non linear digital filters

7/28/2019 non linear digital filters

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Poularikas A. D. Nonlinear Digital Filtering

The Handbook of Formulas and Tables for Signal Processing.

Ed. Alexander D. PoularikasBoca Raton: CRC Press LLC,1999

1999 by CRC Press LLC


2/13

1999 by C RC Press LLC

41N onlinear D igitalFiltering41.1 Definitions

41.2 Mean Filter

41.3 Median Filter41.4 Trimmed Mean Filters

41.5 L-Filters (Order Statistic Filters)

41.6 Weighted Median Filters

41.7 Ranked-Order Filter

41.8 Separable Two-Dimensional Median Filters

41.9 M-Filters

41.10 R-Filters

References

41.1 Definitions

41.1.1 One- and Two-Dimensional Signals

f(t)=continuous, f(n)=discrete; f(x,y)=continuous, f(n,m)=discrete

41.1.2 Filter Length

N=2k+1, N= odd number, range of indices: i k,i k+ 1,,i + k

41.1.3 Data Length

x={x(1),,x(Nd)}

41.2 Mean Filter

41.2.1 Mean Filter

Example

x={4,14,18,40,10}. For k=1, N=21+1=3 and y(1)=4,

y ik

x i j

j k

k

( ) ( )=+

+=12 1


3/13 1999 by C RC Press LLC

Note: 1. Mean filtering is a linear operation 2. Mean filter attenuates noise

41.2.2 Mean Filter Algorithms

Input: data x={x(1),,x(Nd)} Inputs: No of rows NR No of columns NC (image)

moving window wN, N=2k+1 moving window wN=N=2k+1

Output: y={y(1),,y(N)} Output: NRNC (image)

for i = 1 to Nd for i=1 to NRset number at i for j=1 to NCSum=0 set window wN at (i,j)

for every element x(m) of the Sum=0

image inside the window wN for every element x(m,n) of the

Sum=Sum+x(m) image inside wNend Sum=Sum+x(m,n)

y(i)=Sum/N end

end y(i,j)=Sum/N

end

end

41.3 Median Filter

41.3.1 Median Filter

y(i)=MED{x(i2), x(i1), x(i), x(i+1), x(i+2)} i=3,4,,18 for data x={x(1),x(2),,x(20)}

Example

Let x={1,5,3,2,10,2} and k=1 then y(1)=1,y(2)=MED{1,5,3}=MED{1,3,5}=3,

y ( 3 ) = M E D { 5 , 3 , 2} = M E D { 2 , 3 , 5} = 3 ,

y(4)=MED{3,2,10}=MED{2,3,10}=3,y(4)=MED{3,2,10}=MED{2,3,10}=3,y(5)=MED{2,10,2}=MED

{2,2,10}=2,y(6)=2. Hence, y={1,3,3,3,2,2}

Note: a) the filter is nonlinear, b) attenuates noise to some degree c) does not attenuate noise well in

the case of noisy ramp signal d) noisy step signal is rounded at the step e) eliminates impulse noise

f) if impulse noise is close to an edge it may be removed but the edge moves toward the impulse (edge

jitter).

41.3.2 Median Filter Algorithms

Inputs: data x={x(1),,x(Nd)} Inputs: No of rows NR No of columns NCmoving window wN, N=2k+1 moving window wN, N=2k+1

Output: data y={y(1),,y(Nd)} Output: NRNC (image)

for i=1 to Nd for i=1 to NRplace window wN at i for j=1 to NCstore the data inside the window wN in place window wN at (i,j)

x={x(1),,x(N)} store image values inside wN

find the median ofx, MED{x} x={x(1),,x(N)}y(i)=MED{x}; output find median of x, MED{x}

end y(i,j)=MED{x}; output

end

end

y x j y y y

j

( ) ( ) ( ) , ( ) ( / )( ) , ( ) , ( )21

32

1

34 14 18 12 3 1 3 14 18 40 24 4 68 5 10

1

1

= + = + + = = + + = = ==



41.4 Trimmed Mean Filters

41.4.1 Motivation for Trimmed Mean Filters

The idea of trimmed filters is to reject the most probable outliers some of the very small and the

very large values.

41.4.2 (r,s)-Fold Trimmed Mean Filters

Short the samples in the window and omit r+s samples x(1),,x(r) and x(Ns+1),x(Ns+2)

41.4.2.1 (r,s)-Fold Trimmed Mean Filters Algorithm

Inputs: data x={x(1),,x(Nd)} Inputs: No of rows NR No of columns NC (image)

moving window wN, N=2k+1, r and s moving window wN, N=2k+1, integer numbers

integer numbers r,s

Output: y(i) Output: NRNC (image)

for i=1 to Nd for i=1 to NRplace window wN at i for j=1 to NCstore the values inside the window into place window wN at (i,j)

x={x(1),x(2),,x(N)} store data inside window into

sort x to x= x={x(1),,x(N)}

Sum=0 sort x intox =

for m=r+1 to Ns Sum=0Sum=Sum+ for m=r+s to Ns

end Sum=Sum+

y(i)=Sum/(Nrs); output end

end y(i,j)=Sum/(Nrs); output

end end

Note:x(i)x(j) for all i



41.4.4 -Trimmed Mean Filters

If r = s then =j/N, 0 jN/2 is an integer, and the formula is

41.4.5 -WinMean Filters (see 41.4.4)

Note: a) (0,0)-fold trimmed, (0,0)-fold Winsorized, 0-trimmed, and 0-Winsorized mean filters are

the same as the mean filter of the same window size; b) (k,k)-fold trimmed, (k,k)-fold Winsorized, 0.5-

trimmed and 0.5-Winsorized mean filters are the same as the median filter of the same window size

Example

x=(4,14,18,40,10), N=5, x( )=(4,10,14,18,40) a) (2,1)-fold trimmed mean filter (14+18)/2=16, b) (1,2)-

Winsorized mean filter (210+314)/5=12.4, c) 0.2-trimmed mean (14+18+40)/3=14, d) 0.2-Winsorized

mean filter (210+14+218)/5=14

41.4.6 Modified Trimmed Mean Filters

A real valued constant q is first fixed and then all the samples in the range [x(k+1) q, x(k+1) + q] are

averaged, and the results constitute the output

Example

x={4,14,18,40,10}, x( )={4,10,14,18,40}, a) for 0 q < 4 med is 14, b) for 4 q < 10, (10 + 14 +

18)/3=14, c) for 10 q < 26, (4 + 10 + 14 + 18)/4=11.5, d) mean (4 + 10 + 14 + 18 + 40)/5=17.2

41.4.6.1 Modified Trimmed Mean Algorithm


moving window wN, N=2k+1 moving window wN, N=2k+1

positive real number q positive real number q

Output: y={y(1),,(Nd)} Output: NRNC (image)

for i=1 to Nd for i=1 to NRplace window wN at i for j=1 to NCstore the values in the window in place window wN at (i,j)

x={x(1),,x(N)} store the values inside the window wN in

find the median x(k+1)

ofx x={x(1),,x(N)}

Sum1=0 find the median x(k+1) ofx

Sum2=0 Sum1=0

for m=1 to N Sum2=0

if abs(x(m)x(k+1))q then for m=1 to N

= = = +

TrMean( ( )x { ( ), , )};( )x x N N N x

d i

i N

N N

11

21

K

= = + +

+

= +

WinMean( ( )x { ( ), , )}; ( ) ( ) ( )x x N N N x x N xd N ii N

N N

N N1

11

1

K

MTM x x N

a x i

a

ax i x k q

d

i

i

N

i

i

N i( ( )

if

otherwise

x = = = +

=

=

{ ( ), , )}

( )

,( ) ( )

11 1

0

1

1

K



Sum1=Sum1+x(m) if abs(x(m)x(k+1))q then

Sum2=Sum2+1 Sum1=Sum1+x(m)

end Sum2=Sum2+1

y(i)=Sum1/Sum2; output end

end y(i,j)=Sum1/Sum2; output

end end

end

41.4.7 Double Window Modified Trimmed Mean Filters

The idea behind this filter is to suppress additive Gaussian noise with the large window and preserve

the details by rejecting pixels that are far away from the median of the smaller window.

41.4.7.1 Double Window Modified Trimmed Mean Filters Algorithm


moving window wN, N=2k+1 moving window wN, N=2k+1moving window w1, M>N moving window w1, M>N

positive real number q positive real number q

Output: y={y(1),,y(Nd)} Output: NRNC (image)

for i=1 to Nd for i=1 to NRplace window wN and w1 at i for j=1 to NCstore values inside wN in place windows wN and w1 at (i,j)

x={x(1),,x(N)} store the values inside wN in x={x(1),,x(N)}

find the median x(k+1) ofx find the median x(k+1) ofx

sum1=0, sum2=0 sum1=0, sum2=0

store values inside w1

in store values inside w1

in

z={x(1),,x(M)} z={z(1),,z(M)}

for m=1 to M for m=1 to M

if abs(x(m)x(k+1))q then if abs(x(m)x(k+1))q then

sum1=sum1+x(m) sum1=sum1+x(m)

sum2=sum2+1 sum2=sum2+1

end end

y(i)=sum1/sum2 y(i,j)=sum1/sum2

end end

end end

41.4.8 K-Nearest Neighbor Filter

The output is given by the mean of K, 1KN, samples whose values are closest to the value of the

central value x* inside the filter window.

Example

x={4,14,18,40,10}. Center sample is x*=18 a) for K=1, 18 (itself) is the closest sample to 18 and output

is 18/1=18 b) for K=2, 18 and 14 must be used and the output is (18+14)/2=16 c) for K=3, 18, 14 and

10 are closest and output is (18+14+10)/3=14.

41.4.8.1 K-Nearest Neighbor Filter Algorithm



integer number K, 1KN integer number K, 1KN




for i=1 to Nd for i=1 to NRplace window wN at i for j=1 to NCstore values inside the window wN place windows wN at (i,j)

and their differences from x* in store values inside wN

and their differences

z = from x* (pixel (i,j)) in

sort z with respect to and store in z =

t= sort z with respect to abs(x(i)x*), and store

sum=0 the results in

for m=1 to K t=

sum=sum+x(m) sum=0

end for m=1 to K

y(i)=sum/K sum=sum+x(m)

end endend y(i,j)=sum/K

end

end

41.4.8.2 Modified K-Nearest Neighbor Filter

If we change the rule for ai in 41.4.6 to

If q is chosen to be twice the standard deviation of the noise, the filter is known as the sigma filter.

41.5 L-Filters (Order Statistic Filters)

41.5.1 Purpose of the Estimator

L-filters are running estimators making a compromise between a pure nonlinear operation (ordering)

and pure operation (weighting).

41.5.2 L-Filters

41.5.3 L-Filters Algorithms



weight vector a={a1,,aN} weight vector a={a1,,aN}Output: y={y(1),,y(Nd)} Output: NRNC (image)

for i=1 to Nd for i=1 to NR

{( ( ), ( ) ), ,( ( ), ( ) )}* *x x x x N x N x1 1 L

x i x( ) *{( ( ), ( ) ), ,( ( ), ( ) )}* *x x x x N x N x1 1 L

{( , ), ,( , )}

( ) ( )

*

( ) ( )

*x x x x x xN N1 1

L

{( , ), ,( , )}

( ) ( )

*

( ) ( )

*x x x x x xN N1 1

L

ax i x q

i=

1

0

if

otherwise

( ) *

L x x N a x ai i

i

N

i

i

N

( ( ), , ); ) ,( )

1 1

1 1

K ( a = == =



place window wN at i for j=1 to NCstore the data inside the window place window wN at (i,j)

wN in z={x(1),,x(N)} store values inside the window wN in

sort z and store in the vector q z={x(1),,x(N)}

q={x(1)

,,x(N)

} sort z and store in q={x(1)

,,x(N)

}

sum=0 sum=0

for m=1 to N for m=1 to N

sum=sum+amx(m) sum=sum+amx(m)end end

y(i)=sum; output y(i,j)=sum; output

end end

end end

41.6 Weighted Median Filters

41.6.1 Purpose

With this filter we may emphasize the samples that for some reason are supposed to be more reliable,

e.g., center sample x*, and the emphasis is obtained by weighing them more heavily.

41.6.2 Repetition or Duplication Operator &

3&x=x,x,x

41.6.3 Multiset

A multiset is a collection of objects, where the repetition of objects is permitted, e.g.,{3&2,1,2&4}={2,2,2,1,4,4}

41.6.4 Weighted Median Filters

WeightMed{x(1),,x(N);a}=Med{a1&x(1),,aN&x(N)}

41.6.5 Weighted Median Algorithm



weight vector a={a1,,aN} weight vector a={a1,,aN}Output: y={y(1),,y(Nd)} Output: NRNC (image)

halfsum= halfsum=

for i=1 to Nd for i=1 to NRplace window wN at i for j=1 to NCstore values inside the window and weight place window wN and corresponding weighs in

in z={(x1,a1),,(xN,aN)} z={(x1,a1),,(xN,aN)}

sort z with respect to xis and store in q sort z with respect to xis and store in

q={(x(1)a(1)),,(x(N),a(N))} q={(x(1),a(1)),,(x(N),a(N))}

sum=0, m=1 sum=0, m=1

repeat repeat

sum=sum+a(m) sum=sum+a(m)m=m+1 m=m+1

ai

i

N

=

1

2 ai

i

N

=

1

2



until sumhalfsum until sumhalfsum

y(i)=x(m1) y(i,j)=x(m1)end end

end end

Examplex={4,14,18,40,10}, a={0.05,0.1,0.15,0.1,0.05}. After ordering

z={(4,0.05),(10,0.05),(14,0.1),(18,0.15),(40,0.1)}. Halfsum=(0.05+0.05+0.1+0.15+0.1)/2=0.225. But

(0.05+0.05+0.1+0.15)=0.35>0.225 and the filter output is 18.

41.7 Ranked-Order Filter

41.7.1 Purpose

These filters can be used in situations where the noise distribution is not symmetric, e.g., where there

are more positive than negative impulses.

41.7.2 Ranked-Order Filter

RO(x(1),,x(N);r)=x(r), N=2k+1

Note: When r=k+1 we obtain the median filter. If rk+1.

Example

If x={4,14,18,40,10} and the ordered x-ordered={4,10,14,18,40}. Here N=5=2k+1 and k=2. The rth

ranked-order filter gives the outputs: 2 for r=1, 10 for r=2, 14 for r=3, 18 for r=4, and 40 for r=5.

41.7.3 Ranked-Order Filter Algorithms

Inputs: data x={x(1),,x(Nd)} Inputs: No. of rows NR No. of columns NC (image)


rank r rank r


for i=1 to Nd for i=1 to NRplace window wN at i for j=1 to NCstore the values in the window wN place window wN at (i,j)

in z={x(1),,x(N)} store values inside the window wN in

sort z to q={x(1),,x(N)} z={x(1),,x(N)}y(i)=x(r); output sort z to q={x(1),,x(N)}}

end y(i,j)=x(r)end end

end

41.8 Separable Two-Dimensional Median Filters

41.8.1 Purpose

The filter has the same properties as the median filter but is much faster. It is accomplished by usingtwo successive one-dimensional median filters of window size M=2l+1 (MM=N), the first along the

rows and the second one along the columns of the so-obtained image.



41.8.2 Separable Two-Dimensional Median Filters

Med{x(M,1),,x(M,M)}}

41.8.3 Algorithm

Inputs: No. of rows NR time No. of columns NC; (image)

Moving horizontal window w1,M, M=2l+1

Moving vertical window w2,M, M=2l+1

Output: NRNC; (image)

Median filter the image by using horizontal moving window w1,M for each row

Store the resulting values as they are found in a column vector

Median filter the column vector using vertical moving window w2,M

41.9 M-Filters

41.9.1 Purpose

M-estimators are generalizations of maximum likelihood estimators

41.9.2 M-Filters

M(x(1),,x(Nd); ) =arg

41.9.3 M-Filter by the Partial Derivative (with respect to ) of

The estimate M(x(1),,x(Nd)) satisfies the equation

41.9.4 Function in Use

1. Skipped median,

2. Andrews sine,

3. Tukeys biweight,

4. Standard M-filters,

SepMed Med{Med{

x x x M

x x x M

x M x M x M M

x x M

( , ) ( , ) ( , )

( , ) ( , ) ( , )

( , ) ( , ) ( , )

( , ), , ( , )},

1 1 1 2 1

2 1 2 2 2

1 2

1 1 1

L

L

M M M M

L

L L

=

min ( ( ) )

= i

N

x i

1

( ( ) )i

N

x i

= =

1

0

med r

xx x r

r x( )

( )sin( )

=

non linear digital filters

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