12 fourier t xen
Post on 04-Nov-2015
2 Views
Preview:
DESCRIPTION
TRANSCRIPT
-
Fourier transform, in 1D and in 2DVclav Hlav
Czech Technical University in PragueFaculty of Electrical Engineering, Department of Cybernetics
Center for Machine Perceptionhttp://cmp.felk.cvut.cz/hlavac, hlavac@fel.cvut.cz
Outline of the talk: Fourier tx in 1D, computational complexity, FFT. Fourier tx in 2D, centering of the spectrum. Examples in 2D.
-
2/65Initial idea, filtering in frequency domain
Image processing filtration of 2D signals.spatialfilter
frequencyfilter
inputimage
directtransformation
inversetransformation
outputimage
Filtration in the spatial domain. We would say in time domain for 1D signals. Itis 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 transformsserving a similar purpose, e.g., cosine, wavelets.
-
3/651D Fourier transform, introduction
Fourier transforme is one of the most commonlyused techniques in (linear) signal processing andcontrol theory.
It provides one-to-one transform of signalsfrom/to a time-domain representation f(t)to/from a frequency domain representation F ().
It allows a frequency content (spectral) analysis ofa signal.
FT is suitable for periodic signals.
If the signal is not periodic then the Windowed FTor the linear integral transformation with time(spatially in 2D) localized basis function, e.g.,wavelets, Gabor filters can be used.
Joseph Fourier1768-1830
-
4/65Odd, even and complex conjugate functions
Even f(t) = f(t)
f(t)
t
Odd f(t) = f(t)t
f(t)
Conjugagesymmetric f() = f
() f( 5) = 2 + 3if(5) = 2 3i
f denotes a complex conjugate function.
i is a complex unit.
-
5/65
Any function can be decomposedas a sum of the even and odd part
f(t) = fe(t) + fo(t)
fe(t) =f(t) + f(t)
2fo(t) =
f(t) f(t)2
f(t) f (t)e f (t)o
t t t= +
-
6/65Fourier Tx definition: continuous cased
F{f(t)} = F (), where [Hz=s1] is a frequency and 2pi [s1] is the angularfrequency.
Fourier Tx Inverse Fourier Tx
F () =
f(t) e2piit dt f(t) =
F () e2piit d
What is the meaning of the inverse Fourier Tx? Express it as a Riemann sum:
f(t).=(. . .+ F (0) e
2pii0t + F (1) e2pii1t + . . .
) ,
kde = k+1 k pro k . Any 1D function can be expressed as a the weighted sum (integral) of manydifferent complex exponentials (because of Eulers formula ei = cos + i sin ,also of cosinusoids and sinusoids).
-
7/65Existence conditions of Fourier Tx
1.| f(t) | dt
-
8/65Fourier Tx, symmetries
Symmetry with regards to the complex conjugate part, i.e.,F (i) = F (i).
|F (i)| is always even. The phase of F (i) is always odd.
Re{F (i)} is always even. Im{F (i)} is always odd. The even part of f(t) transforms to the real part of F (i).
The odd part of f(t) transforms to the imaginary part of F (i).
-
9/65Convolution, definition, continuous case
Convolution (in functional analysis) is an operation on two functions f andh which produces a third function (f h), often used to create amodification of one of the input functions.
Convolution is an integral mixing values of two functions, i.e., of thefunction h(t), which is shifted and overlayed with the function f(t) orvice-versa.
Consider first the continuous case with general infinite limits
(f h)(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) t0
f() h(t ) d = t0
f(t ) h() d .
-
10/65Cross-correlation and convolution
Convolution defined for 1D signals uses the flipped kernel h
(f h)(t)
f()h(t ) d .
Cross-correlation ? defined for 1D signals uses the (unflipped) kernel h
(f ? h)
f()h(t+ ) d ,
where f denotes the complex conjugate of f .
Cross-correlation is a measure of similarity of two functions as a function of a(time) shift .
-
11/65Convolution, discrete approximation
(f h)(i) = (h f)(i) mO
h(im) f(m) =mO
h(i) f(im) ,
where O is a local neighborhood of a current position i and h is the convolutionkernel (also convolution mask).
-
12/65Fourier Tx, properties (1)
Property f(t) F ()
Linearity af1(t) + bf2(t) aF1() + b F2()
Duality F (t) f()Convolution (f g)(t) F ()G()Product f(t) g(t) (F G)()Time shift f(t t0) e2piit0 F ()
Frequency shift e2pii0tf(t) F ( 0)Differentiation df(t)dt 2piiF ()
Multiplication by t t f(t) i2pidF ()d
Time scaling f(a t) 1|a|F (/a)
-
13/65Fourier Tx, properties (2)
Area in time F (0) =
f(t)dt Area function f(t).
Area in freq. f(0) =
F ()d Area under F ()
Parsevals th.|f(t)|2dt =
|F ()|2d f energy = F energy
-
14/65Basic Fourier Tx pairs (1)
0 0
00
0
0
t
x
t t
x x
1
1
f(t)
Re F( )x
d(t)
d( )x
T-T-2T 2T
f(t)
F( )x
1T
12T
1-T
1-2T
Dirac constant sequence of Diracs
-
15/65Basic Fourier Tx pairs (2)
0 00x x xx0 x0
-x0
-x0
-2x0 2x0-x0 x0
f(t)
F( )x
Re F( )x Re F( )xIm F( )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
-
16/65Basic Fourier Tx pairs (3)
0
00
0
f(t)
F( )x
Re F( )=(sin 2 T)/x px px Re F( )x
f(t)=(sin 2 t)/ tpx p0
t t
f(t) f(t)=exp(-t )2
Re F( )= exp(- )x p p x2 2
t
1
0
0
T-T
1
x x0 x x
rectangle in t rectangle in Gaussian
-
17/65Uncertainty 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) 1pi Observation: Gaussian et2 modulated by a sinusoid (Gabor function) hasthe smallest duration-bandwidth product.
The principle is an instance of the general uncertainty principle introducedby Werner Heisenberg in quantum mechanics.
-
18/65Non-periodic signals
Fourier transform assumes a periodic signal. What if a non-periodic signal has tobe processed? There are two common approaches.1. To process the signal in small chunks (windows) and assume that the signal
is periodic outside the windows. The approach was introduced by Dennis Gabor in 1946 and it is named Short timeFourier 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 discontinuitiesat 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.
-
19/65Discrete Fourier transform
Let f(n) be an input signal (a sequence), n = 0, . . . , N 1. Let F (k) be a Frequency spectrum (the result of the discrete Fouriertransformation) of a signal f(n).
Discrete Fourier transformation
F (k) N1n=0
f(n) e2piiknN
Inverse discrete Fourier transformation
f(n) 1N
N1k=0
F (k) e2piiknN
-
20/65
Computational complexity,a reminder of a notation
While considering complexity, it is abstracted from a specific computer andonly asymptotic behavior of algorithms is concerned.
An asymptotic upper bound for the magnitude of a function (i.e., itsgrowth) in terms of another, usually simpler, function is sought.
Big O notation; for example, O(n2) means that the number of algorithmsteps will be roughly proportional to the square of the number of samples inthe worst case.
Additional terms and multiplicative constants are not taken into accountbecause a qualitative comparison is sought.
The quadratic complexity O(n2) is worse than say O(n) (linear) or O(1)(constant, independent of the length n), but is better than O(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).
-
21/65
Computational complexity ofthe discrete Fourier transform
Let W be a complex number, W e2piiN .
F (k) N1n=0
f(n) e2piiknN =
N1n=0
Wnk f(n)
The vector f(n) is multiplied by the matrix whose element (n, k) is thecomplex constant W to the power N k.
This has the computational complexity O(N2).
-
22/65Fast Fourier transform
A fast Fourier transform (FFT) is an efficient algorithm to compute thediscrete Fourier transform and its inverse.
Statement: FFT has the complexity O(N log2N). Example (according to Numerical recepies in C):
A sequence of N = 106, 1 second 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.
-
23/65Fast Fourier transform, the signal division
A FFT core ideaThe DFT of length N can be expressed as sum of two DFTs of length N/2,the first one consisting of odd and the second of even samples.(Danielson, Lanczos in 1942; further developed by Cooley, Tukey in 1965)
There are two approaches how to split the signal called
Decimation in time (DIT); Decimation in frequency (DIF).
Note 1: FFT exists also for a general length N .
Note 2: The input sequence can be divided to more than two parts ingeneral.
-
24/65Decimation in time (DIT)
The input sequence f(n), n = 1, . . . , N 1 is divided into even fe(n)and odd fo(n) parts, n = 0, 1, . . . , N/2 1.
The Fourier transform of corresponding parts denoted F e, F o can becalculated recursively F (k) = F e(k) +WNk F o(k), wherek = 0, 1, . . . , N .
The signals F e and F o are of a half length. Due to their periodicity,F e(k +N/2) = F e(k), F o(k +N/2) = F o(k) for anyk = 0, 1, . . . , N/2 1.
Courtesy: Pavel Karas.
-
25/65Decimation in frequency (DIF)
The input sequence f of the length N is divided into sequences fr and fsas fr(n) = f(n) + f(n N/2), fs = (f(n) f(n +N/2))WnN .
Their Fourier transform fulfills: F r(k) = F (2k) and F s(k) = 2k + 1 forany k = 0, 1, . . . , N/2 1.
Sequences fr and fs can be processed recursively with inverse equationsf(n) = 12
(fr(n) + fs(n)WN n
),f(n + N2 ) =
12
(fr(n) fs(n)WN n
).
Courtesy: Pavel Karas.
-
26/65FFT, decimation in time, the proof idea
F (k) =N1n=0
e2piiknN f(n)
=
(N/2)1n=0
e2piik(2n)
N f(2n) +
(N/2)1n=0
e2piik(2n+1)
N f(2n+ 1)
=
(N/2)1n=0
e2piiknN/2 f(2n) +Wn
(N/2)1n=0
e2piiknN/2 f(2n+ 1)
= F e(k) +Wn F o(k) , k = 1, . . . , N
The key idea: recursiveness and N is power of 2.
Only log2N iterations needed.
-
27/65FFT, the proof idea (2)
Spectra F e(k) and F o(k) are periodic in k with length N/2.
What is Fourier transform o length 1? It is just identity.
For every pattern of log2N es and os, there is a one-point transform thatis just one of input numbers f(n),
F eoeeoeo...oee(k) = f(n) for some n .
The next trick is to utilize partial results = butterfly scheme ofcomputations.
-
28/65FFT butterfly calculation scheme
f1
F1
f0
F0
f2
F2
f3
F3
f4
F4
f5
F5
f6
F6
f7
F7
Iteration
1
2
3
-
29/652D Fourier transform
The idea. The image function f(x, y) is decomposed to a linear combination ofharmonic (sines and cosines, more generally orthogonal) functions.
Definition of the direct transform. u, v are spatial frequencies.
F (u, v) =
f(x, y) e2pii(xu+yv) dx dy
-
30/65Inverse Fourier tranform
f(x, y) =
F (u, v) e2pii(xu+yv) dudv
f(x, y) is a linear combination of simple harmonic functions (components)e2pii(xu+uv).
Thanks to Euler formulain general eiz = cos z + i sin z, here cos(2piixu) + i sin(2piixu),cos corresponds to the real part and sin corresponds to the imaginary part.
Function F (u, v) (complex spectrum) gives weights of harmoniccomponents in the linear combination.
-
31/652D sinusoid, illustration
2D sinusoids can be imagined as plane waves with the amplitude shown asintensity (gray level).
The analogy to corrugated iron comes from a topographic displaying of a 2Dsinusoid (or cosinusoid).
-
32/652D sinusoid, illustration (2)
=u2 + v2, u = cos , v = sin , = tan1
(vu
).
u
v(u,v)
Q
w
w Qcos
wQ
sin
-
33/65Illustration of 2D FT bases vectors
Analogy corrugated iron.
sin(3x+ 2y) cos(x+ 4y)
-
34/65Linear combination of base vectors
Analogy carton egg tray.
sin(3x+ 2y)+ cos(x+ 4y) different display only
-
35/65Carton egg tray
-
36/652D discrete Fourier transform
Direct transform
F (u, v) =1
MN
M1m=0
N1n=0
f(m,n) exp[2pii
(muM
+nv
N
)],
u = 0, 1, . . . ,M 1 , v = 0, 1, . . . , N 1 ,
Inverse transform
f(m,n) =
M1u=0
N1v=0
F (u, v) exp[
2pii(muM
+nv
N
)],
m = 0, 1, . . . ,M 1 , n = 0, 1, . . . , N 1 .
-
37/652D 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
(2piinvN
)f(m,n)
]exp
(2piimuM
),
u = 0, 1, . . . ,M 1 , v = 0, 1, . . . , N 1 .
The term in square brackets corresponds to the one-dimensional Fouriertransform of the mth line and can be computed using the standard fastFourier transform (FFT).
Each line is substituted with its Fourier transform, and the one-dimensionaldiscrete Fourier transform of each column is computed.
-
38/65Spatial frequencies spectrum
The outcome of the Fourier transform F (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)
-
39/65Displaying spectra, 2D Gaussian example
Gaussian is selected for illustration because it has a smooth spectrum, cf.uncertainty principle.
Gaussian2DanimCinepak.aviMedia File (video/avi)
-
40/65Input intensity image, coordinate system
100 200 300 400 500
50
100
150
200
250
300
350
400
450
5000
50
100
150
200
250
-
41/65Real 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 tospectrum periodicity it can be arbitrarily shifted.
Real part of the spectrum
Spatial frequency u
Spat
ial f
requ
ency
v
100 200 300 400 500
50
100
150
200
250
300
350
400
450
500
real part, image real part, mesh
-
42/65Imaginary part of the spectrum, image and mesh
Imaginary part of the spectrum
Spatial frequency u
Spat
ial f
requ
ency
v
100 200 300 400 500
50
100
150
200
250
300
350
400
450
500
imaginary part, image imaginary part, mesh
-
43/65Log power of the spectrum, image and mesh
log power spectrum
Spatial frequency u
Spat
ial f
requ
ency
v
100 200 300 400 500
50
100
150
200
250
300
350
400
450
500
image mesh
-
44/65Periodic image
-
45/65Periodic spectrum
-
46/65Centered spectra
It is useful to visualize a centered spectrumwith the origin of the coordinate system(0, 0) in the middle of the spectrum.
Assume the original spectrum is divided intofour quadrants. The small gray-filledsquares in the corners represent positions oflow frequencies.
Due to the symmetries of the spectrum thequadrant positions can be swappeddiagonally and the low frequencies locationsappear in the middle of the image.
MATLABu provides function fftshiftwhich converts noncentered centeredspectra by switching quadrants diagonally.
A D
CB
original spectrumlow frequencies in corners
AD
C B
shifted spectrumwith the origin at (0, 0)
-
47/65
Real part of the centered spectrum, image andmesh
Real part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
real part, image real part, mesh
-
48/65
Imaginary part of the centered spectrumimage and mesh
Imaginary part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
imaginary part, image imaginary part, mesh
-
49/65
Log power of the centered spectrumimage and mesh
log power spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
image mesh
-
50/65Prague Castle example, input image 265256
50 100 150 200 250
50
100
150
200
2500
50
100
150
200
-
51/65
Real part of the centered spectrum, image andmesh
Real part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
real part, image real part, mesh
-
52/65
Imaginary part of the centered spectrumimage and mesh
Imaginary part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
imaginary part, image imaginary part, mesh
-
53/65
Log power of the centered spectrumimage and mesh
log power spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
image mesh
-
54/65Rice example, input image 265256
50 100 150 200 250
50
100
150
200
25040
60
80
100
120
140
160
180
200
-
55/65
Real part of the centered spectrum, image andmesh
Real part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
real part, image real part, mesh
-
56/65
Imaginary part of the centered spectrumimage and mesh
Imaginary part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
imaginary part, image imaginary part, mesh
-
57/65
Log power of the centered spectrumimage and mesh
log power spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
image mesh
-
58/65Horizontal line example, input image 265256
-
59/65Horizontal line example, real part of the spectrum
-
60/65
Horizontal line example,imaginary part of the spectrum
-
61/65Horizontal line example, power spectrum
-
62/65Rectangle example, input image 512512
100 200 300 400 500
50
100
150
200
250
300
350
400
450
5000
20
40
60
80
100
120
140
160
180
200
-
63/65
Real part of the centered spectrum, image andmesh
Real part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
real part, image real part, mesh
-
64/65
Imaginary part of the centered spectrumimage and mesh
Imaginary part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
imaginary part, image imaginary part, mesh
-
65/65
Log power of the centered spectrumimage and mesh
log power spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
image mesh
-
spatialfilter
frequencyfilter
inputimage
directtransformation
inversetransformation
outputimage
-
f(t)
t
-
tf(t)
-
f(t) f (t)e f (t)o
t t t= +
-
0 0
00
0
0
t
x
t t
x x
1
1
f(t)
Re F( )x
d(t)
d( )x
T-T-2T 2T
f(t)
F( )x
1T
12T
1-T
1-2T
-
0 00x x xx0 x0
-x0
-x0
-2x0 2x0-x0 x0
f(t)
F( )x
Re F( )x Re F( )xIm F( )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
-
000
0
f(t)
F( )x
Re F( )=(sin 2 T)/x px px Re F( )x
f(t)=(sin 2 t)/ tpx p0
t t
f(t) f(t)=exp(-t )2
Re F( )= exp(- )x p p x2 2
t
1
0
0
T-T
1
x x0 x x
-
f1
F1
f0
F0
f2
F2
f3
F3
f4
F4
f5
F5
f6
F6
f7
F7
Iteration
1
2
3
-
uv(u,v)
Q
w
w Qcos
wQ
sin
-
100 200 300 400 500
50
100
150
200
250
300
350
400
450
5000
50
100
150
200
250
-
Real part of the spectrum
Spatial frequency u
Spat
ial f
requ
ency
v
100 200 300 400 500
50
100
150
200
250
300
350
400
450
500
-
Imaginary part of the spectrum
Spatial frequency u
Spat
ial f
requ
ency
v
100 200 300 400 500
50
100
150
200
250
300
350
400
450
500
-
log power spectrum
Spatial frequency u
Spat
ial f
requ
ency
v
100 200 300 400 500
50
100
150
200
250
300
350
400
450
500
-
A D
CB
-
AD
C B
-
Real part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
-
Imaginary part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
-
log power spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
-
50 100 150 200 250
50
100
150
200
2500
50
100
150
200
-
Real part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
-
Imaginary part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
-
log power spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
-
50 100 150 200 250
50
100
150
200
25040
60
80
100
120
140
160
180
200
-
Real part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
-
Imaginary part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
-
log power spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
100 50 0 50 100
100
50
0
50
100
-
100 200 300 400 500
50
100
150
200
250
300
350
400
450
5000
20
40
60
80
100
120
140
160
180
200
-
Real part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
-
Imaginary part of the spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
-
log power spectrum, centered
Spatial frequency u
Spat
ial f
requ
ency
v
200 100 0 100 200
250
200
150
100
50
0
50
100
150
200
250
-
First pageccmp Initial idea, filtering in frequency domainccmp 1D Fourier transform, introductionccmp Odd, even and complex conjugate functionsccmp Any function can be decomposed\ as a sum of the even and odd partccmp Fourier Tx definition: continuous casedccmp Existence conditions of Fourier Txccmp Fourier Tx, symmetriesccmp Convolution, definition, continuous caseccmp Cross-correlation and convolutionccmp Convolution, discrete approximationccmp Fourier Tx, properties (1)ccmp Fourier Tx, properties (2)ccmp Basic Fourier Tx pairs (1)ccmp Basic Fourier Tx pairs (2)ccmp Basic Fourier Tx pairs (3)ccmp Uncertainty principleccmp Non-periodic signalsccmp Discrete Fourier transformccmp Computational complexity,\a reminder of a notationccmp Computational complexity of \the discrete Fourier transformccmp Fast Fourier transformccmp Fast Fourier transform, the signal divisionccmp Decimation in time (DIT)ccmp Decimation in frequency (DIF)ccmp FFT, decimation in time, the proof ideaccmp FFT, the proof idea (2)ccmp FFT butterfly calculation schemeccmp 2D Fourier transformccmp Inverse Fourier tranformccmp 2D sinusoid, illustrationccmp 2D sinusoid, illustration (2)ccmp Illustration of 2D FT bases vectorsccmp Linear combination of base vectorsccmp Carton egg trayccmp 2D discrete Fourier transformccmp 2D Fourier Tx as twice 1D Fourier Txccmp Spatial frequencies spectrumccmp Displaying spectra, 2D Gaussian exampleccmp Input intensity image, coordinate systemccmp Real part of the spectrum, image and meshccmp Imaginary part of the spectrum, image and meshccmp Log power of the spectrum, image and meshccmp Periodic imageccmp Periodic spectrumccmp Centered spectraccmp Real part of the centered spectrum, image and meshccmp Imaginary part of the centered spectrum\ image and meshccmp Log power of the centered spectrum\ image and meshccmp Prague Castle example, input image 265$imes $256ccmp Real part of the centered spectrum, image and meshccmp Imaginary part of the centered spectrum\ image and meshccmp Log power of the centered spectrum\ image and meshccmp Rice example, input image 265$imes $256ccmp Real part of the centered spectrum, image and meshccmp Imaginary part of the centered spectrum\ image and meshccmp Log power of the centered spectrum\ image and meshccmp Horizontal line example, input image 265$imes $256ccmp Horizontal line example, real part of the spectrumccmp Horizontal line example, \imaginary part of the spectrumccmp Horizontal line example, power spectrumccmp Rectangle example, input image 512$imes $512ccmp Real part of the centered spectrum, image and meshccmp Imaginary part of the centered spectrum\ image and meshccmp Log power of the centered spectrum\ image and meshLast page
top related