SIPC Fourier transform, in 1D and in 2D

8/13/2019 SIPC Fourier transform, in 1D and in 2D

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Fourier transform, in 1D and in 2D

Vclav Hlav

Czech Technical University in Prague

Faculty of Electrical Engineering, Department of Cybernetics

Center for Machine Perceptionhttp://cmp.felk.cvut.cz/hlavac, [email protected]

Outline of the talk:

Fourier tx in 1D, computational complexity, FFT.

Fourier tx in 2D, centering of the spectrum.

Examples in 2D.
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2/59Initial idea, filtering in frequency domain

Image processingfiltration of 2D signals.spatialfilter

frequency

filter

inputimage

direct

transformation

inverse

transformation

outputimage

Filtration in the spatial domain. We would say in time domain for 1D signals.It

is a linear combination of the input image with coefficients of (often local)

filter. The basic operation is called convolution.Filtration in the frequency domain. Conversion to the frequency domain,

filtration there, and the conversion back.

We consider Fourier transform, but there are other linear integral transforms

serving a similar purpose, e.g., cosine, wavelets.
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3/591D Fourier transform, introduction

Fourier transforme is one of the most commonly

used techniques in (linear) signal processing andcontrol theory.

It provides one-to-one transform of signals

from/to a time-domain representationf(t)

to/from a frequency domain representationF().

It allows a frequency content (spectral) analysis of

a signal.

FT is suitable for periodic signals.

If the signal is not periodic then the Windowed FT

or the linear integral transformation with time

(spatially in 2D) localized basis function, e.g.,

wavelets, Gabor filters can be used.

Joseph Fourier

1768-1830


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4/59Odd, even and complex conjugate functions

Even f(t) =f(t)

(t

t

Odd f(t) = f(t)t

f(t)

Conjugage

symmetric f()=f(

)

f( 5) = 2 + 3i

f(5) = 2 3i f denotes a complex conjugate function.

iis a complex unit.
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5/59

Any function can be decomposed

as a sum of the even and odd part

f(t) =fe(t) +fo(t)

fe(t) =f(t) +f(t)

2 fo(t) =

f(t) f(t)2

f(t) f (t)e f (t)o

t t t= +
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6/59Fourier Tx definition: continuous cased

F{f(t)} =F(), where [Hz=s1] is a frequency and2 [s1] is the angularfrequency.

Fourier Tx Inverse Fourier Tx

F() =

f(t) e2it dt f(t) =

F() e2it d

What is the meaning of the inverse Fourier Tx? Express it as a Riemann sum:

f(t) .= . . .+F(0) e

2i0t +F(1) e2i1t +. . . ,

kde=k+1 k prok . Any 1D function can be expressed as a the weighted sum (integral) of manydifferent complex exponentials(because of Eulers formulaej = cos +jsin ,

also of cosinusoids and sinusoids).


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7/59Existence conditions of Fourier Tx

1.

| f(t) | dt < , i.e.f(t)has to grow slower than an exponential curve.

2. f(t)can have only a finite number of discontinuities and maxima, minimain

any finite rectangle.

3. f(t)need not have discontinuities with the infinite amplitude.

Fourier transformation exists always for digital imagesas they are limited andhave finite number of discontinuities.


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8/59Fourier Tx, symmetries

Symmetry with regards to the complex conjugate part, i.e.,F(i) =F(i).

|F(i)|is always even.

The phase ofF(i) is always odd. Re{F(i)}is always even. Im{F(i)} is always odd. The even part off(t)transforms to the real part ofF(i).

The odd part off(t) transforms to the imaginary part ofF(i).
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9/59Convolution, definition, continuous case

Convolution (in functional analysis) is an operation on two functionsf and

hwhich produces a third function(f

h), often used to create a

modification of one of the input functions.

Convolution is an integral mixing values of two functions, i.e., of the

functionh(t), which is shifted and overlayed with the functionf(t)or

vice-versa.

Consider first thecontinuous casewith general infinite limits

(fh)(t) = (h f)(t)

f() h(t ) d =

f(t ) h() d .

The limits can be constraint to the interval[0, t], because we assume zerovalues of functions for the negative argument

(f h)(t) = (h f)(t) t

0

f()h(t ) d = t

0

f(t )h() d .
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10/59Convolution, discrete approximation

(f h)(i) = (h f)(i) mO

h(i m)f(m) = mO

h(i)f(i m),

where O is a local neighborhood of a current positioniandhis the convolutionkernel (also convolution mask).


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11/59Fourier Tx, properties (1)

Property f(t) F()

Linearity af1(t) +bf2(t) a F1() +b F2()

Duality F(t) f()Convolution (f

g)(t) F() G()

Product f(t) g(t) (F G)()Time shift f(t t0) e2it0 F()

Frequency shift e2i0tf(t) F(

0)

Differentiation df(t)dt 2iF()

Multiplication byt t f(t) i2dF()d

Time scaling f(a t) 1

|a|

F(/a)
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12/59Fourier Tx, properties (2)

Area in time F(0) =

f(t)dt Area functionf(t).

Area in freq. f(0) =

F(j)d Area underF(j)

Parsevals th.

|f(t)|2dt=

|F(j)|2d f energy =F energy
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13/59Basic Fourier Tx pairs (1)

0 0

00

0

0

t

x

t t

x x

1

1

f(t)

ReF( )x

d(t)

d( )x

T-T-2T 2T

f(t)

F( )x

1

T

1

2T

1

-T

1

-2T

Dirac constant sequence of Diracs
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0 00x x xx0 x0

-x0

-x0

-2x0 2x0-x0 x0

f(t)

F( )x

ReF( )x ReF( )xImF( )x

t

f(t)= (2 t)cos px0

t t

f(t)= (2 t)sin px0 f(t)= (2 t) +cos px0 cos(4 t)px0

cosine sine two cosines mixture


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0

00

0

f(t)

F( )x

ReF( )=(sin 2 T)/x px px ReF( )x

f(t)=(sin 2 t)/ tpx p0

t t

f(t) f(t)=exp(-t )2

ReF( )= exp(- )x p px2 2

t

1

0

0

T-T

1

x x0 x x

rectangle int rectangle in Gaussian


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16/59Uncertainty principle

All Fourier Tx pairs are constrained by the uncertainty principle.

The signal of short duration must have wide Fourier spectrum and vice versa.

(signal duration) (frequency bandwith)

1

Observation: Gaussianet2

modulated by a sinusoid (Gabor function) has

the smallest duration-bandwidth product.

The principle is an instance of the general uncertainty principle introduced

by Werner Heisenberg in quantum mechanics.


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17/59Non-periodic signals

Fourier transform assumes a periodic signal. What if a non-periodic signal hasto

be processed? There are two common approaches.

1. To process the signal in small chunks (windows) and assume that the signalis periodic outside the windows.

The approach was introduced by Dennis Gabor in 1946 and it is named Short time

Fourier transform.Dennis Gabor, 1900-1979, inventor of holography, Nobel price for physics in 1971,

studied in Budapest, PhD in Berlin in 1927, fled Nazi persecution to Britain in 1933.

Mere cutting of the signal to rectangular windows is not good because discontinuiti esat windows limits cause unwanted high frequencies.

This is the reason why the signal is convolved by a dumping weight function, oftenGaussian or Hamming function ensuring the zero signal value at the limits of thewindow and beyond it.

2. Use of more complex basis function, e.g., wavelets in the wavelet transform.


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18/59Discrete Fourier transform

Letf(n) be an input signal (a sequence),n= 0, . . . , N

1.

LetF(k)be a Frequency spectrum (the result of the discrete Fourier

transformation) of a signalf(n).

Discrete Fourier transformation

F(k) N1n=0

f(n)e2ikn

N

Inverse discrete Fourier transformation

f(n) 1N

N1k=0

F(k)e2iknN

C t ti l l it


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19/59

Computational complexity,

a reminder of a notation

While considering complexity, it is abstracted from a specific computer and

only asymptotic behavior of algorithms is concerned. An asymptotic upper bound for the magnitude of a function (i.e., its

growth) in terms of another, usually simpler, function is sought.

Big

O notation; for example,

O(n2)means that the number of algorithm

steps will be roughly proportional to the square of the number of samplesinthe worst case.

Additional terms and multiplicative constants are not taken into account

because a qualitative comparison is sought.

The quadratic complexityO(n2)is worse than sayO(n) (linear) orO(1)(constant, independent of the lengthn), but is better thanO(n3)(cubic).If the complexity is exponential, e.g.,O(2n), then it often means that thealgorithm cannot be applied to larger problems (in practical terms).

Computational complexity of


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20/59

Computational complexity of

the discrete Fourier transform

LetWbe a complex number,W e2iN .

F(k) N1

n=0

f(n)e2ikn

N =N1

n=0

Wnk f(n)

The vectorf(n) is multiplied by the matrix whose element(n, k) is the

complex constantW to the powerN k. This has the computationalcomplexity

O(N2).
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21/59Fast Fourier transform

A fast Fourier transform (FFT) is an efficient algorithm to compute the

discrete Fourier transform and its inverse. Statement: FFT has the complexityO(Nlog2 N). Example(according to Numerical recepies in C):

A sequence ofN = 106

, 1second computer. FFT 30 seconds of CPU time. DFT 2 weeks of CPU time, i.e., 1,209,600 seconds, which is about

40.000

more.

A FFT idea(Danielson, Lanczos, 1942): The DFT of lengthNcan be

expressed as sum of two DFTs of lengthN/2, the first one consisting ofodd

and the second ofevensamples. Note: FFT exists also for a general

lengthN.


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22/59FFT, the proof

F(k) =N1n=0

e2ikn

N f(n)

=

(N/2)1

n=0

e2ik(2n)

N f(2n) +

(N/2)1

n=0

e2ik(2n+1)

N f(2n+ 1)

=

(N/2)1n=0

e2iknN/2 f(2n) +Wn

(N/2)1n=0

e2iknN/2 f(2n+ 1)

= Fe(k) +Wn Fo(k), k= 1, . . . , N

The key idea:recursivenessandN is power of 2.

Onlylog2 N iterations needed.


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23/59FFT, THE PROOF (2)

SpectraFe(k)andFo(k)are periodic ink with lengthN/2.

What is Fourier transform o length 1? It is just identity.

For every pattern oflog2 Nes and os, there is a one-point transform that

is just one of input numbersf(n),

Feoeeoeo...oee(k) =f(n) for somen .

The next trick is to utilize partial results=

butterfly scheme of

computations.
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24/59FFT butterfly scheme

f1

F1

f0

F0

f2

F2

f3

F3

f4

F4

f5

F5

f6

F6

f7

F7

Iteration

1

2

3


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25/592D Fourier transform

The idea. The image functionf(x, y) is decomposed to a linear combinationof

harmonic (sines and cosines, more generally orthogonal) functions.

Definition of the direct transform. u, v are spatial frequencies.

F(u, v)=

f(x, y)e2i(xu+yv) dx dy


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26/59Inverse Fourier tranform

f(x, y)=

F(u, v)e2i(xu+yv) du dv

f(x, y)is a linear combination of simple harmonic functions (components)e2i(xu+uv).

Thanks to Euler formula (eiz = cos z+i sin z),coscorresponds to the real

part andsincorresponds to the imaginary part.

FunctionF(u, v)(complex spectrum) gives weights of harmonic

components in the linear combination.

2D i id ill i


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27/592D sinusoid, illustration

2D sinusoids can be imagined as plane waves with the amplitude shown as

intensity (gray level).

The analogy to corrugated iron comes from a topographic displaying of a2D

sinusoid (or cosinusoid).

2D i id ill t ti (2)


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28/592D sinusoid, illustration (2)

=

u2 +v2,u= cos,v= sin, = tan1vu

.

u

v(u,v)

Q

w

w Qcos

w

Q

si

n

Ill t ti f 2D FT b t


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29/59Illustration of 2D FT bases vectors

Analogy corrugated iron.

sin(3x+ 2y) cos(x+ 4y)

Li bi ti f b t


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30/59Linear combination of base vectors

Analogy carton egg tray.

sin(3x+ 2y)+cos(x+ 4y) different display only

Carton egg tray


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31/59Carton egg tray

2D discrete Fourier transform


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32/592D discrete Fourier transform

Direct transform

F(u, v) = 1

M N

M1m=0

N1n=0

f(m, n) exp2i

muM

+nv

N

,

u= 0, 1, . . . , M

1, v= 0, 1, . . . , N

1,

Inverse transform

f(m, n) =M1

u=0

N1

v=0

F(u, v) exp 2i muM

+nv

N ,

m= 0, 1, . . . , M 1, n= 0, 1, . . . , N 1.

2D Fourier Tx as twice 1D Fourier Tx


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33/592D Fourier Tx as twice 1D Fourier Tx

2D direct FT can be modified to

F(u, v) = 1

M

M1m=0

1

N

N1n=0

exp

2invN

f(m, n)

exp

2imuM

,

u= 0, 1, . . . , M 1, v= 0, 1, . . . , N 1. The term in square brackets corresponds to the one-dimensional Fourier

transform of themth line and can be computed using the standard fast

Fourier transform (FFT).

Each line is substituted with its Fourier transform, and the one-dimensional

discrete Fourier transform of each column is computed.

Spatial frequencies spectrum


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34/59Spatial frequencies spectrum

The outcome of the Fourier transformF(u, v)is a function of complex variables.

(Complex) spectrum F(u, v) =R(u, v) +i I(u, v)

Amplitude spectrum |F(u, v)| = R2(u, v) +I2(u, v)Phase spectrum (u, v) =tan1

I(u,v)R(u,v)

Power spectrum P(u, v) = |F(u, v)|2 =R2(u, v) +I2(u, v)

Displaying spectra 2D Gaussian example


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35/59Displaying spectra, 2D Gaussian example

Gaussian is selected for illustration because it has a smooth spectrum, cf.

uncertainty principle.

Input intensity image, coordinate system


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36/59Input intensity image, coordinate system

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Real part of the spectrum, image and mesh


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37/59Real part of the spectrum, image and mesh

Problem with the image related coordinate system related to the image:

interesting information is in corners, moreover divided into quarters. Due to

spectrum periodicity it can be arbitrarily shifted.

Real part of the spectrum

Spatial frequency u

Spatialfrequencyv

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real part, image real part, mesh
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Log power of the spectrum, image and mesh
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39/59g p p , g

log power spectrum

Spatial frequency u

Spatialfre

quencyv

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image mesh

Centered spectra
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40/59p

It is useful to visualize a centered spectrum

with the origin of the coordinate system(0, 0) in the middle of the spectrum.

Assume the original spectrum is divided into

four quadrants. The small gray-filled

squares in the corners represent positions oflow frequencies.

Due to the symmetries of the spectrum the

quadrant positions can be swapped

diagonally and the low frequencies locations

appear in the middle of the image.

MATLABu provides functionfftshift

which converts noncenteredcenteredspectra by switching quadrants diagonally.

A D

CB

original spectrum

low frequencies in corners

AD

C B

shifted spectrum

with the origin at(0, 0)

Real part of the centered spectrum, image and

h
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41/59mesh

Real part of the spectrum, centered

Spatial frequency u

Spatialfrequencyv

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Imaginary part of the centered spectrum

image and mesh
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42/59image and mesh

Imaginary part of the spectrum, centered

Spatial frequency u

Spatialfrequencyv

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imaginary part, image imaginary part, mesh

/

Log power of the centered spectrum

image and mesh
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43/59image and mesh

log power spectrum, centered

Spatial frequency u

Spatialfr

equencyv

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mesh


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45/59mesh


Spatial frequency u

Spatialfr

equencyv

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Imaginary part of the centered spectrumimage and mesh


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46/59image and mesh


Spatial frequency u

Spatialfr

equencyv

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Log power of the centered spectrumimage and mesh


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47/59image and mesh


Spatial frequency u

Spatialfr

equencyv

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image mesh

48/59Rice example, input image 265256


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mesh


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49/59mesh


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Spatialfrequencyv

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50/59image and mesh


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Spatialfrequencyv

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51/59g


Spatial frequency u

Spatialfrequencyv

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52/59Horizontal line example, input image 265256


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53/59Horizontal line example, real part of the spectrum


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Horizontal line example,imaginary part of the spectrum


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56/59Rectangle example, input image 512512


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Real part of the centered spectrum, image andmesh


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Spatialfrequencyv

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spatialfilter

frequencyfilter

inputimage

directtransformation

inversetransformation

outputimage


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f( )


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f(t)

t

f(t)


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t

f(t)
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f(t) f (t)e f (t)o

t t t= +
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0 0

00

0

0

t

x

t t

x x

1

1

f(t)

ReF( )x

d(t)

d( )x

T-T-2T 2T

f(t)

F( )x

1

T

1

2T

1

-T

1

-2T


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0 00x x xx0 x0

-x0

-x0

-2x0 2x0-x0 x0

f(t)

F( )x

ReF( )x ReF( )xImF( )x

t

f(t)= (2 t)cos px0

t t

f(t)= (2 t)sin px0 f(t)= (2 t) +cos px0 cos(4 t)px0


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0

00

0

f(t)

F( )x

ReF( )=(sin 2 T)/x px px ReF( )x

f(t)=(sin 2 t)/ tpx p0

t t

f(t) f(t)=exp(-t )2

ReF( )= exp(- )x p p x2 2

t

1

0

0

T-T

1

x x0 x x

f1f0 f2 f3 f4 f5 f6 f7

Iteration


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F1F0 F2 F3 F4 F5 F6 F7

Iteration

1

2

3


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log power spectrum

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A D

CB
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AD

C B

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Spatialf

requency

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S ti l f

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SIPC Fourier transform, in 1D and in 2D

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Transcript of SIPC Fourier transform, in 1D and in 2D