Spatial Spatial Frequencies Frequencies

Why are Spatial Frequencies Why are Spatial Frequencies important?important?

• Efficient data representation

• Provides a means for modeling and removing noise

• Physical processes are often best described in “frequency domain”

• Provides a powerful means of image analysis

What is spatial frequency?What is spatial frequency?

• Instead of describing a function (i.e., a shape) by a series of positions

• It is described by a series of cosines

What is spatial frequency?What is spatial frequency?

A

g(x) = A cos(x)

2

x

g(x)

What is spatial frequency?What is spatial frequency?

Period (L)Wavelength ()Frequency f=(1/ )

Amplitude (A)Magnitude (A)

A cos(x 2/L)g(x) = A cos(x 2/) A cos(x 2f)

x

g(x)

What is spatial frequency?What is spatial frequency?

A

g(x) = A cos(x 2f)

x

g(x)

(1/f)(1/f)

period

But what if cosine is shifted in phase?But what if cosine is shifted in phase?

g(x) = A cos(x 2f + )

x

g(x)

What is spatial frequency?What is spatial frequency?

g(x) = A cos(x 2f + )

A=2 mf = 0.5 m-1

= 0.25 = 45g(x) = 2 cos(x 2(0.5) + 0.25) 2 cos(x + 0.25)

x g(x)0.00 2 cos(0.25) = 0.707106...0.25 2 cos(0.50) = 0.00.50 2 cos(0.75) = -0.707106...0.75 2 cos(1.00) = -1.01.00 2 cos(1.25) = -0.707106…1.25 2 cos(1.50) = 01.50 2 cos(1.75) = 0.707106...1.75 2 cos(2.00) = 1.02.00 2 cos(2.25) = 0.707106...

Let us take arbitrary g(x)

We substitute values of A, f and

We calculate discrete values of g(x) for various values of x

What is spatial frequency?What is spatial frequency?

g(x) = A cos(x 2f + )

x

g(x) We calculate discrete values of g(x) for various values of x

What is spatial frequency?What is spatial frequency?

12/

0

12/

0

/2cos)(Ni

iii

Ni

ii NixAxgxg

g(x) = A cos(x 2f + )

gi(x) = Ai cos(x 2i/N + i), i = 0,1,2,3,…,N/2-1

We try to approximate a periodic We try to approximate a periodic function with standard trivial function with standard trivial (orthogonal, base) functions(orthogonal, base) functions

+

+=

Low frequency

Medium frequency

High frequency

We add values from component We add values from component functions functions point by pointpoint by point

+

+=

g(x)

i=1

i=2

i=3

i=4

i=5

i=63

0 127x

Example of periodic function created by summing standard trivial functions

g(x)

i=1

i=2

i=3

i=4

i=5

i=10

0 127x

Example of periodic function created by summing standard trivial functions

g(x)

g(x)

64 terms

10 terms

Example of periodic function created by summing standard trivial functions

g(x)

i=1

i=2

i=3

i=4

i=5

i=630 127

x

Fourier Decomposition of a step function (64 terms)

Example of periodic function created by summing standard trivial functions

g(x)

i=1

i=2

i=3

i=4

i=5

i=100 63

x

Fourier Decomposition of a step function (11 terms)

Example of periodic function created by summing standard trivial functions

Main concept – summation of base Main concept – summation of base functionsfunctions

12/

0

/2cos)(Ni

iii NixAxg

Any function of x (any shape) that can be represented by g(x) can also be represented by the summation of cosine functions

Observe two numbers for every i

Information is not lost when we Information is not lost when we change the domainchange the domain

gi(x) = 1.3, 2.1, 1.4, 5.7, …., i=0,1,2…N-1

N pieces of information

12/

0

/2cos)(Ni

iiii NixAxg

N pieces of informationN/2 amplitudes (Ai, i=0,1,…,N/2-1) andN/2 phases (i, i=0,1,…,N/2-1) and

SpatialSpatial Domain

Frequency Domain

What is spatial frequency?What is spatial frequency?

gi(x)

Are equivalentThey contain the same amount of information

12/

0

/2cosNi

iii NixA and

The sequence of amplitudes squared is the SPECTRUM

Information is not lost when we Information is not lost when we change the domainchange the domain

EXAMPLE

A cos(x2i/N)frequency (f) = i/Nwavelength (p) = N/I

N=512i f p0 0 infinite1 1/512 51216 1/32 32256 1/2 2

Substitute values

Assuming N we get this table which relates frequency and wavelength of component functions

More examples to give you some intuition….

Fourier Transform NotationFourier Transform Notation• g(x) denotes an spatial domain function of real numbers

– (1.2, 0.0), (2.1, 0.0), (3.1,0.0), …

• G() denotes the Fourier transform

• G() is a symmetric complex function(-3.1,0.0), (4.1, -2.1), (-3.1, 2.1), …(1.2,0.0) …, (-3.1,-2.1), (4.1, 2.1), (-3.1,0.0)

• G[g(x)] = G(f) is the Fourier transform of g(x)

• G-1() denotes the inverse Fourier transform

• G-1(G(f)) = g(x)

Power Spectrum and Phase SpectrumPower Spectrum and Phase Spectrum

• |G(f)|2 = G(f)G(f)* is the power spectrum of G(f)– (-3.1,0.0), (4.1, -2.1), (-3.1, 2.1), … (1.2,0.0),…, (-3.1,-2.1), (4.1, 2.1)– 9.61, 21.22, 14.02, …, 1.44,…, 14.02, 21.22

• tan-1[Im(G(f))/Re(G(f))] is the phase spectrum of G(f)– 0.0, -27.12, 145.89, …, 0.0, -145.89, 27.12

complex

Complex conjugate

1-D DFT and IDFT1-D DFT and IDFT• Discrete Domains

– Discrete Time: k = 0, 1, 2, 3, …………, N-1– Discrete Frequency: n = 0, 1, 2, 3, …………, N-1

• Discrete Fourier Transform

• Inverse DFT

Equal time intervals

Equal frequency intervals

1N

0k

nkN2j

;e ]k[x]n[X

1N

0n

nkN2j

;e ]n[XN1]k[x

n = 0, 1, 2,….., N-1

k = 0, 1, 2,….., N-1