in fourier domain in spatial domain linear filters non...

IE PŁ P. Strumiłło, M. Strzelecki

Image filtering

In Fourier domain In spatial domain

Linear filters Non-linear filters


Imagepreprocessing

Imagepreprocessing

Computer

+

program

+

Knowledge

base

Computer

+

program

+

Knowledge

baseIma

ge

acqu

isitio

n

• contrast enhancement

• noise reduction

• image sharpening

• image smoothing

• edge detection

• correction of geometricdistortions

An image processing system


Image filtering in spectrum domain

g(x,y) = IFFT H(u,v) Ff(x,y)

f(x,y) g(x,y)

Source

imageFFT2 IFFT2 Output

image

H(u,v)


22

0

0

),(

),( 0

),( 1),(

vuvuD

DvuD

DvuDvuH

+=

>

≤=

Ideal low-pass filter

D0


D0=10 D0=70

256x256

FFT


Image after ideal low-pass filtering, D0=70


Image after ideal low-pass filtering, D0=10


D0=70

D0=30

D0=10

Ideal low-pass filter - example

ringing


Butterworth filter

,...,nvu)v,u(D

]D/)v,u(D)[()v,u(H

n

21

121

1

22

20

=+=

−+=

,

D0

n- filter order


n = 1

Low-pass filtering

Butterworth filter


D0=70

D0=30

D0=10

n = 1

Low-pass Butterworth filter - examples

no ringing


22

0

0

1

0

vu)v,u(D

D)v,u(D

D)v,u(D)v,u(H

+=

>

≤=

Ideal high-pass filter

D0


D0=10 D0=70

Ideal high-pass filter - example


Image after ideal high-pass filtering, D0=10


,...,nvu)v,u(D

)]v,u(D/D)[()v,u(H

n

21

121

1

22

20

=+=

−+=

D0

n - filter order

High-pass Butterworth filter 2


D0=10 D0=70

n = 1

High-pass Butterworth filter - examples


n = 1

D0=10 D0=70

High-pass Butterworth filter - examples


Band-pass filter

IFFT

30

20

The filter selects narrow band

of image frequencies

FFT


Image filtering in spatial domain

Inputimage

2D convolutionOutput

image

h(x,y)

g(x,y) = h(x,y)**f(x,y) =

=IF H(u,v) F(u,v)

f(x,y) g(x,y)


0x0x

( )y,xf ( )y,xg

0y0y

The filtering result depends on the processed pixel

neighbourhood. This neighbourhood is call a

filtering window (or a filtering mask).

( )00 y,xfΩ



0x0x

( )lyy,kxxf ±± 00

( )y,xf ( )y,xg

0y0y



The result of filtering an image of size M x N for the filtering

window size (2k+1)x(2l +1) is an image of size (M -2k)x(N -2l)

k=2

l=2

( )y,xg

Boundary elements


f: array[0..M-1,0..N-1] of byte; //source image

g: array[0..M-1,0..N-1] of real; //filtered image

...

for x:=k to M-k-1 do

for y:=l to N-l-1 do

g(x,y):=P(f(x,y)); //perform local computations

end;

end;

( ) ( )1212 +×+ lk

Operator P [.] can be linear or nonlinear.

Correspondingly we define linear or nonlinear image filtering.

k=2

l=2

Filtering algorithm


Salt and pepper noise

0.05, i.e. 5% ofdistorted pixels

Source

Randomly selected pixels are assigned minimum ormaximum value randomly.

0.5, i.e. 20% ofdistorted pixels


σ2= 0.05 σ2=0.2Source

For each image pixel: ( ) ( ) ( )y,xny,xfy,xg +=

Gaussian noise


The Gauss distribution

0 100 200 300 400 500 600 700 800 900 1000-5

-4

-3

-2

-1

0

1

2

3

4

( ) ( )10,N,N =σµ

-5 -4 -3 -2 -1 0 1 2 3 40

5

10

15

20

25

30

35

40

( )( )

−−=

2

2

22

1

σ

µ

πσ

xexpxp


σ2= 0.05 σ2=0.2Source

( ) ( ) ( ) ( )y,xfy,xny,xfy,xg ⋅+=

Uniform noise

Multiplicative noise

For each image pixel:


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

0 200 400 600 800 1000-0.5

0

0.5

Uniform noise


Linear filtering

( ) ( )[ ]( )( )

( )∑Ω∈

Ω++

==

fy,x

y,xflk

y,xfPy,xg1212

1

Operator P [.] is a linear function of neighbour pixels.

Example – an averaging filter

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

( ) ×=25

1l,kh

Filtering mask

∑Why coefficients = 1?

___


1

1

1

1

1

1

1

1

1

3

18

1

15

9

71

2

22

5

3

18

1

15

9

71

2

22

5

1 *

1*

1*

1*

1*

1*

1*

1*

1*

3

18

1

15

146

71

2

22

5

Multiplication of filtercoefficient by thecorrecsponding image pixels

Filtering maskImage

Image filtering – calculations


f, g : array[0..N-1, 0..N-1] of byte;

size2 – half size of the mask

h : array[-size2..size2,-size2..size2] of integer;

...

for i:=1 to N-2 do for j:=1 to N-2 do

begin

g[i,j]:=0;

for k:=-size2 to size2 do for l:=-size2 to size2 do

g[i,j]:=g[i,j] + f[i+k,j+l] * h[i+k,j+l];

end;

...Range check g[i,j] !!!

Image filtering – the algorithm


( ) ( ) ( ) ( ) ( )lx,kxfl,khy,xfy,xhy,xg

k l

−−=∗= ∑ ∑∞

−∞=

∞

−∞=

Linear filtering of an image is given by performing 2D

convolution of an image with the filtering mask

Convolution in 2D

Filteringmask

Image


eg. for a 512x512 image 510*510*9 = 2 340 000 multiplications and additions need to be calculated

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )11111101111

10100101

11111101111

1

1

1

1

−−+−+−+−+

+−+++−+

++−−++−+++−−=

=−−= ∑∑−= −=

y,xf,hy,xf,hy,xf,h

y,xf,hy,xf,hy,xf,h

y,xf,hy,xf,hy,xf,h

ly,kxfl,khy,xg

k l

For a 3x3 filtering mask at each (x,y) image pixel

coordinate the following calculations take place:

Convolution in 2D


( ) ( ) ( )ly,kxfl,khy,xg

k l

−−= ∑∑−= −=

1

1

1

1

( )y,xf

( )00,h

( )11 −− ,h

( )11,h

( )11 −,h

( )11,h −

y

x

−

−

1

1

y

x

+

+

1

1

y

x

+

−

1

1

y

x

−

+

1

1

y

x

y

x

−

−

1

1

y

x

+

+

1

1

y

x

+

−

1

1

y

x

−

+

1

1

y

x

y

x

−

−

1

1

y

x

+

+

1

1

y

x

+

−

1

1

y

x

−

+

1

1

y

x

( )l,kh*

( )ly,kxf −−

Convolution in 2D


Averaging = low pass filter

=

111

111

111

9

11h

=

11111

11111

11111

11111

11111

25

12h

Can one use mask of even size ?


Frequency characteristics of low pass filters

for 5x5 mask for 3x3 mask


Low-pass filtering the image

Source image

5x5 mask

3x3 mask


The Gaussian filter

Gaussian filters on a

discrete grid

1

2

1

2

4

2

1

2

1

×16

1

Integer coefficients

require less computations

0

1

2

1

0

1

6

6

1

16

10

1

6

6

1

0

1

2

1

0

2

2

10

10

10×96

1

( )

+−=

2

22

2 22

1

σπσ

yxexpy,xp


Image filtering using the Gaussian filter

source image filtered image


Smoothing filters - comparison

Gaussian 5x5, σσσσ=1 Averaging 5x5Source


Image low-pass filters - examples

Image distorted by the

Gaussin noise N(0, 0.01)Low pass filter 3x3

Butterworth filter D0=50Gaussian filter 3x3

for grayscale

<0,1>


low-pass filter 5x5

Gaussian filter 5x5 Butterworth filter D0=30



Gaussian noise N(0, 0.01)




Gaussian noise N(0, 0.002)Low pass filter 3x3

Butterworth filter D0=50Gaussian filter 3x3


For ∆x=1:

( )xf

( )x

f

dx

dfx'f

∆

∆≅=

( ) ( ) ( )y,xfy,xfx

fx'f 1+−=

∆

∆≅

Corresponds to a filter mask:

-11

Similarily for y (∆y=1) :

-1

1

Image sharpening filters


Gradient images

-11

-1

1

First order gradient filters

An image of retinal arteries


0

-1

0

-1

4

-1

0

-1

0

How to work out this filter mask?

Hint: compute second order image differences along

columns and rows and add the results

Second order gradient filters


0

-1

0

-1

5

-1

0

-1

0

An original image was added to the second order gradient mask:

0

-1

0

-1

4

-1

0

-1

0

0

0

0

0

1

0

0

0

0

0

-1

0

-1

5

-1

0

-1

0

=+

Second order image sharpening filter


0

-1

0

-1

8

-1

0

-1

0

Please compute coefficients of this filter

The Laplace filter


0

-1

0

-1

9

-1

0

-1

0

-1

-1

-1

-1

8

-1

-1

-1

-1

0

0

0

0

1

0

0

0

0

-1

-1

-1

-1

9

-1

-1

-1

-1

=+

The Laplace filter

An original image was added to the second order gradient mask:


−

−−

−

=

010

151

010

3'

h

−−−

−−

−−−

=

111

191

111

4'

h

Comparison of sharpening filters


( )y,xf

( )y,xf smooth

( ) ( ) ( )y,xfy,xfy,xg smooth−=

( ) ( ) ( ) ( ) ( )y,xfy,xfy,xfy,xgy,xg smoothum −=+= 2

Unsharp masking filter


( )y,xf

( ) ( ) ( ) ( ) 11 <−−= ααα ,y,xfy,xfy,xg smoothum

1-α

α

smoothing

+( )y,xf ( )y,xgum

+

_

Parametr α decides about thesharpening strength.



α=2/3



Blurred image Sharpened image

%MATLABout_image = filter2(filter_mask, in_image);

High-pass filters


Filtering in spectrum domain

FFTIFFT

=

Low-pass filter spectrum

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

×100

1

Filter mask


( ) ( )( )µσ

σµ −

−+= y,xf

ny,xg

2

22

Noisy image(Gaussian noise)

Wiener filter Averaging filter

where:

n2 – noise variance

µ – mean of image pixels in a window

σ2 – image variance

For: ( ) ( )y,xfy,xg,n ≅≅ 0

( ) µσ ≅≅ y,xg,n

The Wiener adaptive filter


Nonlinear filters

NO

The filtered image is defined by a

non-linear function of the source image

Can we compute spectral

characteristics for nonlinear filters?

Because transfer characteristics of nonlinear

filters depend on image content itself!


Median filter (order statistic filter)

The median m of a set of values (e.g.

image pixels in the filtering mask) is such

that half the elements in the set are less

than m and other half are grater than m.

x(n)=1, 5, -7, 101, -25, 3, 0, 11, 7

Sorted sequence of elements:

xs(n)=-25, -7, 0, 1, 3, 5, 7, 11, 101

median


f(x,y) g(x,y)g(x,y)=medianf(x,y); (x,y)∈∈∈∈hf(x,y)

Median filtering the image

source image f output image g

h


1 2 3 4 5 6 7 8 (iterations)

44 06 06 06 06 06 06 0655 44 12 12 12 12 12 1212 55 44 18 18 18 18 1842 12 55 44 42 42 42 4294 42 18 55 44 44 44 4418 94 42 42 55 55 55 5506 18 94 67 67 67 67 6767 67 67 94 94 94 94 94

Unsorted sequence

Bubble-sort algorithm

© N. Wirth, „Algorithms+Data Structures=Programs”


a[k], k=1..N – unsorted sequencefor i:=2 to N dobegin

for j:=N downto i doif a[j-1]>a[j] then

beginx=a[j-1]; a[j-1]:=a[j]; a[j]:=x;end;

end;

Bubble-sort program


Demo – median filter

Source image distorted by

„salt and pepper noise”

Enhanced image using

the median filter (3x3)”

%MATLABout_image = medfilt2(in_image, [m n]);


Median filter

Median filter:

1. Excellent in reducing impulsive noise (of size

smaller than half size of the filtering mask)

2. Keeps sharpness of image edges (as

opposed to linear smoothing filters)

3. Values of the output image are equal or

smaller than the values of the input image

(no rescaling required)

4. Large computing cost involved


Median filter

median average

[1 x 3] 1/3*[1 1 1]


MATLAB Demo – median filter


Filters

Smooth

Sharpen

Convolve

Gaussian Blur

Median

Mean

Unsharp mask

ImageJ

in fourier domain in spatial domain linear filters non...

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