MVlec8

8/11/2019 MVlec8

1/11

Lecture 8: Filters in Frequency Domain

If b is an image, then its Fourier transform will reflect the frequencies of the

periodic parts of the image. By masking or filtering out the unwanted frequencies one can

obtained a new image by applying the inverse Fourier transformation. A filteris a matrix

with the same dimension as the Fourier transform of the padded image. The components

of the filter usually vary from 0 to 1. If the component is 1, then the frequency is allowed

to pass; if the component is 0, then the frequency is tossed out. Let Filterrepresent such a

matrix. Then the filtered imageis given byNewbin

2( .* 2( ,2* 1,2* 1) ).Newb ifft Filter fft b nx ny=

The Matlab codes fftsine.m and fftsine2d.m in the previous lectures illustrated this for alow frequency sine wave with higher frequency sine noise.

Three important types of filters are low pass, high pass and band-reject. They are

depicted in the following graphic where the low frequencies have been shifted to the

center and one is viewing the diagonal section of the filter matrix.

low passhigh

pass

band-reject

frequency

The above filters have jumps and are often called ideal filters. The important

filters that make use of polynomial and exponential approximations are the Butterworth

and Gaussian filters. Let w, d0be given and dist(i,j) = ((i-(nx+1))2+ (j-(ny+1))

2)1/2

8/11/2019 MVlec8

2/11

Butterworth Band-reject Filter:

2 2 2

1( , )

1 ( ( , ) /( ( , ) 0 )) nFilter i j

dist i j w dist i j d =

+

.

Gaussian Bandreject Filter:2 2 21(( ( , ) 0 ) / ( , ) )

2( , ) 1 .dist i j d dist i j w

Filter i j e

=

The parameters wand d0control the width and location of the band. In the Butterworth

band-reject filter the exponent n controls the steepest of the boundaries in the band of

frequencies to be rejected. There are similar versions of high and low pass filters.

The Matlab code filter_ncsu.m uses a low pass filter to mask the noise from sine

and cosine functions with frequency equal to 80 (160 in the padded Fourier transform).

8/11/2019 MVlec8

3/11

Matlab Code filter_ncsu.m

cl ear ;ncsu = [ bi gn( 5) bi gc( 7) bi gs( 9) bi gu( 11) ] ;newncsu = 20*ncsu;

[ nx ny] = si ze( newncsu) ;nxnynewncsu = newncsu( nx: - 1: 1, : ) ;newncsu1 = ui nt8(newncsu) ;i mwr i t e( newncsu1, ' ncsu. j pg' ) ;u = newncsu;f or i = 1: nx % Thi s i s NCSU wi t h per i odi c noi se.

f or j = 1: nyu( i , j ) = u( i , j ) + . . .

15. *( 1+si n( 2*pi *( ( i - 1) / nx) *80) ) +. . .15. *( 1+si n( 2*pi *( ( j - 1) / ny) *80) ) ;

end

endsi nencsu = ui nt 8( u) ;i mwr i t e( si nencsu, ' si nencsu. j pg' ) ;f f t u = f f t 2( u, 2*nx- 1, 2*ny- 1) ;f f t u = f f t s hi f t ( f f t u) ;subpl ot ( 2, 2, 1) ;mesh( u' ) ;subpl ot ( 2, 2, 2) ;mesh( l og( 1+( abs( f f t u) ) ) ) ;f i l t er = ones( 2*nx- 1, 2*ny- 1) ;d0 = 150; % Use i deal l ow pass f i l t er .f or i = 1: 2*nx- 1

f or j =1: 2*ny- 1di st = ( ( i - ( nx+1) ) 2 + ( j - ( ny+1) ) 2) . 5;

i f di st > d0f i l t er ( i , j ) = 0;

endend

endsubpl ot ( 2, 2, 3) ;mesh( f i l t er ) ;f i l _ncsu = f i l t er . * f f tu;subpl ot ( 2, 2, 4) ;mesh( l og( 1+abs( f i l _ncsu) ) ) ;f i l _ncsu = i f f t shi f t ( f i l _ncsu) ;f i l _ncsu = i f f t 2( f i l _ncsu, 2*nx- 1, 2*ny- 1) ;f i l _ncsu = r eal ( f i l _ncsu( 1: nx, 1: ny) ) ;

f i l _ncsu = ui nt 8( f i l _ncsu) ;i mwr i te( f i l _ncsu, ' s i nencsu_f i l . j pg' ) ;

8/11/2019 MVlec8

4/11

The filtered image sinencsu_fil.jpg is not entirely satisfactory. Another approach

is to use a band-reject filter where one may need to experiment with the width, w, and the

location, d0, of the band. The Matlab code filter_bu.m uses the Butterworth band-reject

filter and may be applied to either the noisy big sine wave or the noisy ncsu.

8/11/2019 MVlec8

5/11

Matlab Code filter_bu.m

cl ear ;ncsu = [ bi gn( 5) bi gc( 7) bi gs( 9) bi gu( 11) ] ;newncsu = 20*ncsu;[ nx ny] = si ze( newncsu) ;

nxnynewncsu = newncsu( nx: - 1: 1, : ) ;newncsu1 = ui nt8( newncsu) ;i mwr i t e( newncsu1, ' ncsu. j pg' ) ;u = newncsu;f or i = 1: nx

f or j = 1: ny% Thi s i s t he bi g wave wi t h per i odi c noi se.u( i , j ) = 0+100*( 1+si n( 2*pi *( ( j - 1) / ny) *5) ) + . . .


% Thi s i s NCSU wi t h per i odi c noi se.%u( i , j ) = u( i , j ) + . . .


endendsi nencsu = ui nt 8( u) ;i mwr i t e( si nencsu, ' si nencsu. j pg' ) ;f f t u = f f t 2( u, 2*nx- 1, 2*ny- 1) ;f f t u = f f t s hi f t ( f f t u) ;subpl ot ( 2, 2, 1) ;mesh( u' ) ;subpl ot ( 2, 2, 2) ;mesh( l og( 1+( abs( f f t u) ) ) ) ;f i l t er = ones( 2*nx- 1, 2*ny- 1) ;

d0 = 160; % Use But t er wor t h band- r ej ect f i l t er .n = 4;w = 20;f or i = 1: 2*nx- 1

f or j =1: 2*ny- 1di st = ( ( i - ( nx+1) ) 2 + ( j - ( ny+1) ) 2) . 5;i f di st ~= d0

f i l t er ( i , j ) = 1/ ( 1 + ( di st*w/ ( di st 2 - d0 2) ) ( 2*n) ) ;el se

f i l t er ( i , j ) = 0;end

endendsubpl ot ( 2, 2, 3) ;

mesh( f i l t er ) ;f i l _ncsu = f i l t er . * f f tu;subpl ot ( 2, 2, 4) ;mesh( l og( 1+abs( f i l _ncsu) ) ) ;f i l _ncsu = i f f t shi f t ( f i l _ncsu) ;f i l _ncsu = i f f t 2( f i l _ncsu, 2*nx- 1, 2*ny- 1) ;f i l _ncsu = r eal ( f i l _ncsu( 1: nx, 1: ny) ) ;f i l _ncsu = ui nt 8( f i l _ncsu) ;i mwr i te( f i l _ncsu, ' s i nencsu_f i l . j pg' ) ;

8/11/2019 MVlec8

6/11

Another application of the band-reject filter is to the noisy aerial photograph. This

image suffers from too much light an exposure and from banded sine and cosine noise.

The light is modified by use of the power transformation with the power equal to two,and then the Butterworth band-reject filter is used to reduce to noise.

8/11/2019 MVlec8

7/11

Matlab Code filter_aerial.m

cl ear ;aer i al = i mr ead( ' Fi g3. 09( a) . j pg' ) ;aer i al = doubl e( aer i al ) ;

[ nx ny] = si ze( aer i al ) ;nxnyu = aer i al ;

f or i = 1: nxf or j = 1: ny

% Thi s i s aer i al wi t h per i odi c noi se.u( i , j ) = u( i , j ) + . . .

5. *( 1+si n( 2*pi *( ( i - 1) / nx) *200) ) +. . .5. *( 1+si n( 2*pi *( ( j - 1) / ny) *200) ) +. . .5. *( 1+cos( 2*pi *( ( i - 1) / nx+( j - 1) / ny) *141) ) +. . .5. *(1+si n( 2*pi *(( i - 1) / nx- ( j - 1) / ny) *141) ) ; ;

endendsi neaer i al = ui nt 8( u) ;i mwr i t e( s i neaer i al , ' s i neaer i al . j pg' ) ;

c = 1. ; % Use t he power t r ansf ormat i on t o dar ken.gamma = 2;f _f p =255*c* ( u/ 255) . gamma;u = f _f p;f f t u = f f t 2( u, 2*nx- 1, 2*ny- 1) ;f f t u = f f t s hi f t ( f f t u) ;subpl ot ( 1, 2, 1)mesh( l og( 1+( abs( f f t u) ) ) ) ;

f i l t er = ones( 2*nx- 1, 2*ny- 1) ;d0 = 400; % Use But t er wort h band r ej ect f i l t er .n = 4;w = 20;f or i = 1: 2*nx- 1

f or j =1: 2*ny- 1di st = ( ( i - ( nx+1) ) 2 + ( j - ( ny+1) ) 2) . 5;i f di st ~= d0

f i l t er ( i , j ) = 1/ ( 1 + ( di st*w/ ( di st 2 - d0 2) ) ( 2*n) ) ;el se

f i l t er ( i , j ) = 0;end

end

endf i l _ aer i al = f i l t er . * f f t u;subpl ot ( 1, 2, 2)mesh( l og( 1+abs( f i l _aer i al ) ) ) ;

f i l _aer i al = i f f t shi f t ( f i l _aer i al ) ;f i l _aer i al = i f f t2( f i l _aer i al , 2*nx- 1, 2*ny- 1) ;f i l _aer i al = real ( f i l _aer i al ( 1: nx, 1: ny) ) ;f i l _aer i al = ui nt8( f i l _aer i al ) ;i mwr i te( f i l _aer i al , ' si neaer i al _ f i l . j pg' ) ;

8/11/2019 MVlec8

8/11

8/11/2019 MVlec8

9/11

The Matlab code filter_micro.m uses a high pass filter to give emphasis to the

higher frequencies in the image of a damaged electronic chip. The high pass image is

then added to the original image so as to obtain a sharper image. The reader may find it

interesting to experiment with width and frequency threshold of the Butterworth or the

Gaussian high pass filters. Also, it is interesting to compare this sharpening, which is

done in the frequency domain, with the sharpening done in the space domain as in the

third lecture .

FFT of Image High Pass Filter of FFT Image

8/11/2019 MVlec8

10/11

Matlab Code filter_micro.m

cl ear ;

mi cro = i mr ead( ' Fi g4. 04( a) . j pg' ) ;mi cr o = doubl e(mi cr o) ;[ nx ny] = si ze( mi cr o);nxnyu = mi cr o;mi cr o = ui nt 8( u) ;i mwr i t e( mi cro, ' mi cro. j pg' ) ;f f t u = f f t 2( u, 2*nx- 1, 2*ny- 1) ;f f t u = f f t s hi f t ( f f t u) ;subpl ot ( 1, 2, 1)mesh( l og( 1+( abs( f f t u) ) ) ) ;% Use But t er wor t h or Gaussi an hi gh pass f i l t er .f i l t er = ones( 2*nx- 1, 2*ny- 1) ;

d0 = 100;n = 4;f or i = 1: 2*nx- 1

f or j =1: 2*ny- 1di st = ( ( i - ( nx+1) ) 2 + ( j - ( ny+1) ) 2) . 5;% Use But t er wor t h hi gh pass f i l t er .f i l t er ( i , j ) = 1/ ( 1 + ( di st / d0) ( 2*n) ) ;f i l t er ( i , j ) = 1. 0 - f i l t er ( i , j ) ;% Use Gaussi an hi gh pass f i l t er .%f i l t er ( i , j ) = exp( - di st 2/ ( 2*d0 2) ) ;%f i l t er ( i , j ) = 1. 0 - f i l t er ( i , j ) ;

endend

% Updat e i mage wi t h hi gh f r equenci es.f i l _mi cro = f f tu + f i l t er . * f f tu;subpl ot ( 1, 2, 2)mesh( l og( 1+abs( f i l _mi cro- f f t u) ) ) ;f i l _mi cro = i f f t shi f t ( f i l _mi cro) ;f i l _mi cro = i f f t 2( f i l _mi cro, 2*nx- 1, 2*ny- 1) ;f i l _mi cro = r eal ( f i l _mi cro( 1: nx, 1: ny) ) ;f i l _mi cro = ui nt 8( f i l _mi cro) ;i mwr i te( f i l _mi cro, ' mi cro_ f i l . j pg' ) ;

8/11/2019 MVlec8

11/11

Documents

MVlec8