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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