Antal Nagy Department of Image Processing and Computer Graphics University of Szeged 17th SSIP 2009, 2 - 11 July, Debrecen, Hungary1

Antal NagyDepartment of Image Processing

and Computer GraphicsUniversity of Szeged

17th SSIP 2009, 2 - 11 July, Debrecen, Hungary 1

Human perception Image degradation Convolution, Furier Transform Noise Image operations

◦ Frequency filters◦ Spatial filtering

Inverse filtering Wiener filtering


Aim◦ to improve the perception

of information images for human viewers

◦ to provide ‘better’ input for other automated image

processing techniques


No general theory for determining what is good image enhancement◦ If it looks good, it is good!?




http://www.youtube.com/watch?v=_d_l5nsnIvM


Focus◦ Noise reduction techniques

Quantitative measures can determine which techniques are most appropriate◦ How does it improve e.g.

the result of the next automated image processing step? E.g. image segmentation



Non-linear mapping◦ E.g., non-linear sensitivity, image of the straight

line is not straight e.t.c. Blurring

◦ Image of a point is blob Moving during the image acquisition Probabilistic noise



),(),(),(),(

),(),(),(),(

vuNvuFvuHvuG

yxyxfyxhyxg

Frequency domain

Spatial domain

◦ Multiplication point by point


)()()*( hFfFhfF


* =

=·

Multiplication

Convolution

Fourier transf.Inverse

Fourier transf.


dvduvyuxgvufyxgf

duuxgufxgf

,),(),)((

)( ))((

Definition


ationdifferenti '*'*)'*(

vitydistributi )*()*()(*

ityassociativ *)*()*(*

multipl.scalar ity withassociativ )(**)()*(

itycommutativ **

gfgfgf

hfgfhgf

hgfhgf

gafgfagfa

fggf


otherwise,0

,2/1if,1)(

xxg

2/1

2/1

)()(x

x

dfxgf

x

x

x

)(xf

Even functions that are not periodic can be expressed as the integrals of sines and/or cosines multiplied by a weighting function.

The formulation in this case is the Fourier transform.


∑∑ ==

Taught mathematics in Paris

Eventually traveled to Egypt with Napoleon to become the secretary of the Institute of Egypt

After fall of Napoleon worked at Bureau of Statistics

Elected to National Academy of Sciences in 1817



dueuFxf

dxexfuF

ixu

ixu

2

2

)()(

)()(

),(1 Lf

(invers transform)(invers transform)

(continous)(continous)

base-functionsbase-functions


dudvevuFyxf

dydxeyxfvuF

yvxui

yvxui

)(2

)(2

),(),(

),(),(

base-functionsbase-functions

(invers transform)(invers transform)


))(2cos(

Re )(2

vyuxi

e vyuxi

u=0, v=0 u=1, v=0 u=2, v=0u=-2, v=0 u=-1, v=0

u=0, v=1 u=1, v=1 u=2, v=1u=-2, v=1 u=-1, v=1

u=0, v=2 u=1, v=2 u=2, v=2u=-2, v=2 u=-1, v=2

u=0, v=-1 u=1, v=-1 u=2, v=-1u=-2, v=-1 u=-1, v=-1

u=0, v=-2 u=1, v=-2 u=2, v=-2u=-2, v=-2 u=-1, v=-2

u

v

wavelength:

22/1 vu

F(0,0) - value is by far the largest component of the image,

Other frequency components are usually much smaller,

The magnitude of F(X,Y) decreases quickly◦ Instead of displaying the |F(u,v)| we display

log( 1 + |F(u,v)| ) real function usually



x

y

v

u

),(1log vuF),( yxf

The 2D Fourier transform can be separated

The edges on the image appears as point series in perpendicular direction in Fourier transform of the image and vice versa.



Image-spaceImage-space

Frequency spaceFrequency spaceooriginal rotation linearity riginal rotation linearity shiftshift scale scale

Noise unknown subtraction not possible Periodic noise

◦ N(u,v) can be estimated from G(u,v)


),(),(),(

and

),(),(),(

vuNvuFvuG

yxyxfyxg


Gaussian◦ In an image due to

factors Electronic circuit

noise Sensor noise due to

poor illumination High temperature

Rayleigh◦ Range imaging

Exponential and gamma◦ Laser imaging

Impulse◦ Faulty switching

Uniform density◦ Practical situations


22 2/)(

2

1)(

noiseGaussian

zzezp

azfor 0

for )(2

)(

noiseRayleigh

/)( 2

azeazbzp

baz

4

)4(

and

4/

2

b

baz

0zfor 0

0for )!1(

ap(z)

noise (gamma) Erlang1b

zeb

z azb

22

and

a

b

a

bz

0a where

0zfor 0

0for )(

noise lExponentia

zae

zpaz

22 1

and

1

a

az


otherwise 0

if 1

)(

Noise Uniform

bzaabzp

12

and2

22 ab

baz

otherwise 0

for

for

)(

noise peper)-and-(salt Impulse

bzP

azP

zp b

a




Electrical and electromechanical interference

Spatial dependent noise Can be reduced via

frequency domain filtering ◦ Pair of impulses



T: A=[a(i,j)] → B=[b(i,j)]

b(i,j)=T{a(i,j), S(i,j), i, j}

intensity enviroment position

Global: b(i,j)=T{A} (S(i,j)=A) (e.g. Fourier-transformation)

Local: T{a(i,j), S} given size of S and independent from the position (e.g. convolution with a mask)

Local, adaptive: T{a(i,j), S(i,j), i, j} the size of S(i,j) is independent from the size of image (e.g. adaptive thresholding)

Point operation: T{a(i,j)} (e.g. gamma-correction, histogram-equalization)



D0: cutoff frequency

All frequencies less than D0 will be passed,

Other frequencies will be filtered out.Bluring and ringing propertiesScope: noise filtering

otherwise,0

if,1),(

20

22 DYXYXH ILPF


F

F-1.

Input image

Frequency-mask


Original 5

15 30

80 230

Cutoff frequencies


n: order of the filter

Properties: Smooth transition in blurring No ring effect (continouos filter) Smoothed edges

nBLPF

DYX

YXH 2

0

22

1

1),(


Original 5

15 30

80 230

Cutoff frequenciesCutoff frequencies


022 2/)(),( Dvu

GLPF evuH


Original 5

15 30

80 230

Cutoff frequenciesCutoff frequencies


),(1),( vuLPFvuHPF


Bandreject and Bandpass Filters Notch Filters

◦ Rejects or passes frequencies in a predefined neighborhood about the frequency rectangle

◦ Zero-phase-shift filters Symmetric about the origin

(u0,v0) (-u0,-v0)

◦ Product of highpass filters whose centers have been translated to the centers of the notches.


Q

kkkNR vuHvuHvuH

1

),(),(),(

),(1),( vuHvuH NRNP


noisyimage

frequencymask

frequencyimage

filteredimage

11

00


Mean Filter


where g input image, S(x,y) neighborhood of (s,t) point,

mn number of pixels in neighborhood.

3x3 neighborhood

1

1

1

1

),(9

1),(ˆ

u v

vyuxgyxf

xySts

tsgmn

yxf),(

),(1

),(ˆ


111

111

111

9

1

:where

),,)((

),(),(

),(9

1),(ˆ

1

1

1

1

1

1

1

1

h

jihf

vuhvyuxg

vyuxgyxf

u v

u v

AvAveerraagingging ◦ same weight for every pixels in neighborhood,same weight for every pixels in neighborhood,

Weighted averageWeighted average ◦ weights for pixels in the neighborhood (generally decreasing with weights for pixels in the neighborhood (generally decreasing with

the distance).the distance).

The sum of the Noise Filtering/smoothing The sum of the Noise Filtering/smoothing masks elements is 1! masks elements is 1!


121

242

121

16

1

111

111

111

9

1

Smooth when the difference between the intensity value of the given pixel and the mean of the neighborhood is larger than threshold value defined in advance.


otherwise),,(

,),(),( if),,(),('

jif

Tjifjigjigjig

Arithmetic mean filter

Geometric mean filter◦ Lose less image details

Harmonic mean filter◦ Works well for

salt noise, Gaussian◦ Fails for pepper noise

Contraharmonic mean filter◦ Q>0 pepper, Q<0 salt


xySts

tsgmn

yxf),(

),(1

),(ˆ

mn

Sts xy

tsgyxf

1

),(

),(),(ˆ

xySts tsg

mnyxf

),( ),(1

),(ˆ

xy

xy

Sts

Q

Sts

Q

tsg

tsg

yxf

),(

),(

1

),(

),(

),(ˆ




Median filter

Max and min filters


),(median),(ˆ),(

tsgyxfxySts

),(min),(ˆ

),(max),(ˆ

),(

),(

tsgyxf

tsgyxf

xy

xy

Sts

Sts

50%

0%

100%



Behavior changes based on statistical characteristic in the filter region◦ Improved filtering power◦ Increase in filter complexity◦ Noise only!


Adaptive, Local noise reduction filter◦ Mean◦ Variance

Local region Sxy

◦ g(x,y) intensity value◦ the variance of the noise corrupting f(x,y) to

form g(x,y) ◦ mL local mean◦ local variance


2

2L

Adaptive, Local noise reduction filter1. If is zero, the filter should return the value of

g(x,y) Zero noise case

2. If the local variance is high relative to the filter should return a value close to g(x,y)

Edges should be preserved3. If the variances are equal the filter should

return the arithmetic mean value Local noise averaging


2

2

Simplest approach to restoration is direct inverse filtering


experience:H: quickly decreasing function,N: not (so quickly) decreasinglet us cut the high frequencies

),(

),(),(ˆ

vuH

vuGvuF

),(

),(),(

),(

),(),(

),(

),(),(ˆ

vuH

vuNvuF

vuH

vuNvuF

vuH

vuGvuF

No additive noise

If we only know the degradation function◦ Can not recover the undegraded image

H(u,v ) is not known Get around the zero or small value problem

◦ To limit the filter frequencies to values near the origin H(0,0) is usually the highest value of H(u,v) in the

frequency domain


Inverse filtering makes no explicit provision for handling noise

Approach incorporates◦ Degradation function◦ Statistical characteristics of noise


Method◦ Considering images and noise as random variable◦ Objective is to find an estimate of the

uncorrupted image f such that the mean square error between them is minimized.


f̂

argument theof value

expected theis where

ˆ 22

E

ffEe

17th SSIP 2009, 2 - 11 July, Debrecen, Hungary

66

spectrum

Wiener filter

input

result

original

spectrum of the result

Edge preserving filters◦ E.g. anisotropic diffusion

Wavelet denoising Point operations

◦ E.g. gamma correction, histogram operations Iterative filtering E.t.c.


We always should consider some kind of noise model◦ Even when working on phantom data

Should do in automatic way◦ Have to chose carefully the method

Depends on the given task◦ Determining the parameters

What we gain?◦ Less problem afterwards◦ Better final result

Even no other technique will be applied


Digital Image Processing◦ Gonzalez and Woods◦ www.ImageProcessingPlace.com

Course on Image Processing at University of Szeged◦ Attila Kuba, Kálmán Palágyi

Image Restoration presentation◦ Attila Kuba◦ SSIP 2006


Documents

Antal Nagy Department of Image Processing and Computer Graphics University of Szeged 17th SSIP 2009, 2 - 11 July, Debrecen, Hungary1