mathematics - fouriertransform in image processing

8/14/2019 Mathematics - FourierTransform in Image Processing

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The Fourier Transform in

Image ProcessingAnd a little

Digital Signal Processing


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The Basics

Image as a function f(m,n) = grey level

Alternative transforms make certain types of image manipulation easier

The Fourier Transform Image processing

Image restoration

Image filtering

Image analysis


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This is the 2-D Fourier transform:

1 and 2 are frequencies (- =< =< radians) F(1,2) is the frequency domain

representation of the image

F(0,0) is the sum of all frequencies

+=

=

+=

==

n

n

njmjm

meenmfF 21),(),( 21


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Inverse Fourier Transform

= =

=

1 2

21

2121 )(21),( ddeeFnmf njmj


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Examples


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Examples


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Examples


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MatLab Code

r2=imread('d:\images\2rect.jpg');

r2f=fft2(r2);

imshow(log(abs(r2f)),[3 10]); colormap(jet); colorbar


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Discrete Fourier Transform

Digital sampling of FT

Fast Fourier Transform

f(m,n) defined over 0 =< m,n =< M-1,N-1 DFT

=

=

=

1

0

1

0

22

),(),(M

p

N

n

qnN

jpmM

j

eenmfqpF


10/23

First 56 periodic basis patterns in the DFT


11/23

Inverse DFT

( ) ( )

=

==

1

0

1

0

22

),(1

),(

M

p

N

q

qn

N

jpm

M

j

eeqpFMNnmf


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Why use Fourier?

Fast convolution

Convolution in image domain is multiplication inFourier domain

Linear filter convolution

Filters designed according to frequency response

Locating features (template match)

Rotate filter by 180 degrees

Convolve with test image

Peaks indicate matches


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Phase and Amplitude

In Fourier space an image has twopieces of information

Amplitude (real) Strength of the wave front

Information on the frequencies in the the image

Phase (complex) Position within the wave front

Information in the structure within the image

aj

a er


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Amplitude AmplitudePhase


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RANDOM SIGNALS

A Random Process

A model for a digital signal

A signal is a probabilistic combination ofmultiple random signals (c.f. Fourier)

Given a signal we need to estimate itsunderlying Probability law.


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Correlation and Power Spectra

We often want to analyse images accordingto their statistical content

Signal Noise can be produced by

Signal Processing with finite length buffers Systems with stochastic outputs

Noise cannot be modelled by a function

Images represented in terms statistical

combination of discrete time signals These do not have a Fourier Transform

Some properties can be modelled


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Autocorrelation

Probability of measuring grey value q at (m,n) andq at (m,n)

Autocovariance

Remove the average of the signal and thencalculate correlation

),;,;,(),;,(1

0

1

0

nmnmqqpffnmnmRQ

q

Q

q

qqff=

=

=

))((1

),( ,,

1

0

1

0

nnmmnnmm

M

m

N

n

nmnmff FFFFMN

nmC ++++

=

= =


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We can get Fourier transform of

Autocovariance Describes the distribution of power across

the frequency range

Effect of linear filters on stochastic signals

can be described in terms of the effect onthe autocovariance function

Digital Signal Processing

Design and analysis of filters for signals

and systems.


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Discrete Cosine Transform

Inverse Discrete Cosine Transform


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DCT Image seen as the combination of MxN

functions

If we perform the DCT in 8*8 functionwindows the Bpq are the weights applied tothe 64 basis functions

Used in JPEG Image broken into 8*8

DCT

Coded Coefficients sent

Inverse DCT Try dctdemo in matlab


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Radon Transform

Compute Projections of objects inimages

Similar to the Hough transform find

lines and shapes


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Radon Transform for any

+= ydyxyxfxR )cossin,sincos()(

Where:

=

y

x

y

x

cossin

sincos


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Find straight lines

Edge detect image

Perform radon transform for all angles

Peaks in Radon Transform representstraight lines

Matlab function radon(Image, angle)

mathematics - fouriertransform in image processing

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