noise and restoration of images · outline 1 noise models 2 restoration from noise degradation 3...

Noise and Restoration of Images

Dr. Praveen Sankaran

Department of ECE

NIT Calicut

February 24, 2013

Dr. Praveen Sankaran (Department of ECE NIT Calicut )DIP Winter 2013 February 24, 2013 1 / 35

Outline

1 Noise Models

2 Restoration from Noise Degradation

3 Estimation of Degradation

4 Restoration from DegradationInverse and Wiener FilteringConstrained Least Squares Filtering


Noise Models

Degradation Model

All digital images have some amount of noise and/or some form ofdegradation in them.

Some have more due to varying sources

Acquisition

Environmental conditions.

quality of sensing elements - CCD noise.

TransmissionFormation

Quantization noise

Blur


Noise Models

Degradation Model

g [x ,y ] = f [x ,y ]∗h [s, t]+η [x ,y ] (1)

η → uncorrelated noise (no relation between noise and pixel value ofthe image).

think of an example where both could be correlated?


Noise Models

Noise Models


Noise Models

Noise Models - Explained


Noise Models

In Color - Gaussian


Noise Models

In Color - Uniform


Noise Models

Periodic Noise


Noise Models

Identifying System Noise

Imaging system available

Capture a set of images of �at environments - image a solid gray boardthat is illuminated uniformly.

Imaging system NOT available, we have only the images

Estimate parameters from small patches of roughly constant background.We already know how to calculate mean and variance from a histogram.


Restoration from Noise Degradation

Model - Simpli�ed

g [x ,y ] = f [x ,y ]+η [x ,y ] (2)

G [u,v ] = F [u,v ]+N [u,v ] (3)

Noise term is unknown → No blind subtraction possible.

Spatial �ltering is used mostly.



Typical Filters

We have come across a lot of these already. Filter of size m×nArithmetic mean: Noise reduced as a result of blurring.

Geometric mean:

f [x ,y ] =

{∏s,t

f [s, t]

}1/mn

(4)

Harmonic mean: works for salt, fails for pepper

f [x ,y ] =mn

∑s,t

1

g [s, t]

(5)

Contra-harmonic mean: works well to remove salt&pepper

f [x ,y ] =

∑s,t

g [s, t]Q+1

∑s,t

g [s, t]Q(6)



Illustration



Order Static Filters

1 Median �lter:f [x ,y ] =medians,t∈Sxy {f [s, t]} (7)

2 MinMax �lters:

3 Mid point �lter:

f [x ,y ] =1

2

{maxs,t∈Sxy {f [s, t]}+mins,t∈Sxy {f [s, t]}

}(8)



Adaptive Median Filter

Key idea → variable window area Sxy to work with.

So how do we decide what area to work with? → provide conditions.

Assumptions

zmin = minimum intensity,zmax = maximum intensity,zmed = medianzxy = value at point [x ,y ]Smax = maximum threshold of window size.



Adaptive - Conditions

Stage 1

A1 = zmed − zmin

A2 = zmed − zmax

If A1 > 0 and A2 < 0, goto Stage 2.

Else increase the windowsize.

If window size ≤ Smax ,repeat Stage 1.

Else output zmed .

Stage 2

B1 = zxy − zmin

B2 = zxy − zmax

If B1 > 0 and B2 < 0,output zxy .

Else, output zxy .



Periodic Noise - Frequency Filtering

Idea → selectively �lter out frequencies relating to noise. Can haveband-reject, band-pass or notch �lters.

Example: band-reject


Illustration

Estimation of Degradation


1 Observation

2 Experimentation

3 Mathematical modeling



Degradation Model

g [x ,y ] = H [f [x ,y ]] = H [x ,y ]∗F [x ,y ] (9)

G [u,v ] = H [u,v ]F [u,v ] (10)

Let's assume for now,η [x ,y ] = 0 (11)


Modeling Atmospheric Turbulence

H [u,v ] = e−k[u2+v2]

5/6

(12)

Restoration from Degradation Inverse and Wiener Filtering

Outline

1 Noise Models






Inverse Filtering

F [u,v ] =G [u,v ]

H [u,v ](13)

Issue

Presence of noise in the image.

F [u,v ] = F [u,v ]+N [u,v ]

H [u,v ](14)

The degradation function may have zero or small values - a realpossibility as we move away from the center point of the DFT image.

N [u,v ]

H [u,v ]⇒ large!

Have to cut-o� small values, so have to �nd an optimal cut-o�frequency in the DFT image.



Inverse Filtering Problem Illustration



Wiener Filtering - Variables

Idea → minimize mean square error between the uncorrupted image andthe corrupted image.

e2 = E

{(f − f

)2}(15)

H [u,v ] = degradation function,

H∗ [u,v ] = complex conjugate of H [u,v ],

Sη [u,v ] = |N [u,v ]|2 = power spectrum of the noise, (auto correlationof noise)

Sf [u,v ] = |F [u,v ]|2 = power spectrum of the undegraded image.(auto correlation of the image)



Wiener Filter Equation

F [u,v ] =

[H∗ [u,v ]Sf [u,v ]

Sf [u,v ] |H [u,v ]|2+Sη [u,v ]

]G [u,v ] (16)

=

[1

H [u,v ]

|H [u,v ]|2

|H [u,v ]|2+ Sη [u,v ]/Sf [u,v ]

]G [u,v ] (17)

≈

[1

H [u,v ]

|H [u,v ]|2

|H [u,v ]|2+K

]G [u,v ] (18)

The last equation approximates the wiener �lter equation since inmost cases we do not know accurately the power spectrum of bothnoise and the un-degraded image.



Comparative Illustration between Inverse Filtering and

Wiener Filtering



Comparative Illustration 2


Restoration from Degradation Constrained Least Squares Filtering

Outline

1 Noise Models






Variables - Introduction

g =Hf+η (19)

g, f, η are vectorized form of the the 2D structure, of size MN×1.

H⇒MN×MN.

Objective is to minimize,

Criterion function,

C =M−1

∑x=0

N−1

∑y=0

[∇2f [x ,y ]

]2(20)

Laplacian again. So sort of reduce sharpness.With constraint: ∥∥∥g− Hf∥∥∥2 = ‖η‖2 (21)



Final Frequency Domain Form

F [u,v ] =

[H∗ [u,v ]

|H [u,v ]|2+ γ |P [u,v ]|2

]G [u,v ] (22)

P [u,v ] = ℑ

p [x ,y ] =

0 −1 0−1 4 −10 −1 0

(23)

Okay! So we end up �nding an approximation to γ here instead of Kin Wiener �ltering.

γ is a scalar value,

K was the ratio of two unknown frequency functions.

Adjusting γ is easier and better.

Note that we did not really make use of the constraint function tillnow. That can be made use of, if in need of optimal results.



Questions to solve

1-9 (you may write simple matlab codes and observe the e�ect foreach).

10, 11

18 (We havent gone through this, you may solve this out of interest).

22,23



Reference

Lecture Notes: Reduction of Uncorrelated Noise: Richard Alan PetersII

http://www.owlnet.rice.edu/~elec539/Projects99/BACH/proj2/inverse.html


noise and restoration of images · outline 1 noise models 2 restoration from noise degradation 3...

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