1

Methods in Image Analysis – Lecture 3

Fourier

U. Pitt Bioengineering 2630

CMU Robotics Institute 16-725

Spring Term, 2006

George Stetten, M.D., Ph.D.

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Frequency in time vs. space

• Classical “signals and systems” usually temporal signals.

• Image processing uses “spatial” frequency.• We will review the classic temporal description first,

and then move to 2D and 3D space.

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Phase vs. Frequency

• Phase, , is angle, usually represented in radians.• (circumference of unit circle)• Frequency, , is the rate of change for phase.

• In a discrete system, the sampling frequency, , is the amount of phase-change per sample.

€

θ

°=360 radians 2πω

t ωθ =sω

ns ωθ =

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Euler’s Identity

€

e jθ = cosθ + j sinθ

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Phasor = Complex Number

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multiplication = rotate and scale

( )( )( )( )

( )

.by scale and by rotate

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21

21

21

2211

r

err

erer

jyxjyx

rejyxz

j

jj

j

θ

θθ

θθ

θ

+=

=

++

=+=

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Spinning phasor

f 2πω =

8

9

10

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Continuous Fourier Series

( ) ∑+∞

−∞=

=k

tjkkeatx 0ω ( )∫ −=

0

0

0

1

T

tjkk dtetx

Ta ω

Synthesis Analysis

0ω is the Fundamental Frequency

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Selected properties of Fourier Series

( ) kFs atx ⏐ →← ( ) k

Fs bty ⏐ →←

( ) ( ) kkFs BbAatBytAx +⏐ →←+

*kk aa −= ( )txfor real

( )k

Fs ajkdt

tdx0ω⏐ →←

( )∫ ⏐ →← kFs a

jkdttx

0

1

ω

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Differentiation boosts high frequencies

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Integration boosts low frequencies

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Continuous Fourier Transform

Synthesis Analysis

( ) ( )ωXtx F⏐ →←

( ) ( )∫+∞

∞−

= ωωπ

ω deXtx tj

2

1 ( ) ( ) dtetxX tjωω −+∞

∞−∫=

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( ) ⎟⎠⎞

⎜⎝⎛⏐ →←αω

αα Xtx F 1

Selected properties of Fourier Transform

( ) ( ) ( ) ( )ωω YXtytx F⏐ →←∗

( ) ( )∫∫+∞

∞−

+∞

∞

= ωωπ

dXdttx2

-

2

2

1

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Special Transform Pairs

• Impulse has all frequences

• Average value is at frequency = 0

• Aperture produces sync function

( ) ( ) ( )ωπω ∂=⏐ →←= 2 1 Xtx F

( ) ( ) ( ) 1 =⏐ →←∂= ωXttx F

( ) ( ) ( )ωωω 1

1

1 sin2

0

,1 TX

Tt

Tttx F =⏐ →←

⎩⎨⎧

>

<=

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Discrete signals introduce aliasing

Frequency is no longer the rate of phase change in time, but rather the amount of phase change per sample.

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Sampling > 2 samples per cycle

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Sampling < 2 samples per cycle

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Under-sampled sine

( ) stx ωat sampled For

frequency.Nyquist theis 2

sω

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Discrete Time Fourier Series

[ ] [ ]Nnxnx +=

πωω 2 assume 0 == Ns

Sampling frequency is 1 cycle per second, and fundamental frequency is some multiple of that.

Nkk aa +=Synthesis Analysis

[ ] ∑=

=Nk

tjkkeanx 0ω [ ]∑

=

−=Nk

tjkk enx

Na 0

1 ω

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Matrix representation Nj

ewπ2

= 1=Nw

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Fast Fourier Transform

• N must be a power of 2• Makes use of the tremendous

symmetry within the F-1 matrix• O(N log N) rather than O(N2)

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Discrete Time Fourier Transform

[ ] ( )∫+

−

=π

π

ωω ωπ

deeXnx njj 2

1

( ) ( )( )πωω 2+= jj eXeX

Synthesis Analysis

( ) [ ] nj

n

j enxeX ωω −+∞

−∞=∑=

πω 2 assume =s

Sampling frequency is still 1 cycle per second, but now any frequency are allowed because x[n] is not periodic.

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The Periodic Spectrum

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Aliasing Outside the Base Band

sω41

sin−Perceived as

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2D Fourier TransformAnalysis

Synthesis

or separating dimensions,

€

F u,v( ) = f x, y( ) e− j 2π ux +vy( )dx dy−∞

+∞

∫−∞

+∞

∫

€

F u,v( ) = f x, y( ) e− j 2π uxdx −∞

+∞

∫ ⎡

⎣ ⎢

⎤

⎦ ⎥

−∞

+∞

∫ e− j 2π vydy

€

f x,y( ) = F u,v( ) e j 2π ux +vy( )du dv−∞

+∞

∫−∞

+∞

∫

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Properties

• Most of the usual properties, such as linearity, etc.• Shift-invariant, rather than Time-invariant• Parsevals relation becomes Rayleigh’s Theorem• Also, Separability, Rotational Invariance, and

Projection (see below)

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Separability

( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )vFyf

uFxf

vuFvFuFyfxf

then

vuFyxf

yfxfyxf

F

F

F

F

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2121

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,

,,

,

if

⏐ →←

⏐ →←

=⏐ →←

⏐ →←

=

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Rotation Invariance

( )( )θθθθ

θθθθcossin ,sincos

cossin ,sincos

vuvuF

yxyxf F

+−+

⏐ →←+−+

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡′′

y

x

y

x

θθ

θθ

cossin

sincos

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Projection

( ) ( )

( ) ( )0 ,

,

uFuP

dyyxfxp

=

= ∫+∞

∞−

Combine with rotation, have arbitrary projection.

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Gaussian

( )2

2

2

2

2

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222 σσσ

yxyx

eee−−+−

=seperable

Since the Fourier Transform is also separable, the spectra of the 1D

Gaussians are, themselves, separable.

€

g x( ) F← → ⏐ G u( )

g1 x( )∗g2 x( ) = g3 x( )

G1 u( )G2 u( ) = G3 u( )

g3 x( ) F← → ⏐ G3 u( )

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Hankel TransformFor radially symmetrical functions

( ) ( )( ) ( )

( ) ( ) ( )

( ) ( )qFdrrderf

dydxeyxfvuF

vuqqFvuF

yxrrfyxf

rqrj

r

vyuxj

r

r

=⎥⎦

⎤⎢⎣

⎡

==

+==

+==

∫ ∫

∫ ∫∞

−

+−∞+

∞−

∞+

∞−

,,

,,

,,

0

2

0

cos2

2

222

222

θπ

θπ

π

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Elliptical Fourier Series for 2D Shape

( ) kFs atx ⏐ →←

( ) kFs bty ⏐ →←

Parametric function, usually with constant velocity.

( )00,center ba=

Truncate harmonics to smooth.

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Fourier shape in 3D

• Fourier surface of 3D shapes (parameterized on surface).

• Spherical Harmonics (parameterized in spherical coordinates).

• Both require coordinate system relative to the object. How to choose? Moments?

• Problem of poles: singularities cannot be avoided

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Quaternions – 3D phasors

4321 kajaiaaa +++=

1222 −==== ijkkji

4321* kajaiaaa −−−=

Product is defined such that rotation by arbitrary angles from arbitrary starting points become simple multiplication.

( )2

12

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32

22

1 aaaaa +++=

( ) ( ) ( ) ( )44332211 bakbajbaibaba +++++++=+

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Summary

• Fourier useful for image “processing”, convolution becomes multiplication.

• Fourier less useful for shape.• Fourier is global, while shape is local.• Fourier requires object-specific coordinate system.