AUTO-FOCUS
BANDWIDTH
USING
PSD
ESTIMATION
EFFECTIVE
By Laurence G. Hassebrook
2-22-2012
Please follow the tutorial and reproduce the figures with your own code.
We demonstrate how to use the effective bandwidth of the PSD estimate to determine the image
that is most in focus. The first part of this visualization uses control noise generated from a white
Gaussian noise image filtered by a Gaussian function to simulate 2-D blurring. Since the noise is
stationary we use the rows of the colored noise to estimate a 1-D PSD of the blurred noise image.
From this we estimate an effective bandwidth [1] by first removing dc, then taking the peak
value of the PSD and forming a rectangle that is equal in area as the PSD estimate. The rectangle
height is equal to the peak PSD value and the width yields the effective bandwidth. The image
with the maximum effective bandwidth is considered to be most in focus.
The second part of the tutorial uses real data that is blurred by repositioning the target under a
microscope. The same algorithm is applied except we use a second type of effective bandwidth
definition [2] better suited to signals with a fast frequency drop off. In this second case we find
the bandwidth that contains 98% of the total area of the estimated PSD. We chose 98% because
that is what is commonly used in “Carsons rule” to define the bandwidth of wideband systems
having a fast drop off.
1. AUTOFOCUS SIMULATION
Form an input noise image from White Gaussian Noise.
clear all;
% dimensions of image
Nx=512;
My=Nx;
% form the control noise image from Gaussian distribution
w=randn(My,Nx);
W=fft2(w);
%
figure(1)
imagesc(w)
colormap gray;
axis image;
axis off;
title('Input White Noise Pattern')
xlabel('Spatial X dim (pixels)');
ylabel('Spatial Y dim (pixels)');
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Figure 1: Gaussian white noise input.
Estimate the PSD by using each row as a separate “run” of the data. We can do this because the
data is stationary in all directions.
% estimate PSD along 1 dimension of the input noise pattern by averaging
% the PSD of each row
Wy=zeros(1,Nx);
wy=zeros(1,Nx);
for m=1:My
wy(1:Nx)=w(m,1:Nx);
Wy=Wy+abs(fft(wy)).^2;
end;
Wy=Wy/My;
Then estimate an effective bandwith based on the maximum value and total area of the PSD
estimate such that:
% Determine equivalent ideal lowpass
E=sum(Wy);
Amax=max(Wy);
B2=floor(E./(2*Amax));
B2=2*B2+1;
if B2>Nx
B2=Nx;
end;
Heff=Amax*irect(1,B2,1,Nx);
figure(2)
k=1:Nx;
plot(k,Wy,k,Heff)
axis([1,Nx,0,1.1*max(Wy)]);
title('Estimated White Noise PSD')
xlabel('Discrete Frequency');
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Figure 2: Effective Bandwidth and PSD estimate of Fig. 1.
Now loop through a range of blurring by varying sigma of a Gaussian function based filter. Each
time, estimate the PSD and effective bandwidth. Keep track of the largest bandwidth which is an
indicator of which image is the most in focus.
% Simulate blurring process
nmax=10;
deltadev=60
Nindex=floor(2*nmax+1);
EffBW=zeros(1,Nindex);
sigmaAll=zeros(1,Nindex);
EffBWindex=0;
EffBWmax=0;
istore=0;
for n=-nmax:nmax % -nmax <n1<
n1=floor(abs(nmax-(abs(n)-1)));
sigma=n1*deltadev;
sigmaAll(n+nmax+1)=sigma;
Hblurr=igauss(sigma,sigma,My,Nx);
wcolor=real(ifft2(Hblurr.*W));
% the PSD of each row
Wy=zeros(1,Nx);
wy=zeros(1,Nx);
for m=1:My
wy(1:Nx)=wcolor(m,1:Nx);
Wy=Wy+abs(fft(wy)).^2;
end;
Wy=Wy/My;
% Determine equivalent ideal lowpass
E=sum(Wy);
Amax=max(Wy);
B2=E./(2*Amax);
B2=floor(B2);
B2=2*B2+1;
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if B2>Nx
B2=Nx;
end;
Heff=Amax*irect(1,B2,1,Nx);
% store effective bandwidth
EffBW(n+nmax+1)=B2;
if B2>EffBWmax
EffBWmax=B2;
EffBWindex=n+nmax+1;
istore=1;
end;
Notice how we store only the best so far of the color filter, blurred image and the PSD by using
the istore variable.
figure(3)
imagesc(Hblurr)
colormap gray;
title('Blurring Filter')
xlabel('DT Frequency (pixels)');
ylabel('DT Frequency (pixels)');
if istore==1
print -djpeg fig3
end;
Figure 3: Of the range of sigma used, this transfer function gave the least blurring.
%
figure(4)
imagesc(wcolor)
colormap gray;
title('Output Colored Noise')
xlabel('Spatial X dim (pixels)');
ylabel('Spatial Y dim (pixels)');
if istore==1
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print -djpeg fig4
end;
Figure 4: Least blurred noise image.
Figure 5: Least blurred noise PSD estimate.
%
figure(5)
plot(k,Wy,k,Heff)
axis([1,Nx,0,1.1*max(Wy)]);
title('Estimated Colored Noise PSD and Eff. BW')
xlabel('Discrete Frequency');
ylabel('PSD');
if istore==1
print -djpeg fig5
istore=0;
end;
end;
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Figure 6: Sigma and Eff. Bandwidth correspondence.
In Fig. 6 we show how the effective bandwidth and sigma varied with image index. As expected,
the larger the sigma, the larger the effective bandwidth.
Bw=1:Nindex;
figure(6);
plot(Bw,EffBW,Bw,sigmaAll);
title('Effective Bandwidth and Sigma')
xlabel('Image Index');
ylabel('Eff. Bandwidth and sigma (pixels)');
legend('Eff. Bandwidth','sigma');
2. EXPERIMENTAL AUTOFOCUS
Download the set of test images. The images were blurred by changing their distance from the
camera lens using a Z stage adjustment. This particular type of Z stage also introduced a change
in Y position but the technique is relatively invariant to position changes because it uses a PSD
estimate.
%% REAL DATA
Pathname='ee640data' % path or folder name in reference to your default
path
Filename='autofocusdataB_' % name of files not including the index or
suffix
Filesuffix='jpg' % suffix or image type
% get size of images
index=0;
Fullname=sprintf('%s%c%s%d%c%s',Pathname,'\',Filename,index,'.',Filesuffix)
A_bmp=double(imread(Fullname)); % load pattern#.bmp
[My, Nx, Pz] =size(A_bmp);
Nindex=20;
k=1:Nx;
EffBW=zeros(1,Nindex);
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EffBWindex=0;
EffBWmax=0;
istore=0;
Figure 7: Blurred and in focus sample images from data set.
The image names were indexed to allow automated input of each image. Note that we also
zeroed out the dc component.
for index=0:(Nindex-1)
Fullname=sprintf('%s%c%s%d%c%s',Pathname,'\',Filename,index,'.',Filesuffix)
A_bmp=double(imread(Fullname)); % load pattern#.bmp
[My, Nx, Pz] =size(A_bmp);
Ar=A_bmp(:,:,1);
Ag=A_bmp(:,:,2);
Ab=A_bmp(:,:,3);
Abw=Ar+Ag+Ab;
Abw=Abw/max(max(Abw));
% the PSD of each row
Wy=zeros(1,Nx);
wy=zeros(1,Nx);
for m=1:My
wy(1:Nx)=Abw(m,1:Nx);
Wyfft=abs(fft(wy));
% zero out dc;
Wyfft(1)=0+i*0;
Wy=Wy+abs(Wyfft).^2;
end;
Wy=Wy/My;
Amax=max(Wy);
An accumulated PSD area is formed so that the effective bandwidth is found by correspondence
with 98% of the PSD area. Note that we only use half the PSD vector for this due to symmetry.
% form accumulated energy
Nx2=Nx/2;
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Eaccum=zeros(1,Nx2);
for n=1:Nx2
for m=1:n
Eaccum(n)=Eaccum(n)+Wy(m);
end;
end;
% Determine equivalent ideal lowpass
% let B2 be the index for 0.98 of total
Bthresh=0.98
for n=1:Nx2
if Eaccum(n)/Eaccum(Nx2) < Bthresh
B2=n;
end;
end;
B2=2*B2+1;
if B2>Nx
B2=Nx;
end;
Heff=Amax*irect(1,B2,1,Nx);
% store effective bandwidth
EffBW(index+1)=B2;
if B2>EffBWmax
EffBWmax=B2;
EffBWindex=index+1;
istore=1;
end;
Figure 8: Image 10 was found to have the highest value of effective bandwidth.
figure(7); % figure 8 in this document
imagesc(Abw);
colormap gray;
title('Input Image')
if istore==1
title('Input Image with Best Focus')
print -djpeg fig7
end;
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Figure 9: PSD with highest Effective Bandwidth value.
%
figure(8)% figure 9 in document
plot(k,Wy,k,Heff)
%axis([1,Nx,0,1.1*max(Wy)]);
title('Estimated Image PSD and Eff. BW')
xlabel('Discrete Frequency');
ylabel('PSD');
if istore==1
title('Estimated Image PSD and Eff. BW of Best Focus')
print -djpeg fig8
istore=0;
end;
end; % loop index through figures
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Figure 10: Correspondence between image index and effective bandwidth.
%plot final results for effective bandwidth
Bw=1:Nindex;
figure(9); % image 10 in this document
plot(Bw,EffBW);
title('Effective Bandwidth of 0.98 Emax')
The final results show that the algorithm works well with the real data which had considerable
structure and was much less colored than the control noise. In practice, the PSD of the test data
could be used to color the test noise for a closer correspondence of the PSDs.
3. REFERENCES
1. Random Signals Detection, Estimation and Data Analysis by K. Sam Shannugan and A.
M. Breipohl. John Wiley& Sons, New York. 1988
2. “Carsons Rule” http://en.wikipedia.org/wiki/Carsons_rule
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