Trasformata di Fourier

Michele Nappi, Ph.D

Università degli Studi di Salerno

mnappi@unisa.it

biplab.unisa.it

089-963334

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IndexLinear Systems

• Definitions & Properties

• Shift Invariant Linear Systems

• Linear Systems and Convolutions

• Linear Systems and sinusoids

• Complex Numbers and Complex Exponentials

• Linear Systems - Frequency Response

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Linear System

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Linear System Properties

Linear Shift Invariant Systems (LSIS)

Linearity:

1f 1g 2f 2g

21 ff 21 gg

Shift invariance:

axf axg

a a

Example of LSIS

Defocused image ( g ) is a

processed version of the

focused image ( f )

g f

Ideal lens is a LSIS xf xgLSIS

Linearity: Brightness variation

Shift invariance: Scene movement

(not valid for lenses with non-linear distortions)

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Shift Invariant Linear System (cont.)

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Convolution in Continous Case

1 2-1-2

xc

-1 1

1

xb

-1 1

1 xa

bac

1

Convolution - Example

Convolution Kernel – Impulse Response

f gh hfg

• What h will give us g = f ?

Dirac Delta Function (Unit Impulse)

x

21

0

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Impulse Sequence

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Shift-Invariant Linear

System and Convolution

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Complex Number

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Complex Number (cont.)

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Complex Number (cont.)

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The (Co-)Sinusoid

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The (Co-)Sinusoid (cont.)

• The wavelength of

sin(2πωx) is 1/ω .

• The frequency is ω .

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The (Co-)Sinusoid (cont.)

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The (Co-)Sinusoid (cont.)

• If we add a Sine wave

to a Cosine wave with

the same frequency

we get a scaled and

shifted (Co-)Sine

wave with the same

frequency

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The (Co-)Sinusoid (cont.)

Phase tg

Amplitude R

sin cossinsincos

:obtain we(1) from Thus

sin and cos

:such that exists There

1

Since

(1) cossinbcosasin

:Proof

1-

22

2222

2222

2

22

2

22

2222

22

a

b

ba

xbaxxba

ba

b

ba

a

ba

b

ba

a

xba

bx

ba

abaxx

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The response of Shift-Invariant

Linear System to a Sine wave

• The response of a

shift-invariant linear

system to a sine wave

is a shifted and

scaled sine wave with

the same frequency.

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The response of Shift-Invariant

Linear System to a Sine wave (cont.)

Jean Baptiste Joseph Fourier (1768-1830)

• Had crazy idea (1807):

• Any periodic function

can be rewritten as a

weighted sum of Sines and

Cosines of different

frequencies.

• Don’t believe it?

– Neither did Lagrange,

Laplace, Poisson and

other big wigs

– Not translated into

English until 1878!

• But it’s true!

– called Fourier Series

– Possibly the greatest tool

used in Engineering

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The Fourier Transform (cont.)

Time and Frequency

• example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)

Time and Frequency

• example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)

= +

Frequency Spectra

• example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)

= +

Frequency Spectra

• Usually, frequency is more interesting than the phase

= +

=

Frequency Spectra

= +

=

Frequency Spectra

= +

=

Frequency Spectra

= +

=

Frequency Spectra

= +

=

Frequency Spectra

= 1

1sin(2 )

k

A ktk

Frequency Spectra

Frequency Spectra

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Frequency Analysis

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The Fourier Transform

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The Fourier Transform (cont.)

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The Fourier Transform (cont.)

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The Fourier Transform (cont.)

Frequency Domain

Filtering

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2D Fourier Transform

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Discrete Fourier Transform

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Discrete Fourier Transform (cont.)

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Discrete Fourier Transform (cont.)

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Discrete Fourier Transform (cont.)

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Discrete Fourier Transform (cont.)

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Discrete Fourier Transform (cont.)

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Discrete Fourier Transform (cont.)

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Discrete Fourier Transform (cont.)

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Spatial vs Frequency Domain

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Spatial vs Frequency Domain (cont.)

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Spatial vs Frequency Domain (cont.)

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Low Pass Filter

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Sharpening (High Pass) Filter

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Band Pass Filter

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Filtering Example (cont.)

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Filtering Example

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Filtering Example (cont.)

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Filtering Example (cont.)

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Frequency Bands

• Percentage of image

power enclosed in

circles (small to

large):

90, 95, 98, 99, 99.5, 99.9

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Fourier Transform

• Computing Time

– O(n2)

• Fast Fourier Transform (FFT)

– O(nlogn)