Fourier Transform

Naveen Sihag

Mathematical Background:Complex Numbers

• A complex number x is of the form:

a: real part, b: imaginary part

• Addition

• Multiplication

Mathematical Background:Complex Numbers (cont’d)

• Magnitude-Phase (i.e.,vector) representation

Magnitude:

Phase:

φPhase – Magnitude notation:

Mathematical Background:Complex Numbers (cont’d)

• Multiplication using magnitude-phase representation

• Complex conjugate

• Properties

Mathematical Background:Complex Numbers (cont’d)

• Euler’s formula

• Propertiesj

Mathematical Background:Sine and Cosine Functions

• Periodic functions

• General form of sine and cosine functions:

Mathematical Background:Sine and Cosine Functions

Special case: A=1, b=0, α=1

π

π

Mathematical Background:Sine and Cosine Functions (cont’d)

Note: cosine is a shifted sine function:

• Shifting or translating the sine function by a const b

cos( ) sin( )2

t t

Mathematical Background:Sine and Cosine Functions (cont’d)

• Changing the amplitude A

Mathematical Background:Sine and Cosine Functions (cont’d)

• Changing the period T=2π/|α| consider A=1, b=0: y=cos(αt)

period 2π/4=π/2

shorter period higher frequency(i.e., oscillates faster)

α =4

Frequency is defined as f=1/T

Alternative notation: sin(αt)=sin(2πt/T)=sin(2πft)

Image Transforms

• Many times, image processing tasks are best performed in a domain other than the spatial domain.

• Key steps:(1) Transform the image

(2) Carry the task(s) in the transformed domain.

(3) Apply inverse transform to return to the spatial domain.

Transformation Kernels

• Forward Transformation

• Inverse Transformation

1

0

1

0

1,...,1,0,1,...,1,0),,,(),(),(M

x

N

y

NvMuvuyxryxfvuT

1

0

1

0

1,...,1,0,1,...,1,0),,,(),(),(M

u

N

v

NyMxvuyxsvuTyxf

inverse transformation kernel

forward transformation kernel

Kernel Properties

• A kernel is said to be separable if:

• A kernel is said to be symmetric if:

),(),(),,,( 21 vyruxrvuyxr

),(),(),,,( 11 vyruxrvuyxr

Notation

• Continuous Fourier Transform (FT)

• Discrete Fourier Transform (DFT)

• Fast Fourier Transform (FFT)

Fourier Series Theorem

• Any periodic function can be expressed as a weighted sum (infinite) of sine and cosine functions of varying frequency:

is called the “fundamental frequency”

Fourier Series (cont’d)

α1

α2

α3

Continuous Fourier Transform (FT)

• Transforms a signal (i.e., function) from the spatial domain to the frequency domain.

where

(IFT)

Why is FT Useful?

• Easier to remove undesirable frequencies.

• Faster perform certain operations in the frequency domain than in the spatial domain.

Example: Removing undesirable frequencies

remove highfrequencies

reconstructedsignal

frequenciesnoisy signal

To remove certainfrequencies, set theircorresponding F(u)coefficients to zero!

How do frequencies show up in an image?

• Low frequencies correspond to slowly varying information (e.g., continuous surface).

• High frequencies correspond to quickly varying information (e.g., edges)

Original Image Low-passed

Example of noise reduction using FT

Frequency Filtering Steps

1. Take the FT of f(x):

2. Remove undesired frequencies:

3. Convert back to a signal:

We’ll talk more about this later .....

Definitions

• F(u) is a complex function:

• Magnitude of FT (spectrum):

• Phase of FT:

• Magnitude-Phase representation:

• Power of f(x): P(u)=|F(u)|2=

Example: rectangular pulse

rect(x) function sinc(x)=sin(x)/x

magnitude

Example: impulse or “delta” function

• Definition of delta function:

• Properties:

Example: impulse or “delta” function (cont’d)

• FT of delta function:

1

ux

Example: spatial/frequency shifts

)()()2(

)()()1(

),()(

02

20

0

0

uuFexf

uFexxf

thenuFxf

xuj

uxj

Special Cases:

020 )( uxjexx

)( 02 0 uue xuj

Example: sine and cosine functions

• FT of the cosine function

cos(2πu0x)

1/2

F(u)

Example: sine and cosine functions (cont’d)

• FT of the sine function

sin(2πu0x)-jF(u)

Extending FT in 2D

• Forward FT

• Inverse FT

Example: 2D rectangle function

• FT of 2D rectangle function

2D sinc()

Discrete Fourier Transform (DFT)

Discrete Fourier Transform (DFT) (cont’d)

• Forward DFT

• Inverse DFT

1/NΔx

Example

Extending DFT to 2D

• Assume that f(x,y) is M x N.

• Forward DFT

• Inverse DFT:

Extending DFT to 2D (cont’d)

• Special case: f(x,y) is N x N.

• Forward DFT

• Inverse DFT

u,v = 0,1,2, …, N-1

x,y = 0,1,2, …, N-1

Visualizing DFT

• Typically, we visualize |F(u,v)|

• The dynamic range of |F(u,v)| is typically very large

• Apply streching: (c is const)

before scaling after scalingoriginal image

DFT Properties: (1) Separability

• The 2D DFT can be computed using 1D transforms only:

Forward DFT:

Inverse DFT:

2 ( ) 2 ( ) 2 ( )ux vy ux vy

j j jN N Ne e e

kernel isseparable:

DFT Properties: (1) Separability (cont’d)

• Rewrite F(u,v) as follows:

• Let’s set:

• Then:

• How can we compute F(x,v)?

• How can we compute F(u,v)?

DFT Properties: (1) Separability (cont’d)

)

N x DFT of rows of f(x,y)

DFT of cols of F(x,v)

DFT Properties: (1) Separability (cont’d)

DFT Properties: (2) Periodicity

• The DFT and its inverse are periodic with period N

DFT Properties: (3) Symmetry

• If f(x,y) is real, then

(see Table 4.1 for more properties)

DFT Properties: (4) Translation

f(x,y) F(u,v)

) N

• Translation is spatial domain:

• Translation is frequency domain:

DFT Properties: (4) Translation (cont’d)

• Warning: to show a full period, we need to translate the origin of the transform at u=N/2 (or at (N/2,N/2) in 2D)

|F(u-N/2)|

|F(u)|

DFT Properties: (4) Translation (cont’d)

• To move F(u,v) at (N/2, N/2), take

Using ) N

DFT Properties: (4) Translation (cont’d)

no translation after translation

DFT Properties: (5) Rotation

• Rotating f(x,y) by θ rotates F(u,v) by θ

DFT Properties: (6) Addition/Multiplication

but …

DFT Properties: (7) Scale

DFT Properties: (8) Average value

So:

Average:

F(u,v) at u=0, v=0:

Magnitude and Phase of DFT

• What is more important?

• Hint: use inverse DFT to reconstruct the image using magnitude or phase only information

magnitude phase

Magnitude and Phase of DFT (cont’d)

Reconstructed image using

magnitude only

(i.e., magnitude determines the

contribution of each component!)

Reconstructed image using

phase only

(i.e., phase determines

which components are present!)

Magnitude and Phase of DFT (cont’d)