Affine Projection Algorithm David Straight

7/31/2019 Affine Projection Algorithm David Straight

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AFFINE PROJECTION ALGORITHM: VARIATIONS AND

APPLICATION TO 2-D IMAGE PROCESSING

David A. Straight, Santa Clara University

ABSTRACT

A two dimensional affine projection algorithm has

been realized by augmenting a one dimensional affine

projection algorithm using a two dimensional state

equation. Application of the algorithm was then applied

to an interference cancellation configuration for image

processing. Convergence behavior was briefly explored

in simulation to show how numerical errors influence

minimum mean squared error.

1. INTRODUCTION

Interference in a system has the potential to degradeperformance and therefore it would be advantageous to

remove the interfering signal by some means. Simple

subtraction of such a signal is attractive when an exact

copy of the interference signal is known or available. In

many systems the interfering source is known, but an

exact copy is not realizable or varies over time. Some

examples would include adjacent transmitters and

receivers that must operate simultaneously and medical

imaging systems such as MRI and ultrasound. Adaptive

filtering is a viable solution to this type of interference

since, quite often, the interfering source is available. A

few examples of this include 60Hz interference,

mechanical vibration, magnetic fluctuation anddisturbance of power supply lines. In an MRI system any

of these disturbances can degrade the resultant image due

to the sensitivity of the system as a whole. This paper

provides a method for cancelling noise in two

dimensional images.

There are five sections in this article. 1) the introduction

2) methods for developing the algorithm 3) results of

simulation 4) MATLAB code used and 5) references

used.

Notation used

),( nmy scalar (also a function ofm and n)

),( nmw vector (also a function ofm and n)

),( nmY matrix (also a function ofm and n)

knu a vector that is a function ofkand n

T)( transpose operator

2.1 IMPLEMENTATION OF 2D APA

A two dimensional convolution of the form

= =

=M

i

N

j

jnimnmnm1 1

),(),(),( WXY

can be represented as

),(),(),( nmwnmxnmyT

=

where is a single element of and

and have been mapped into column-

ordered vectors

),( nmy ),( nmY),( nmX ),( nmW

),( nmx and ),( nmw respectively [6] as

shown below.

=

=

),(

)1,3(

),2(

)1,2(),1(

)3,1(

)2,1(

)1,1(

),(,

),(

),2(

),1(

),1(),(

)2,(

)1,(

),(

),(

NNw

w

Nw

wNw

w

w

w

nmw

NnNmx

nmx

Nnmx

nmxNnmx

nmx

nmx

nmx

nmx

M

M

M

M

M

M


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After mapping a block of data of size N x N to a column

vector of size N2 x 1 the affine projection algorithm can

be implemented as two one dimensional algorithms that

update the weight equation along the horizontal and

vertical directions. Updating the weights along both

horizontal and vertical directions is what makes this a true

two dimensional algorithm [2]. Because of the vector

mapping Gram Schmidt orthogonalization is no different

than the one-dimensional case.

For the one dimensional case Sankaran [1] uses the

following weight update equation

M

nMnnnn uuuww ++++=+ L1

1

0

01

Where,

1

+nw is the updated weight.

nw is the previous weight.M10

nnn uuu ,,, L are a set of vectors all orthogonal

to each other.

M ,,, 10 L are functions of the error and data

Mkfor

u

e

k

n

nk

,,2,1,0

2

L=

=

k

The error function isk

n

T

knkn

k

n wxde = . The

superscript, k, in the weight vector, knw , represents the

estimated weight vector generated for each

orthogonalized vectork

nu , using

kkk+1

nknn uww += ++ 11 where00

nn xu = . and

Mknu L3,2,1,0for =k

are vectors that are orthogonal to

each other and originate from the set of data vectors

Mnnnn xxxx L,,, 21 using the Gram Schmidt

process as described by [8].

Following the work of Ohiki & Hashiguchi [2] as well as

Pinto [7] a two dimensional algorithm using a twodimensional state equation gives

M

MvMv

M

MhMhnvvnhh

nvvnhh

nmvnmhnm

uuuu

xx

wfwfw

,,,,

1

,1,

1

,1,

0

,0,

0

,0,

,11,1,1

++++

++

+= ++++

L

Where,

h and v represent horizontal and vertical respectively

are constants related byvh ff and, 1=+ vh ff

nmnm ww ,11, and ++ are the weight update vectors

updated along the horizontal and

vertical directions respectfully

As pointed out by Ohiki and Hashiguchi [2] both

nmnm ww ,11, and ++ are updated in one dimension along

the horizontal and vertical directions respectively and this

is why the following equations are considered one

dimensional even though they operate on two dimensionaldata

vh ff and, are the state equations.


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2.2 - ALGORITHM CHART

(1)nh

T

nhnhnh wxde ,,,, =

nv

T

nvnvnv wxde ,,,, =

(2)nh

T

nhnh xxx ,,2

, =

nv

T

nvnv xxx ,,2

,=

(3)

2

,

,

,0

nh

nh

nh

x

e =

2

,

,

,0

nv

nv

nv

x

e =

(4)nhnhnh xww ,0,

1

1, +=+

nvnvnv xww ,0,1

1, +=+

(5)

nhnh

xu,

0

,

=

nvnv xu ,0

, =

For k = 1,2,3 M repeat steps (6) through (9)

(6)

=

=

1

0

,2

,

,,

,,

k

i

i

nhi

nh

i

nh

T

knh

knh

k

nh u

u

uxxu

=

=

1

0

,2

,

,,

,,

k

i

i

nvi

nv

i

nv

T

knv

knv

k

nv uu

uxxu

(7)nh

T

knhknh

k

nh wxde ,,,, =

nv

T

knvknv

k

nv wxde ,,,, =

(8)

=

else

uif

u

e knh

k

nh

k

nh

kh

0

02

,2

,

,

,

=

else

uifu

e knv

k

nv

k

nv

kv

0

02

,2

,

,

,

(9) knhk

k

nh

k

nh uww ,1,1

1, += +

++

k

nvk

k

nv

k

nv uww ,1,1

1, += +

+

+

(10) 11,1,

+

++ =M

nhnh ww

1

1,1,

+

++ =M

nvnv ww

The chart to the left shows every step needed to perform

the affine projection algorithm.

Figure 1 and Figure 2 provide an example of how data is

gathered from the original data. For this example M=2.

The horizontal data set at any iteration, n, is

. The diagram below showsfor the black, blue and green dashed

rectangles respectively.

MkforL

3,2,1,0k-nm,=

X2,22,32,4 ,, XXX

n

m

Index 1,1x(1,1)

2D Image

direction of

weight update

horizontal

Figure 1

Similarly, denotes data for

iteration m. The diagram below is shown for

for the black, blue and green dashed

rectangles respectively.

Mkfor L3,2,1,0nk,-m =X

2,23,24,2 ,, XXX

Index 1,1x(1,1)

n

m

direction of

weight updatevertical

Figure 2

Column ordered vectors are formed from and

giving

k-nm,X

nk,-mX nhx , and nvx , respectively as seen in the

algorithm chart on the left.


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2.3 NOISE REDUCTION

Following a simple derivation from Widrow [5] we showhow an interference cancellation circuit using an adaptive

filter works. Figure 3 shows an application for this

adaptive filter as a noise canceling circuit. In the diagramwe have a signal, s, that has been contaminated with

noise, n, where n is not correlated with s. n and s have

zero mean.

+

AdaptiveFilter

+

-

e

y

System

output

H

2

nwith variance

nnoise

2

H

2)var( yy =

Input Imagecontaminated

with noise

ns +22

ns +with variance

Figure 3

By first taking the z transform of the autocorrelation of

the noise and signal and then forming a ratio of signal tonoise ratios between the output and input. Letting

)(zout represent the signal to noise ratio of the system

output and )(zin represent the signal to noise ratio of

the input contaminated by noise the improvement in

signal to noise ration is then found by [5]

[ ]2)()(1)(

)(

)(

)(

)(

)(

zWzHz

z

z

z

z

z

nn

nn

noiseoutput

nn

in

out

=

=

Where,

is the power spectral density of the

interfering signal

is the noise power spectral

density at the output.

is the transfer function of the adaptive

filter.

)(znn

)(znoiseoutput

)(zW

When the adaptive filter converges then )(1)( zHzW = and the denominator of the above equation will go to zero

making the output signal to noise ratio infinitely large.

Of course there are other sources of noise that preventinfinite improvement. In this simple diagramH(z)

represents an unknown channel that provides a correlated

source to the original interferer, but not an exact copy of

the noise. The adaptive filter helps by inverse modeling

this channel and upon doing so provides the exact copy

(for the noisless case) of the correlated disturbance.

2.4 CONVERGENCE

There are many resources (papers) available on the topic

of convergence [1]. This paper is not about convergence,however a few simple formulas for specific cases are

provided as a convenience here. From Sankaran [1] the

flowing relationships were derived on the assumption is

that white Gaussian noise with zero mean was used as theinput.

The convergence rate is approximately

[ ])1/(5dB20

+

MN

The minimum mean squared error.

+= R)tr(

)(

11

2

2

nxEMMSE v

3.1 - MSE learning curves

Prior to application of a noise cancelling system the

simulation was tested to verify proper operation

The algorithm was simulated in a system identification

configuration as shown in Figure 4 below.

Adaptive

Filter

xinput

+

-

e

d

Plant

y

+

+

+

+

output

additive

noisen

12=x

20210

=n

Figure 4

Initial results for eight simulations appear in Figure 5.

Simulation of the two dimensional affine projection

algorithm show the benefit of faster convergence with theaddition of additional orthogonal data vectors. The impact


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)(2)( 222 yndynd ++=

])[(][

)]([2])[(][][

22

222

ynEdE

yndEynEdEE

+=

++=

The second term on the right hand side of the equation isthe only term minimized by the adaptive filter leaving the

power of the uncontaminated alone. This is the basis of

how this type of system works.

In simulation is estimated by squaring the error

and averaging over the number of iterations performed.

The error term was generated using

][ 2E

1,1

++ nmw in the

following error equation.

1,1, ++=nm

T

nmn wxde

Where, is a given pixel of the imagemnd ,

1,1

++ nmw is the adaptive weight

coefficient of a 2D mask (that has been

mapped as a vector (described insection 2.1 and 2.2)

Figure 11 shows the test image representing . The

variance of is 0.01.

0nd+

0n

Figure 12 and Figure 13 are two examples of noiseremoved from the original picture ofFigure 11. For eachof these pictures 2000 iterations were run, in Figure 12

M=1and in Figure 13 M=3.

Figure 11 Image corrupted by noise (variance of the addednoise was 0.01)

Figure 12 2000 iterations, M=1

Figure 13 2000 iterations, M=3

The simulation ofFigure 12 had the following estimatedweight values.

=

0234.00330.00010.0

0310.09657.00158.0

0530.00447.00112.0

W

The simulation ofFigure 13 had the following estimated

weight values.

=

0602.00033.00493.0

0285.09812.00789.0

0634.00257.00051.0

W

It is easy to see (and prove) that the ideal weights would

be an impulse of the form

=

000

010

000

oW


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simply because a simple subtraction will remove allnoise, perfectly because no = n1 (assuming a perfectly

linear process).

0=+= ynde

Therefore we would need ny = , but since n was fedinto the adaptive filter then an impulse response would bean appropriate coefficient.

3.3 CLIPPING

Distortion in a system can prevent a cancellation

system from working properly. This was discoveredduring simulation in the image shown in Figure 14.

Limited by the dynamic range of the system both the

noise and the image exceeded a maximum limit leading

to clipping of the image. This occurred in areas where

the nose and mouth of the pumpkin are located. In theseareas, information was lost during the process of adding

noise resulting in loss of noise in these regions.

Without the added noise the adaptive circuit minimized

these areas

Figure 15

3.4 CONCLUSION

An application for a two dimensional affine algorithm was

developed and simulated for the purpose of noise

cancellation. The Affine algorithm is well suited for this

purpose because of its convergence speed even with highlycorrelated inputs commonly found in images. The adaptive

filter has the ability to channel equalize the path of the

noise source making it suitable for image processing inchanging environments. Applying image enhancement

using this algorithm would benefit systems such as MRI

that fall vulnerable to disturbances of known origin, but

unknown channel paths. It was found through oursimulations that limits in dynamic range as well as

numerical computation can influence overall system

design of an adaptive 2D image interference cancellationsystem.

Figure 14


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4.1 MATLAB CODE (2D AFFINE)

% 2D Affine Application to image processing - interference cancellation%

% for best results set the variable "times" to 1000, however this will take

% a very long time therefore it is recommended that lower values be used.% times=100 will take about 15 minutes to run.

close all;

clear all;

fh = 0.5;

fv = 1-fh;

% getimpage() is a function that loads an image with noise

% into the first element of the row vector (d) and the noise used

% to contaminate the image into the second element(x).

[clnImg d x] = getimage();

L=size(d)*[1;0]; % lenth of input data (horz. & vert.)

% == Filter settingsN=3 ; % horz. & vert. length of vector x (w & wo are size NxN)

% N must be odd

mu = 1; % set to 0 < mu < 2 (mu=1 is best)

r = (N-1)/2; %used repeatedly

%represents "pixel" lost around perimeter of X upon%convolving using 'valid' command in conv2()

s = (N+1)/2; % used repeatedly (center of mask)

error = zeros(L,L); %

wav = zeros(N*N,1);% == averaging learning curves

times = 100;

% == filter setting

M=1; % number of orthogonal vectors used (total number of orthogonal vectors is M+1

mse=zeros(L*L,1);

for t=1:times; % this loop used for averaging learning curve == t Loop

% == initialize variables, vectors and matices

sqmagXh = 0;

sqmagXv = 0;w=zeros(N*N,1); %initialize adaptive weights

wh=zeros(N*N,1);

wv=zeros(N*N,1);wtemp=zeros(N*N,1); % used for Gram Schmidt calculation

Uh=zeros(N*N,1); % M orthogonal vectors (length N)

Uv=zeros(N*N,1); % M orthogonal vectors (length N)

wtemph=zeros(N*N,1);wtempv=zeros(N*N,1);


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temph=zeros(N*N,1);

tempv=zeros(N*N,1);

[clnImg d x] = getimage();

%flipud(fliplr())

c=0; % used for averaging mse (learning) curve - counts total interations

for n=s+M:L-r %=== loop 1 ===

for m=s+M:L-r

xhvec = reshape(x(m-r-M:m+r-M,n-r:n+r), N*N, 1);

eh = d(m-M,n) - xhvec'*wh;

xvvec = reshape(x(n-r:n+r,m-r-M:m+r-M), N*N, 1);

ev = d(n,m-M) - xvvec'*wv;

sqmagXh = xhvec'*xhvec; % (2h)

sqmagXv = xvvec'*xvvec; % (2v)

muoh = (mu*eh)/sqmagXh; %(3h)muov = (mu*ev)/sqmagXv; %(3v)

wtemph = wh + muoh*xhvec; % temp weight vector %(4h)

wtempv = wv + muov*xvvec; % temp weight vector %(4v)

Uh(:,1) = xhvec; %(5h) first of M+1 orthogonal vectors

Uv(:,1) = xvvec; %(5v)

for k=1:M % if M>0 then the algorithm is Affine, otherwise it is NLMS

% if you plan to use M=0 then you must add an if

% condition so that the program does not enter

% into this loop.

% ======== Begin (6) =========for i=1:k % == Gram-Schmidt -- begin --

xhvec = reshape(x(m-r-M:m+r-M,n-r-k:n+r-k), N*N, 1);temph = temph + (( xhvec'*Uh(:,i) ) / (Uh(:,i)'*Uh(:,i) + 10^-100))*Uh(:,i);

xvvec = reshape(x(n-r-k:n+r-k,m-r-M:m+r-M), N*N, 1);

tempv = tempv + (( xvvec'*Uv(:,i) ) / (Uv(:,i)'*Uv(:,i) + 10 -100))*Uv(:,i);end %== Gram-Schmidt -- end --

Uh(:,k+1) = xhvec - temph;

Uv(:,k+1) = xvvec - tempv;

temph = zeros(N*N,1);

tempv = zeros(N*N,1);

% ======== End (6) =========

eh = d(m-M,n-k) - xhvec'*wtemph; %(7)ev = d(n-k,m-M) - xvvec'*wtempv;

UmagSqh = Uh(:,k+1)'*Uh(:,k+1);

UmagSqv = Uv(:,k+1)'*Uv(:,k+1);


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if UmagSqh ~= 0 %(8h)

mu0h = (mu*eh)/UmagSqh;else

mu0h = 0 ;

end

if UmagSqv ~= 0 %(8v)

mu0v = (mu*ev)/UmagSqv;

else

mu0v = 0 ;end

wtemph = wtemph + mu0h*Uh(:,k+1); % (9h)wtempv = wtempv + mu0v*Uv(:,k+1); % (9v)

end % end k Loop (same as loop 2)wh = wtemph;

wv = wtempv;

wtemph = zeros(N*N,1);wtempv = zeros(N*N,1);

evh = d(m,n) - reshape(x(m-r:m+r,n-r:n+r),N*N,1)'*w;c=c+1;

error(m,n) = error(m,n) + evh;

w = wh*fh + wv*fv; % update equation - left side is w(1+n) (10)

end % == end m loop (part of loop 1)

% initialize variables before next n (and m) or iteration

end% end n loop (part of loop 1) loop 1 has ended

wav = wav + w;w = wav./t;

end% end loop t

% graphs go here

wav = wav./times; %average weight coefficient

error = error./times; %average MSE

colormap('gray')

ma = max(max(error));

mi = min(min(error));error = error./(ma-mi);

error = error - min(min(error));error = error*64;subplot(2,2,1)

image(error)

title('adaptive filter')

subplot(2,2,2)

image(d*50)

title('original picture contaminated by noise')


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temp = conv2(d,ones(3,3)*(1/9));

ma = max(max(temp));mi = min(min(temp));

temp = temp./(ma-mi);

temp = temp - min(min(temp));temp = temp*64;

subplot(2,2,3)

image(temp)

title('picture processed with a LPF')

subplot(2,2,4)

image(clnImg*0.23)

title('pure image')


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5.1 - REFERENCE

[1] Sankaran, S.G., Beex, A.L., "Convergence Behavior

of Affine Projection Algorithms,'' IEEE Transactions onSignal Processing, vol. 48, No. 4, pp. 10861096, April

2000.

[2] Makoto Ohki, Sumihisa Hashiguchi, "Two-

dimensional LMS Adaptive Filters,'' IEEE Transactions

on Consumer Electronics, vol. 37, No. 1, pp. 6673, Feb.

1991.

[3] Slock, Dirk T.M., "On the Convergence Behavior of

LMS and NLMS Algorithms,'' IEEE Transactions on

Signal Processing, vol.41, pp. 2811-2825, Sept. 1993.

[4] R. C. Gonzalez, R. E. Woods, Digital Image

Processing, second edition Upper Saddle River, N.J.07458, Prentice Hall, 2002

[5] B. Widrow, S. Stearns, Adaptive Signal Processing

Englewood Cliffs, N.J. 0763: Prentice-Hall, Inc.1985

[6] Strait, Jenkins Orthogonalization Techniques forTwo-Dimensioanl Adaptive Fitlers CoordinatedCoordinated Science Laboratory, University of Illinois,

Urbana, IL 61801

[7] S. S. Pinto Convergence Behavior of 2-Dimensional

Affine Projection Algorithms, MSEE Thesis, Santa ClaraUniversity, Santa Clara, CA 95053.

[8] G. H. Golub, C. F. Van Loan, Matrix computations -

third edition The Johns Hopkins University Press,

Baltimore, Maryland, 21218-4319, The Johns HopkinsUniversity Press

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Affine Projection Algorithm David Straight