image D;
@MTF, 8PlotRange
ÆAll , PlotJoined
10
200
8
20
40
60
80
100
120
6
1000
4
800
2
600
20
400
Ö Graphics Ö
200
20
40
60
80
100
120
22.058 - lecture 4, Convolution and Fourier Convolution
40
60
80
100
120
ÆTrue <
FT to determine PSF of Pinhole
MTF = Abs @pinhole
ListPlot
Dê Abs @image D;
@MTF, 8PlotRange
ÆAll , PlotJoined
ÆTrue <D
10
8
6
4
2
20
Ö Graphics Ö
40
60
80
PSF = Fourier
ListPlot
100
120
@RotateLeft
@RotateLeft
@MTF, 64 DD;
@Abs @PSF D, 64 D, 8PlotRange
7
6
5
4
3
2
1
20
22.058 - lecture 4, Convolut
Ö Graphics Ö
40
60
80
100
120
ÆAll , PlotJoined
ÆTrue <D
Filtered FT to determine PSF of Pinhole
MTF = Abs @pinhole
ListPlot
Dê Abs @image D;
@MTF, 8PlotRange
ÆAll , PlotJoined
ÆTrue <D
10
1.5
8
1.25
1
6
0.75
4
0.5
2
0.25
20
Filter
ListPlot
40
60
80
100
20
120
= Table @Exp @- Abs @x - 64 Dê 15 D, 8x , 0 , 128 <D êê N;
@Filter
, 8PlotRange
ÆAll , PlotJoined
1
0.8
0.6
0.4
0.2
20
Ö Graphics Ö
40
60
80
100
120
ÆTrue <D
40
60
80
100
120