fouriertransform...

FOURIER TRANSFORM

APPLICATION TO SIGNAL PROCESSING

Pawel A. Penczek

The University of Texas – Houston Medical School, Department of Biochemistry

Wednesday, August 24, 2011

FOURIER TRANSFORMS

1.Fourier Transform

2.Fourier Series

3.Discrete Fourier Transform (DFT)


FOURIER TRANSFORMSThe Fourier transform is a generalization of the complex Fourier series in the limit of .L→∞

F s( ) = f x( )e−2π isx dx−∞

+∞

∫

f x( ) = F s( )e2π isx ds−∞

+∞

∫


FOURIER TRANSFORMSF s( ) = f x( )e−2π isx dx

−∞

+∞

∫

f x( ) = F s( )e2π isx ds−∞

+∞

∫Euler's formula: eix = cos x + i sin x

Since any function can be split into even and off portions:fE x( ) = 1

2f x( ) + f −x( )⎡⎣ ⎤⎦ fO x( ) = 1

2f x( ) − f −x( )⎡⎣ ⎤⎦

f x( ) = fE x( ) + fO x( )

a Fourier transform can always be expressed in terms of the Fourier cosine transform and Fourier sine transform as

F s( ) = fE x( )cos 2πisx( )dx−∞

+∞

∫ − i fO x( )sin 2π sx( )dx−∞

+∞

∫Wednesday, August 24, 2011

COMPLEX ALGEBRA

i2 = −1z,u ∈a,b ∈z = a + ibz∗ = a − ib

z 2 = a2 + b2

zu = ?zu* = ?z / u = ?

Complex numbers versus vectors


DIRAC (DELTA) FUNCTION*

*a misnomer, Dirac's delta is not a function, it is a distribution!

The Dirac delta function as the limit (in the sense of distributions) of the sequence of Gaussians:

δa x( ) =a→0

1a π

e− x

2

a2

The Dirac delta function, or δ function, is (informally) a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It is also known as unit impulse function.

Fourier transform of a delta function

δ x( )e−2π isx dx−∞

+∞

∫ = 1

e2π isx ds−∞

+∞

∫ = δ x( )


http://en.wikipedia.org/wiki/Distribution_(mathematics)

http://en.wikipedia.org/wiki/Distribution_(mathematics)

http://en.wikipedia.org/wiki/Generalized_function

http://en.wikipedia.org/wiki/Generalized_function

FOURIER PAIRS

Gaussian function top-hat function


FOURIER PAIRS

cosine function

sine function


PROPERTIES OF FOURIER TRANSFORM

Linearity

Rotation

Translation

Scaling

Conjugation

Parseval's theorem

If h x( ) = af x( ) + bg x( ), then H s( ) = aF s( ) + bG s( )

If R is a rotation matrix, FT f Rx( )( ) = H Rs( )

If h x( ) = f x − t( ), then H s( ) = e− i2π tsF s( )

If h x( ) = f ax( ), a ≠ 0, then H s( ) = 1aF s

a⎛⎝⎜

⎞⎠⎟

f x( ) 2 dx−∞

∞

∫ = F s( ) 2 ds−∞

∞

∫

FT −1 H ∗ s( )( ) = f −x( )

Density scaling

Rotation of FT,of power spectrum

Lossless shift

Scaling

1D - mirror2D - rotation by 180o

Preservation of energy,power spectrum


CROSS-CORRELATION THEOREM

c t( ) = f x( )g x + t( )dx−∞

∞

∫ = FT −1 F∗ s( )G s( )( )

SPECIAL CASE OF CCF, AUTOCORRELATION FUNCTION

a t( ) = f x( ) f x + t( )dx−∞

∞

∫ = FT −1 F∗ s( )F s( )( ) = FT −1 F s( ) 2( )

CONVOLUTION THEOREM

h = f ∗ g

h t( ) = f x( )g x − t( )dx−∞

+∞

∫ = FT −1 F s( )G s( )( )


FOURIER SERIESA Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.

Fourier series make use of the orthogonality relationships of the sine and cosine functions.

f x( ) = 12a0 + an cos nx( )

n=1

∞

∑ + bn sin nx( )n=1

∞

∑-π period +π

a0 =1π

f x( )dx−π

π

∫

an =1π

f x( )cos nx( )dx−π

π

∫

bn =1π

f x( )sin nx( )dx−π

π

∫


FOURIER SERIESGIBBS PHENOMENON

Ringing artifact near sharp edges that is due to truncation of Fourier series. Note the amplitude of oscillation is constant, does not depend on the number of Fourier coefficients included. Often observed when filter is incorrect (too sharp). Ringing can be reduced by making filter "smooth" - an extreme is a Gaussian filter. Regrettably, the cut-off frequency of a Gaussian filter is poorly defined and much noise in high frequencies passes through. There are many filters designed that offer a trade-off between sharpness (and thus artifacts) and smoothness (fewer artifacts but more noise): Butterworth, tangent..


Effects of top-hat, Gaussian, and Butterworth filters on a step function. Comparison of Gaussian and Butterworth filters.

Digital filters.


FOURIER SERIESAS AN EXPANSION IN AN ORTHONORMAL BASIS

Using Euler's formula

f x( ) = 12a0 + an cos nx( )

n=1

∞

∑ + bn sin nx( )n=1

∞

∑

einx = cosnx + i sinnx

f x( ) = cneinx

n=−∞

∞

∑ with Fourier coefficients(complex!) given by

cn =12π

f x( )e− inx dx−π

π

∫

an = cn + c−n , n = 0,1,2,…bn = i cn − c−n( ), n = 1,2,…

The set of functions forms an orthonormal basis for the Hilbert space

(of squared integrable functions on ) with an inner product

Indeed,

en = einx , n ∈{ } L2 −π ,π[ ]

−π ,π[ ] f ,g =def 12π

f x( )g∗ x( )dx−π

π

∫ .

en ,em =12π

einxe− imx dx−π

π

∫ =12π

ei n−m( )x dx−π

π

∫ = δnm .


DISCRETE FOURIER TRANSFORM (DFT)AS A REPRESENTATION IN AN ORTHONORMAL BASIS

DFT is a transformation of a finite (and presumably periodic) sequence of real or complex number into a finite space whose orthonormal basis is spanned by complex exponentials (discretized).

Xk = xne−

2π iN

kn

n=0

N

∑ k = 0,…,N −1

xk =1N

Xke2π iN

kn

k=0

N

∑ n = 0,…,N −1

xn

Re X0( ) 0 Re X1( ) Im X1( ) Re X2( ) Im X3( ) Re X3( ) Im X3( ) Re X4( ) 0⎡⎣⎢

⎤⎦⎥


DISCRETE FOURIER TRANSFORM (DFT)AS A REPRESENTATION IN AN ORTHONORMAL BASIS

DFT is a transformation of a finite (and presumably periodic) sequence of real or complex number into a finite space whose orthonormal basis is spanned by complex exponentials (discretized).

Xk = xne−

2π iN

kn

n=0

N

∑ k = 0,…,N −1

xk =1N

Xke2π iN

kn

k=0

N

∑ n = 0,…,N −1

xn

Re X0( ) 0 Re X1( ) Im X1( ) Re X2( ) Im X3( ) Re X3( ) Im X3( ) Re X4( ) 0⎡⎣⎢

⎤⎦⎥

X0X1

XN −1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

wN0i0 wN

0i1 wN0i(N −1)

wN1i0 wN

1i1 wN1i(N −1)

wN(N −1)i0 wN

(N −1)i1 wN(N −1)i(N −1)

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

x0x1xN −1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

wN = e−2π i

N


FAST FOURIER TRANSFORM (FFT) ALGORITHM FOR DFTRADIX-2

Cooley, James W., and John W. Tukey, "An algorithm for the machine calculation of complex Fourier series," Math. Comput. 19, 297–301 (1965)Gauss, Carl Friedrich, "Nachlass: Theoria interpolationis methodo nova tractata", Werke, Band 3, 265–327 (Königliche Gesellschaft der Wissenschaften, Göttingen, 1866) (1805)

X0X1

XN −1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

wN0i0 wN

0i1 wN0i(N −1)

wN1i0 wN

1i1 wN1i(N −1)

wN(N −1)i0 wN

(N −1)i1 wN(N −1)i(N −1)

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

x0x1xN −1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

wN = e−2π i

N

FN - N2 fps




X0X1

XN −1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

wN0i0 wN

0i1 wN0i(N −1)

wN1i0 wN

1i1 wN1i(N −1)

wN(N −1)i0 wN

(N −1)i1 wN(N −1)i(N −1)

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

x0x1xN −1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

wN = e−2π i

N

FN - N2 fps

F8=

w w 0 0 0 0 0 00 0 w w 0 0 0 00 0 0 0 w −w 0 00 0 0 0 0 0 w −ww −w 0 0 0 0 0 00 0 w −w 0 0 0 00 0 0 0 w w 0 00 0 0 0 0 0 w w

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

w 0 w 0 0 0 0 00 w 0 w 0 0 0 00 0 0 0 w 0 −w 00 0 0 0 0 w 0 −ww 0 −w 0 0 0 0 00 w 0 −w 0 0 0 00 0 0 0 w 0 w 00 0 0 0 0 w 0 w

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

w 0 0 0 w 0 0 00 w 0 0 0 w 0 00 0 w 0 0 0 w 00 0 0 w 0 0 0 ww 0 0 0 −w 0 0 00 w 0 0 0 −w 0 00 0 w 0 0 0 −w 00 0 0 w 0 0 0 −w

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥




X0X1

XN −1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

wN0i0 wN

0i1 wN0i(N −1)

wN1i0 wN

1i1 wN1i(N −1)

wN(N −1)i0 wN

(N −1)i1 wN(N −1)i(N −1)

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

x0x1xN −1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

wN = e−2π i

N

FN - N2 fps

F8=

w w 0 0 0 0 0 00 0 w w 0 0 0 00 0 0 0 w −w 0 00 0 0 0 0 0 w −ww −w 0 0 0 0 0 00 0 w −w 0 0 0 00 0 0 0 w w 0 00 0 0 0 0 0 w w

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

w 0 w 0 0 0 0 00 w 0 w 0 0 0 00 0 0 0 w 0 −w 00 0 0 0 0 w 0 −ww 0 −w 0 0 0 0 00 w 0 −w 0 0 0 00 0 0 0 w 0 w 00 0 0 0 0 w 0 w

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

w 0 0 0 w 0 0 00 w 0 0 0 w 0 00 0 w 0 0 0 w 00 0 0 w 0 0 0 ww 0 0 0 −w 0 0 00 w 0 0 0 −w 0 00 0 w 0 0 0 −w 00 0 0 w 0 0 0 −w

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

FFT - 2N log2N fps!Wednesday, August 24, 2011



X0X1

XN −1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

wN0i0 wN

0i1 wN0i(N −1)

wN1i0 wN

1i1 wN1i(N −1)

wN(N −1)i0 wN

(N −1)i1 wN(N −1)i(N −1)

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

x0x1xN −1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

wN = e−2π i

N

FN - N2 fps

FFT - 2N log2N fps!Wednesday, August 24, 2011

FOURIER

DECOMPOSITION SYNTHESIS

Amplitudes

8.2

3.1

2.1

1.0

1.5

2.0

0.6

1.0


2D FT

⊗

c0,0 c0,1 c0,2 c0,3 c0,4 c0,5 c0,6 c0,7c1,0 c1,1 c1,2

c2,0

c3,0

c4,0

c5,0

c6,0

c7,0 c7,1 c7,7

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

=

amplitude: Ai, j = ci, j = Re ci, j( )2+ Im ci, j( )2

phase: ϕ i, j = atan2 Im ci, j( ),Re ci, j( )( )Wednesday, August 24, 2011

DFT CASE STUDY:POWER SPECTRUM (PS)

Stationary random process (signal + noise): f(x)

The power spectrum S(f) is the Fourier transform of the autocorrelation function of the process.

Sf s( ) = a τ( )e−2π isτ dτ−∞

∞

∫

Periodogram is the squared amplitude of the Fourier transform of the process.

Periodogram is a very poor estimator of the power spectrum: its relative error is 100% and it is biased. Finally, it does not converge to the true power spectrum with increased window size.

Pf s( ) = f x( )e−2π isx dx−∞

∞

∫2

Ansemble average (expected value) of periodogram approximates power spectrum.

E Pf s( )⎡⎣ ⎤⎦→ Sf s( )Wednesday, August 24, 2011

DFT CASE STUDY:POWER SPECTRUM (PS) ESTIMATION

AVERAGING OF PERIODOGRAMS - WELCH METHOD

.

.

.

+

Zhu, J., Penczek, P.A., Schröder, R., and Frank, J. (1997). Three-dimensional reconstruction with contrast transfer function correction from energy-filtered cryoelectron micrographs: procedure and application to the 70S Escherichia coli ribosome. Journal of Structural Biology 118, 197-219.

Average of periodogramsreduced variance

50% overlap

Rotational average


DFT CASE STUDY:CROSS-CORRELATION

c t( ) = f x( )g x + t( )dx−∞

+∞

∫ = FT −1 F∗ s( )G s( )( )

ck = flgl+ k−∞

∞

∑ = FT −1 F∗G( )



Discrete FT is periodic, thus it "sees" a single image as periodic.

wrap-around effect

... ...

ck = flg l+ k( )mod N0

N −1

∑ = FT −1 F∗G( )k = −N

2,…,N 2



To avoid wrap-around effect, pad image with zeroes (or something else??).

ck = flpadgl+ k

pad

l=0

N −1

∑ = FT −1 F pad∗Gpad( )k = −N

2,…,N 2Each ccf coefficient is computed using

different number of image points!Wednesday, August 24, 2011


To avoid uneven normalization problem, normalize by lag padded ccf.

ck =1

N − kflpadgl+ k

pad

l=0

N −1

∑ =1

N − kFT −1 F pad∗Gpad( )

k = −N2,…,N 2

The variance of ccf coefficients increases with lags!



Which ccf should I use?

ck = flg l+ k( )mod N0

N −1

∑ = FT −1 F∗G( )

ck = flpadgl+ k

pad

l=0

N −1

∑ = FT −1 F pad∗Gpad( )

ck =1

N − kflpadgl+ k

pad

l=0

N −1

∑ =1

N − kFT −1 F pad∗Gpad( )


Tomographyhistorical background

1956 - Bracewell reconstructed sun spots from multiple projection views of the Sun from the Earth.

1967 - Medical Research Council Laboratory, Cambridge, England: Aaron Klug and grad student David DeRosier reconstructed three-dimensional structures of viruses.

1969 – W. Hoppe (Germany) proposed three-dimensional high resolution electron microscopy of non-periodic biological structures.

1972 - British engineer Godfrey Hounsfield of EMI Laboratories, England, and independently South African born physicist Allan Cormack of Tufts University, Massachusetts, invented CAT (Computed Axial Tomography) scanner. Tomography is from the Greek word τοµή meaning "slice" or "section" and graphia meaning "describing".


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