g52iip, school of computer science, university of nottingham 1 image transforms fourier transform...

G52IIP, School of Computer Science, University of Nottingham

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Image Transforms

Fourier TransformBasic idea


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Image Transforms

Fourier transform theory Let f(x) be a continuous function of a real variable x.

The Fourier transform of f(x) is

Given F(u), f(x) can be obtained by using the inverse Fourier transform

dxuxjxfuF 2exp)(

duuxjuFxf 2exp)(


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Image Transforms

Fourier transform theoryThe Fourier transform F(u) is in general

complex

It is often convenient to write it in the form

uieuFujuIuRuF exp)()()( 2

122

)()()( ujIuRuF


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Image Transforms

Fourier transform theoryMagnitude and Phase

21

22 )()( uIuRuF

uR

uIu 1tan

Fourier Spectrum of f(x)

Phase angle

)()( 22 uIuRuP Power Spectrum

(spectrum density function) of f(x)


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Image Transforms

Fourier transform theory Frequency

Euler’s formula

21

22 )()( uIuRuF

uR

uIu 1tan

u is called the

frequency variable

uxjuxuxj 2sin2cos2exp


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Image Transforms

Fourier transform theory Intuitive interpretation

dxuxjxfuF 2exp)(

duuxjuFxf 2exp)(

An infinite sum of sine and cosine terms,

each u determines the frequency of its

corresponding sine cosine pair


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Image Transforms

Fourier transform


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Image Transforms

Fourier transformWhen W become smaller, what will

happen to the spectrum?


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Image Transforms

Discrete Fourier transformContinuous function f(x) is discretized

into a sequence

by taking N samples x units apart

xNxfxxfxxfxf )1(,,2,, 0000


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Image Transforms

Discrete Fourier transform pair of the sampled function

1,...,2,1,0

2exp

1 1

00

Nufor

N

uxjxxxf

NuF

N

x

1,...,2,1,0

2exp

1

0

Nxfor

N

uxjuFxf

N

u


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Image Transforms

Fourier transform of unit impulse function

0

00)(

t

tt and 1)(

dtt

0 t


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Image Transforms

Fourier transform of unit impulse function

dxexx jux)()]([ F 10

x

juxe

0 x

(x)

0 u

1

F(ju)

F


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Image Transforms

Fourier transform of unit impulse train

Here t = x and = u


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Convolution

ConvolutionThe convolution of two functions f(x) and

g(x), denote f(x)*g(x)

daaxgafxgxf )()()()(


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Convolution

ConvolutionAn example


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Convolution

Convolution and Spatial Filtering

f(x,y)

w(x,y)

f(x,y)*w(x,y)


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Convolution

Convolution theorem

uGuFxgxf

uGuFxgxf


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Sampling

Sampling

FT

FT

FT

t

-w w

t t

t

f(t) F(u)

s(t) S(u)

s(t)f(t) S(u)*F(u)


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Sampling

Sampling

FT

FT

t

t-w w

G(u)

G(u)[S(u)*F(u)]= F(u)]f(t)

-w w


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Sampling Theorem

Bandwidth, Sample Rate, and Nyquist Theorem

The sampling rate (Nyquist rate) must be at least two times the bandwidth of a bandlimited signal

wtw

t 2

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1

t

-w w

G(u)

G(u)[S(u)*F(u)]= F(u)]

-w w


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Aliasing

Over- and under-samplingAnti-aliasing filtering


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Aliasing

Consider an image with 512 alternating vertical black and white stripes. (You may not even be able to see the alternating stripes because of poor screen resolution. But take my word for it, they are there.) Source:http://www.cs.unm.edu/~brayer/vision/perception.html


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Aliasing

The image is created by sampling an image with 512 alternating values of black (gray = 0) and white (gray = 255). Starting in row 0, 512 samples of the image are taken. For each successive row, 1 fewer sample is taken from row 0, (i.e. for row 1, take 511 samples, for row 2, take 510 samples, ... for row 511, take 1 sample). The whole row is then reconstructed from the samples by pixel replication. The result is a colossal aliasing pattern.

Source:http://www.cs.unm.edu/~brayer/vision/perception.html


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Aliasing

More examples


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Aliasing

More examples


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Aliasing

More examples


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Image Transforms

2D Fourier Transform (Fourier Transform of Images)

dxdyvyuxjyxfvuF )(2exp),(,

dudvvyuxjvuFyxf )(2exp),(,


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Image Transforms

2D Fourier Transform (Fourier Transform of Images)

21

22 ),(),(, vuIvuRvuF

vuR

vuIvu

,

,tan, 1

Fourier Spectrum of f(x)

Phase angle

),(),(, 22 vuIvuRvuP Power Spectrum

(spectrum density function) of f(x)


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Image Transforms

2D Discrete Fourier Transform (Fourier Transform of Digital Images)

1,...,2,1,01,...,2,1,0

2exp,1

,1

0

1

000

NvMufor

N

vy

M

uxjyyxxxxf

MNvuF

M

x

N

y

1,...,2,1,01,...,2,1,0

2exp,,1

0

1

0

NyMxfor

N

vy

M

uxjvvuuFyxf

M

u

N

v


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Frequency Domain Processing

What does frequency mean in an image?


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High frequency components – fast changing/sharp features

Low frequency components – slow changing/smooth features


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The foundation of frequency domain techniques is the convolution theorem

vuGvuFyxgyxf ,,,,


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H(u, v) is called the transfer

function


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Typical lowpass filters and their transfer functions


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Example


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Example


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Example


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Example


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Example


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Example


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Typical highpass filters and their transfer functions


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Examples


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Examples


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Examples


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More examples


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Examples


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Examples


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Spatial vs frequency domain


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Spatial vs frequency domain


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Examples

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image transforms fourier

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