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22.058 - lecture 4, Convolution and Fourier Convolution Convolution Fourier Convolution ne 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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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

22.058 - lecture 4, Convolution and Fourier Convolution

Instrument Response FunctionThe Instrument Response Function is a conditional mapping, the form of the map depends on the point that is being mapped.

This is often given the symbol h(r|r’).

Of course we want the entire output from the whole object function,

and so we need to know the IRF at all points.

IRF(x,y|x0,y0)=S δ(x−x0)δ(y−y0){ }

E(x,y)=

∫∫−∞

∫∫−∞

I (x,y)S δ(x−x0)δ(y−y0){ }dxdydx0dy0

E(x,y)=

∫∫−∞

∫∫−∞

I (x,y)IRF(x,y|x0,y0)dxdydx0dy0

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.

IRF(x,y|x0,y0)=PSF(x−x0,y−y0)

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)⇒ PSF(x−x0,y−y0)

E(x,y)=

∫∫−∞

I (x0,y0)PSF(x−x0,y−y0)dx0dy0

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,

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

f(x)=g(x)⊗h(x)=−∞

∞∫ g(x')h(x−x')dx'

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 x

22.058 - lecture 4, Convolution and Fourier Convolution

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:

associative:

multiple convolutions can be carried out in any order.

distributive:

f ⊗g=g⊗ f

f ⊗(g⊗h)=( f ⊗g)⊗h

f ⊗(g+h)=f ⊗g+f ⊗h

22.058 - lecture 4, Convolution and Fourier Convolution

Recall that we defined the convolution integral as,

One of the most central results of Fourier Theory is the convolution theorem (also called the Wiener-Khitchine theorem.

where,

Convolution Integral

f ⊗g= f(x)g(x'−x)dx−∞

∞∫

ℑ f ⊗g{ }=F(k)⋅G(k)

f(x)⇔ F(k)

g(x)⇔ G(k)

22.058 - lecture 4, Convolution and Fourier Convolution

Convolution Theorem

In other words, convolution in real space is equivalent to multiplication in reciprocal space.

ℑ f ⊗g{ }=F(k)⋅G(k)

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 1 2 3

5

10

15

=

-3 -2 -1 1 2 3

5

10

15 ?

22.058 - lecture 4, Convolution and Fourier Convolution

Proof of the Convolution Theorem

The inverse FT of f(x) is,

and the FT of the shifted g(x), that is g(x’-x)

f ⊗g= f(x)g(x'−x)dx−∞

∞∫

f(x)=12π

F(k)eikxdk−∞

∞∫

g(x'−x)=12π

G(k')eik'(x'−x)dk'−∞

∞∫

22.058 - lecture 4, Convolution and Fourier Convolution

Proof of the Convolution Theorem

So we can rewrite the convolution integral,

as,

change the order of integration and extract a delta function,

f ⊗g= f(x)g(x'−x)dx−∞

∞∫

f ⊗g=1

4π2 dx F(k)eikxdk−∞

∞∫

−∞

∞∫ G(k')eik'(x'−x)dk'

−∞

∞∫

f ⊗g=12π

dkF(k) dk'G(k')eik'x'1

2πeix(k−k')dx

−∞

∞∫

δ(k−k')1 2 4 4 3 4 4 −∞

∞∫

−∞

∞∫

22.058 - lecture 4, Convolution and Fourier Convolution

Proof of the Convolution Theorem

or,

Integration over the delta function selects out the k’=k value.

f ⊗g=12π

dkF(k) dk'G(k')eik'x'1

2πeix(k−k')dx

−∞

∞∫

δ(k−k')1 2 4 4 3 4 4 −∞

∞∫

−∞

∞∫

f ⊗g=12π

dkF(k) dk'G(k')eik'x'δ(k−k')−∞

∞∫

−∞

∞∫

f ⊗g=12π

dkF(k)G(k)eikx'

−∞

∞∫

22.058 - lecture 4, Convolution and Fourier Convolution

Proof of the Convolution Theorem

This is written as an inverse Fourier transformation. A Fourier transform of both sides yields the desired result.

f ⊗g=12π

dkF(k)G(k)eikx'

−∞

∞∫

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 reciprocal space

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.

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 (kx,ky)=O(kx,ky)⋅MTF(kx,ky)+ℑ noise{ }

E(x,y)=

∫∫−∞

∫∫−∞

I (x,y)S δ(x−x0)δ(y−y0){ }dxdydx0dy0

E(x,y)=

∫∫−∞

∫∫−∞

I (x,y)IRF(x,y|x0,y0)dxdydx0dy0

22.058 - lecture 4, Convolution and Fourier Convolution

Optics with lens‡ Input bit mapped image

sharp =Import @"sharp. bmp"D;

Shallow @InputForm@sharpDD

Graphics@Raster@<< 4>>D, Rule@<< 2>>DD

s = sharp @@1, 1DD;

Dimensions @sD

8480, 640<

ListDensityPlot @s, 8PlotRange ÆAll , MeshÆFalse <D

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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22.058 - lecture 4, Convolution and Fourier Convolution

Optics with lens

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üFourier transform of vertical line to show modulation

ftline =Fourier @line D;

ListPlot @RotateLeft @Abs@ftline D, 64D, 8PlotJoined ÆTrue <D

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ListPlot @RotateLeft @Abs@ftline D, 64D, 8PlotRange ÆAll , PlotJoined ÆTrue <D

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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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22.058 - lecture 4, Convolution and Fourier Convolution

Optics with lens

Projections

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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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22.058 - lecture 4, Convolution and Fourier Convolution

Optics with pinhole

22.058 - lecture 4, Convolution and Fourier Convolution

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22.058 - lecture 4, Convolution and Fourier Convolution

2D FT

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22.058 - lecture 4, Convolution and Fourier Convolution

Optics with pinhole

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22.058 - lecture 4, Convolution and Fourier Convolution

Projections

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22.058 - lecture 4, Convolution and Fourier Convolution

Deconvolution to determine MTF of Pinhole

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MTF=Abs@pinhole DêAbs@imageD;

ListPlot @MTF, 8PlotRange ÆAll , PlotJoined ÆTrue <D

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22.058 - lecture 4, Convolution and Fourier Convolution

FT to determine PSF of PinholeMTF=Abs@pinhole DêAbs@imageD;

ListPlot @MTF, 8PlotRange ÆAll , PlotJoined ÆTrue <D

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Ö Graphics Ö PSF =Fourier @RotateLeft @MTF, 64DD;

ListPlot @RotateLeft @Abs@PSFD, 64D, 8PlotRange ÆAll , PlotJoined ÆTrue <D

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22.058 - lecture 4, Convolution and Fourier Convolution

Filtered FT to determine PSF of PinholeMTF=Abs@pinhole DêAbs@imageD;

ListPlot @MTF, 8PlotRange ÆAll , PlotJoined ÆTrue <D

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Ö Graphics ÖFilter = Table @Exp@-Abs@x - 64Dê15D, 8x, 0, 128<DêêN;

ListPlot @Filter , 8PlotRange ÆAll , PlotJoined ÆTrue <D

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