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8/11/2019 School.com

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22.058 - lecture 4, Convolution and

Fourier Convolution

Convolution

Fourier Convolution

OutlineReview linear imaging modelInstrument 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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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 )

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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,y

0) PSF(x x

0,y y

0)

E(x,y)

I(x0,y0 )PSF(x x0,y y0 )dx0dy0

Image(x,y) Object(x,y)PSF(x,y) nois

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Fourier Convolution

Convolution Integrals

Letslook 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'

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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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Fourier Convolution


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Fourier Convolution


Consider the following two functions:

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Fourier Convolution


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Fourier Convolution

Some Properties of the Convolution

commutative:

associative:

multiple convolutions can be carried out in any order.

distributive:

fgg f

f (g h) (fg) h

f (g h) fg f h

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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

fg F(k) G(k)

f(x)F(k)

g(x) G(k)

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Fourier Convolution

Convolution Theorem

In other words, convolution in real space is equivalent to

multiplication in reciprocal space.

fg F(k)G(k)

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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

?

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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)

fg f(x)g(x'x)dx

f(x) 1

2

F(k)eikx

dk

g(x'x) 1

2 G(k' )eik'(x'x)dk'

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Fourier Convolution


So we can rewrite the convolution integral,

as,

change the order of integration and extract a delta function,

fg f(x)g(x'x)dx

fg1

42 dx F(k)e

ikxdk

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

fg

1

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

2 eix(kk')

dx

(kk')

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Fourier Convolution


or,

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

fg1

2

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

2

eix(kk')

dx

(kk')

fg1

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

fg 12

dkF(k)G(k)eikx'

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Fourier Convolution


This is written as an inverse Fourier transformation. A Fourier

transform of both sides yields the desired result.

fg1

2

dkF(k)G(k)eikx'

fg F(k) G(k)

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Fourier Convolution

Fourier Convolution

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Fourier Convolution

Reciprocal Space

real space reciprocal space


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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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Fourier Convolution

Modulation transfer function

i(x,y) o(x,y) PSF(x,y) noise

I(kx ,ky ) O(kx ,ky ) MTF(kx ,ky ) noise

E(x,y)

I(x,y)S (x x0 )(y y0 ) dxdydx0dy0

E(x,y)

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

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Fourier Convolution

Optics with lens Input bit mapped image

sharp = Import @"sharp . bmp" D;

Shallow @InputForm @sharp DD

Graphics@Raster@>D, Rule@>DD

s = sharp @@1 , 1 DD;

Dimens ions @s D

8480, 640

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