Ip Image Restoration

8/3/2019 Ip Image Restoration

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Module IV (13 hours)

Image restoration - image observation

models - inverse filtering - wiener filteringImage compression - pixel coding -predictive coding - transform coding -basic ideas



IMAGE RESTORATION The process of recovering an image which has beendegraded by using a prior knowledge of degradation

phenomenon. Degradation comes in many forms such as motion blur or

camera misfocus Image degradation model is important in image

restoration. We have to find what is the image degradation model and

once it is found then we can apply inverse process torecover the image.

in image enhancement we donot use any degradationmodel



What is degradation model?

Degradation function H filters

Noise termn(x,y)

f(x,y)

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



f(x,y) is a 2 D function and degraded bydegradation function H

g(x,y) is the degraded image From g(x,y) we have to get f(x,y) using

some image restoration function So for recovering f(x,y) , perform filtering

operation and this filter is derived usingdegradation function H and o/p of filter isf^(x,y)



Reconstruced image is f^(x,y) bcozrestoring of original image is difficult in

many cases Using goodness criteria we can getapproximate image of original

Blocks from f(x,y) to g(x,y) -----process of degradation From Block of filter is restoration process



Important task in restoration is to estimatethe degradadtion model which has

degraded the original image There are various techniques to estimatethe degradation function



Conversion from f(x,y) to g(x,y)

In spatial domaing(x,y) = h(x,y)*f(x,y) + (x,y)

In frequency domainG(u,v)= H(u,v)F(u,v) + N(u,v)

G(u,v) FT of g(x,y)

H(u,v) FT of h(x,y)N(u,v) FT of (x,y)



Convolution in spatial domain ismultiplication in frequency domain.

Before we proceed we will look into somedefinitions used in image restorationg(x,y) = H[f(x,y)] + (x,y)

H ---- degradation operator



What is linearity?H is linear operator.If f 1(x,y) & f 2(x,y) are 2 functions

We say H(k1f 1(x,y) +k2f 2(x,y)) =k1 H(f 1(x,y)) + k2 H (f 2(x,y))If this relation is true then H is a linear

operator If k1,k2=1 then the above equation leads

H(f 1(x,y) +f 2(x,y)) = H (f 1(x,y)) + H (f 2(x,y))

This is called additive property



If f2(x,y) =0

Then the above equation becomesH(k1f 1(x,y)) =k1 H (f 1(x,y))This property is homogenety property

These are properties of linear system &system is called position invariant if itsatisfies the following property

If an operator is having the input-outputrelationship g(x,y)=H(f(x,y)) and it is saidto be position invariant if



H(f(x-,y-)) = g(x-,y-) This says that the response at any point in

the image will depend only on the value of the input at that point not on its position . What is degradation model incase of

continuous function? We will make use of the old mathematicfn.



(x,y) = 1 x=0 & y=0

0 otherwise

We use shifted version as(x-x0,y-y0) = 1 if x=x0

0 otherwiseIf we have image f(x,y)

-f(x,y) (x-x0,y-y0) dxdy = f(x0,y0)this says that if I multiply a 2D function with delta

function and integrate the product, the result is f (x,y)at position (x0,y0) , so slightly modify the expression



-f(,) (x-,y-) dd = f(x,y)So we get equivalent expression f(x,y) in terms

of &For time being consider noise term =0Therefore

g(x,y) = H[f(x,y)]

= H[ -f(,) (x-,y-) dd]



Now apply linearity & additive property onthis we get

g(x,y)== -

H[f(,) (x-,y-) dd] As f(,) are independent variables of x andy

Therefore same expression can berewritten asg(x,y)== -f(,) H[ (x-,y-) ] dd



H[ (x-,y-) ] can be written as h(x,,y,)& this is known as the impulse response of H

Using this impulse response g(x,y) can bewritten asg(x,y)== -f(,) h(x,,y,)] dd

this says that response of H is known,response to any input f(,) can becalculated using the above equation



Impulse response characterises the system In addition to this if H is position invariant then

obviously

H[ (x-,y-) ] = h (x-,y-)Using this position inavariant property we can writedegraded image g(x,y) asg(x,y)= -f(,) h(x-,y-)] dd

This expression is nothing but convolutionoperation of f(x,y) & h(x,y)



Earlier we had taken noise term is 0

But now consider noise termdegradation model if H is positioninvariant and linear isg(x,y)=

-f(,) h(x-,y-)] dd + (x,y)

the above equation can be written asg(x,y) = h(x,y) *f(x,y) + (x,y)

this in frequency domain isG(u,v) = H(u.v)F(u,v) + N(u,v)



How to do restoration?

If operation is done in frequency domain or spatial domain, knowledge of degradation

function is essential Th simplest approach to restoration isdirect inverse filtering



Objective is to estimate original image function from adegraded image g and some knowledge about H and n

We know

g(x,y) = h(x,y)*f(x,y) + (x,y)G(u,v)= H(u,v)F(u,v) + N(u,v)in matrix form the above eqn. g=Hf+n

Where g , H and F are column vectorsn=g-Hf

In the absence of knowledge of nf^ is found such that



Hf^ §g

We want to find f^ such thatn2 = g - Hf^2 is min

by def

(i) n2 = nTntherefore

(ii) g - Hf^2 = (g-Hf^ )T (g-Hf^ )

the (i) and (ii) are squared norms of n andg - Hf^



Equalent view of this problem can be done byminimizing the criterion fn

J (f ̂ ) = g - Hf^2

for minimizing the above equation differentiate Jw.r.t f ^ and set the result =0 J (f ̂ ) / (f ̂ ) =0

= -2HT(g - Hf^) = -2HTg +2HTH f^thereforef^ =HTg (HTH) -1



Assume H is square matrix and H -1 existsf^ = H-1(HT) -1 HTg

f^=H-1g ------- unconstrainedrestorationConstrained restoration:

Here the minimizing function takes theform Qf^2where Q is linear operator onf



Here the constraint is expressed in theform

(g - Hf^2 -n2 ) and appendingQf^2 to the function

We seek an f that minimizes the criterionfunction

J (f ^) = Qf^2 + (g - Hf^2 -n2 ) is constant called the langrangesmultiplier



differentiate J w.r.t f ^ and set the result =0 J (f ̂ ) / (f ̂ ) =0 =2 QTQf^ - 2 HT(g-Hf^)

we getF=(HTH+QTQ)-1HTg where = 1/



Inverse filtering

Concept is simpleOur expression isG(u,v)= H(u,v)F(u,v)

H(u,v) is degradation function in frequencydomain

now Bcoz H(u,v) F(u,v) is point by pointmultiplication, from this expression we can get

F(u,v) = G(u,v)/H(u,v)and since H(u,v) is estimated value and it will

never be exact



Therefore F(u,v) is appropriate imageF(u,v) = F^(u,v)

If we consider noise termG(u,v)= H(u,v)F(u,v)+N(u,v)F^(u,v) =G(u,v)/H(u,v)

=H(u,v)F(u,v)/H(u,v) +N(u,v)/H(u,v)= F(u,v) + N(u,v)/H(u,v)



From the above expression we cannot

recover the original image even if we knowdegradation function bcoz N(u,v) is arandom function whose FT is not known.

If degradation has 0 or small values thethe ratio N(u,v)/H(u,v) will dominate theF^(u,v) , so noise term dominates.

To overcome this problem limit the filter frequencies to values near origin, or instead of taking entire frequency planewe have to restrict our frequency



Wiener filter

This is min mean suqare error approach Uses estimated value of H

Tries to restore by minimizing an error If original iamge is f and reconstructed

image is f^

Winier filter tries to minimize error functionwhich is given by



E[(f-f^)2] and this is error value e wenier filtering minimizes this expectation

value. Here assumption is iamge and noise

intensity are uncorrelated & using thisassumption wenier filter works

It can be shown that if e is min,corresponding F(u,v) in freq domain isgiven by



F^(u,v) = H*(u,v)Sf (u,v)Sf(u,v)H(u,v) 2 +S(u,v)

G(u,v)



H* -- it is complex conjugate of H(u,v) Sf(u,v) --- power spectrum of original

image S(u,v) --- nose power spectrum If I simplify the above expression we get

F^(u,v) = [1/H(u,v) . |H(u,v)|2

/ (|H(u,v)|2

+S(u,v)/Sf (u,v))] G(u,v)



This is the Ft of reconstructed image whenusing wiener filter

If image has no noiseS(u,v) = 0Therefore wiener filter will become identical

to inverse filtering

if noise term is present tehn weiner filter inverse filetring



Weiner filter considers ratio of power spectrum of Sand Sf

If noise present is white noise for whichSn is constant but it is not possible to findwhat is power spectrum of original imagefor that we take

S(u,v)/Sf (u,v) = constant K



F^(u,v) = [1/H(u,v) . |H(u,v)|2/ (|H(u,v)|2 +K)]G(u,v)

K is to adjusted manually for best visualization or

appearance Therefore weiner filter is slightly better than

inverse filtering Advantage

need not consider to what extend of frequencycomponents are to be used for reconstruction



Disadvantage:manually adjust K and this is not a

justified approach in all cases as K variesfor different imageso go in for filtering operation called

constrained least square filter



Pseudo inverse filtering

Stabilized version of inverse filter Defined as

F^(u,v) = 1/H(u,v) H00 H=0



Consider the result of weiner filter F^(u,v) = H*(u,v)Sf (u,v)

Sf (u,v)H(u,v) 2 +S(u,v)



This can be written as|H|Sf (u,v) / (|H|2 Sf (u,v) +Sn(u,v))

weiner smoothing filter In the absence of blur H=1Therefore above equation becomes

Sf (u,v) / ( Sf (u,v) +Sn(u,v)) ± this issmoothing filter



In the absence of noise|H|Sf (u,v)

Sf (u,v)|H|2

= 1/H which is inverse filter On the other hand if S 0 we getLim S 0 G = 1/H(u,v) H0

0 H=0 this is pseudo inversefilter

so we can say smoothing and pseudoinverse filters arespecial cases of weiner filter



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Ip Image Restoration