basis beeldverwerking (8d040) dr. andrea fuster prof.dr. bart ter haar romeny dr. anna vilanova...

Basis beeldverwerking (8D040)

dr. Andrea FusterProf.dr. Bart ter Haar Romenydr. Anna VilanovaProf.dr.ir. Marcel Breeuwer

The Fourier Transform I

Contents

• Complex numbers etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Introduction• Jean Baptiste

Joseph Fourier (*1768-†1830)

• French Mathematician• La Théorie Analitique

de la Chaleur (1822)

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Fourier Series

• Any periodic function can be expressed as a sum of sines and/or cosines

Fourier Series

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(see figure 4.1 book)

Fourier Transform

• Even functions that • are not periodic • and have a finite area under curve

can be expressed as an integral of sines and cosines multiplied by a weighing function

• Both the Fourier Series and the Fourier Transform have an inverse operation:

• Original Domain Fourier Domain

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Contents

• Complex numbers etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Complex numbers

• Complex number

• Its complex conjugate

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Complex numbers polar

• Complex number in polar coordinates

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Euler’s formula

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Sin (θ)

Cos (θ)

?

?

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Re

Im

Complex math

• Complex (vector) addition

• Multiplication with i

is rotation by 90 degrees in the complex plane

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Contents

• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Unit impulse (Dirac delta function)

• Definition

• Constraint

• Sifting property

• Specifically for t=0

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Discrete unit impulse

• Definition

• Constraint

• Sifting property

• Specifically for x=0

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What does this look like?

Impulse train

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ΔT = 1

Note: impulses can be continuous or discrete!

Contents

• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Fourier Series

with

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Series of sines and cosines,

see Euler’s formula

Periodic with

period T

Fourier transform – 1D cont. case

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Symmetry: The only difference between the Fourier transform and its inverse is the sign of the exponential.

Fourier

Euler

Fourier and Euler

• If f(t) is real, then F(μ) is complex• F(μ) is expansion of f(t) multiplied by sinusoidal terms• t is integrated over, disappears• F(μ) is a function of only μ, which determines the

frequency of sinusoidals• Fourier transform frequency domain

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Examples – Block 1

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-W/2 W/2

A

Examples – Block 2

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Examples – Block 3

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?

Examples – Impulse

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constant

Examples – Shifted impulse

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Euler

Examples – Shifted impulse 2

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Real part Imaginary part

impulse constant

• Also: using the following symmetry

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Examples - Impulse train

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Periodic with period ΔT

Encompasses only one impulse, so

Examples - Impulse train 2

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• So: the Fourier transform of an impulse train with period is also an impulse train with period

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Contents

• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Fourier + Convolution

• What is the Fourier domain equivalent of convolution?

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• What is

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Intermezzo 1

• What is ?

• Let , so

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Intermezzo 2

• Property of Fourier Transform

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Fourier + Convolution cont’d

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Convolution theorem

• Convolution in one domain is multiplication in the other domain:

• And also:

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And:

• Shift in one domain is multiplication with complex exponential (modulation) in the other domain

• And:

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Contents

• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Sampling

• Idea: convert a continuous function into a sequence of discrete values.

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(see figure 4.5 book)

Sampling

• Sampled function can be written as

• Obtain value of arbitrary sample k as

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Sampling - 2

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Sampling - 3

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FT of sampled functions

• Fourier transform of sampled function

• Convolution theorem

• From FT of impulse train

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(who?)

FT of sampled functions

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• Sifting property

• of is a periodic infinite sequence of

copies of , with period

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Sampling

• Note that sampled function is discrete but its Fourier transform is continuous!

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Contents

• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Discrete Fourier Transform

• Continuous transform of sampled function

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• is continuous and infinitely periodic with period 1/ΔT

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• We need only one period to characterize• If we want to take M equally spaced samples from

in the period μ = 0 to μ = 1/Δ, this can be done thus

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• Substituting

• Into

• yields

61Note: separation between samples in F. domain is

By now we probably need some …

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