Basis beeldverwerking (8D040)
dr. Andrea Fuster
Prof.dr. Bart ter Haar Romeny
dr. Anna Vilanova
Prof.dr.ir. Marcel Breeuwer
The Fourier Transform I
Contents
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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
(see figure 4.1 book)
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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
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•
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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
Sin (θ) ?
Cos (θ)?
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Im
Re
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Complex math
• Complex (vector) addition
• Multiplication with i
is rotation by 90 degrees in the
complex plane
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Contents
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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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Impulse train
 What does this look like? ΔT = 1
Note: impulses can be continuous or discrete!
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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 Series
Periodic
with
period T
with
Series of
sines and
cosines,
see
Euler’s
formula
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Fourier transform – 1D cont. case
Symmetry: The only difference between the Fourier
transform and its inverse is the sign of the exponential.
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Fourier and Euler
 Fourier
 Euler
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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
A
-W/2
W/2
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Examples – Block 2
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Examples – Block 3
?
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Examples – Impulse
constant
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Examples – Shifted impulse
Euler
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Examples – Shifted impulse 2
impulse
constant
Real part
Imaginary part
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• Also: using the following symmetry
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Examples - Impulse train
Periodic with period ΔT
Encompasses only one impulse, so
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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.
(see figure 4.5 book)
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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
(who?)
• From FT of impulse train
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FT of sampled functions
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• Sifting property
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of
copies of
is a periodic infinite sequence 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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•
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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
Note: separation between samples in F. domain is
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By now we probably need some …
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