Fourier Transform – Chapter 13

Image space

• Cameras (regardless of wave lengths) create images in the spatial domain

• Pixels represent features (intensity, reflectance, heat, etc.) from the “real 3D world”

Image space

• All operations we’ve looked at so far are applied in the spatial domain– Histogram (statistical) operations

– Point operations

– Filter (convolution) operations

– Edge operations

– Corner (feature) operations

– Line (curve) detection

– Morphological operations

– Region operations

– Color space

Frequency domain

• As it turns out, all the spatial domain “signals” can be represented in the frequency domain

• And, some of the previously mentioned operations can be performed more efficiently in the frequency domain– Convolution– Filtering – Compression

The Fourier Transform

• The Fourier Transform provides the means from moving between the spatial and frequency domains

• Developed in the area of sound processing– Decomposition of sound waves into

elementary harmonic functions

Preliminaries

Sine and Cosine functions• You’ve seen these many times before

)cos()( xxf )2cos()2cos()cos()( kxxxxf

Sine and Cosine functions• Frequency – cycles on horizontal axis

)3cos()( xxf

frequencyangularperiodTT 2

Sine and Cosine functions• Angular frequency and “common”

frequency (f)

fT

f

22

1

Sine and Cosine functions• Amplitude – increasing on vertical axis

)cos(2)( xxf

Sine and Cosine functions• Phase – shifting on horizontal axis

)4cos()( xxf

)cos()( xxf

Sine and Cosine functions• Orthogonality

– We can combine sine and cosine waveforms with varying frequency, amplitude, and phase parameters to create other sine and cosine waveforms

)cos()sin()cos( xCxBxA

A

BC BA tan

122

Sine and Cosine functions• Orthogonality example

)sin(2)cos(3 xx

)cos(3 x)sin(2 x

Sine and Cosine functions• Vector representation

• Complex number representation

• Euler notation (complex numbers on the unit circle)

biaz

71828.2)sin()cos( eiz ei

A

B C

)cos( x

)sin( x )cos()sin( xBxA

vector length ≡ amplitude

Euler notation• Euler notation

• This brings us to the “complex-valued sinusoid”

• Since it’s on a unit circle the amplitude is

• And the phase is

• That is, multiplying by a real value alters the amplitude, multiplying by a complex value alters the phase

71828.2)sin()cos( eiz ei

)sin(}Im{

)cos(}Re{

eei

i

aaaso eeeiii

1

eeeiii )(

Fourier Series

• Not only can sinusoidal functions [of varying frequency, amplitude, and phase] be combined to create other sinusoidal functions but…

• They can be combined to create almost and periodic function

frequencylfundamenta

xkxkxgk

kk BA

0

000)sin()cos()(

Fourier Series

• Frequencies kω0x are harmonics (multiples) of the fundamental

• Ak and Bk are derived via Fourier Analysis

frequencylfundamenta

xkxkxgk

kk BA

0

000)sin()cos()(

Fourier Integral

• But that wasn’t enough…Fourier wanted to cover non-periodic functions too

• Requires more than just integer multiples (harmonics) of the fundamental frequency

• Requires infinitely many frequencies

dxxxg BA )sin()cos()(0

Fourier Integral

• To solve for the amplitudes Aω and Bω we need the following integrals

• Aω and Bω form continuous functions of coefficients (corresponding to infinitely many, densely spaced frequencies)

• Aω and Bω form the Fourier Spectrum

dxxxgB

dxxxgA

B

A

)sin()(1

)(

)cos()(1

)(

Fourier Transform

• Apply the Fourier Series to complex-valued functions using Euler’s notation to get the Fourier Transform

• And the inverse Fourier Transform

dxxgG exi

)(

2

1)(

dxxGg exi

)(

2

1)(

Fourier Transform

• In general– A real-valued function yields a complex-

valued Fourier Transform– A complex-valued function yields a read-

valued Fourier Transform

Fourier Transform

• For example, lots of sinusoidal waves can make up a square wave

Fourier Transform

• Transform pairs are unique, one-to-one

Dirac (delta) function

Fourier Transform

• Properties– There are a bunch of properties that you

can read about, but only one is “surprising”

• Convolution Property– Convolution in the spatial domain is point-

by-point multiplication in the frequency domain

Convolution Property

• Spatial domain– Perform a slide/multiply-accumulate

operation with a kernel and the image

• Frequency domain– Fourier Transform kernel → spectrum– Fourier Transform image → spectrum– Multiply the two spectrums– Inverse Fourier Transform product →

filtered image

Discrete Signals

• So how do we apply these integrals to digital (discrete) images?

Discrete Signals

Next time