Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr. Bart ter Haar Romeny dr. Anna Vilanova...
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Transcript of 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
5
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!
50
Contents
• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform
57
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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