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analysis_fmi.m
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analysis_fmi.m
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function nfmi = analysis_fmi(ima, imb, imf, feature, w)
% FMI calculates the Feature Mutual Information (FMI), the non-reference
% performance metric for fusion algorithms, proposed in:
%
% M.B.A. Haghighat, A. Aghagolzadeh, H. Seyedarabi, "A Non-Reference Image
% Fusion Metric Based on Mutual Information of Image Features," Computers
% and Electrical Engineering, vol. 37, no. 5, pp. 744-756, Sept. 2011.
% http://dx.doi.org/10.1016/j.compeleceng.2011.07.012
%
%
% This code is the implementation of FAST-FMI, presented in:
%
% M. Haghighat, M.A. Razian, "Fast-FMI: non-reference image fusion metric,"
% 8th International Conference on Application of Information and
% Communication Technologies (AICT), pp. 1-3, 2014.
%
%
% Inputs:
% ima : First source image
% imb : Second source image
% imf : Fused image
% feature : Feature extraction method: gradient, edge, dct, wavelet, none (raw pixels) (default: image with no feature extraction)
% w : Sliding window size w by w (default: 3)
%
% Output:
% nfmi : Normalized Feature Mutual Information
%
%
% Sample use:
% nfmi = fmi(ima,imb,imf,'none',3);
% % Or simply:
% nfmi = fmi(ima,imb,imf);
%
%
% (C) Mohammad Haghighat, University of Miami
% haghighat@ieee.org
% IF YOU USE THIS CODE IN YOUR WORK, PLEASE CITE THE ABOVE PAPERS.
if nargin < 3 % Check correct number of arguments
error('There should be three input images (2 source images and 1 fused image)!');
end
if size(ima) ~= size(imb) % Check if the source images are of the same size
error('Size of the source images must be the same!');
end
if size(ima) ~= size(imf) % Check if the source images are of the same size
error('Size of the source and fused images must be the same!');
end
if ~exist('feature', 'var')
feature = 'edge'; % Default feature extraction
end
if ~exist('w', 'var')
w = 3; % Default window size
end
ima = double(ima);
imb = double(imb);
imf = double(imf);
% Feature Extraction
switch feature
case 'none' % Raw pixels (no feature extraction)
aFeature = ima;
bFeature = imb;
fFeature = imf;
case 'gradient' % Gradient
aFeature = gradient(ima);
bFeature = gradient(imb);
fFeature = gradient(imf);
case 'edge' % Edge
aFeature = edge(ima);
bFeature = edge(imb);
fFeature = edge(imf);
case 'dct' % DCT
aFeature = dct2(ima);
bFeature = dct2(imb);
fFeature = dct2(imf);
case 'wavelet' % Discrete Meyer wavelet
[cA,cH,cV,cD] = dwt2(ima,'dmey');
aFeature = rerange([cA,cH;cV,cD]);
[cA,cH,cV,cD] = dwt2(imb,'dmey');
bFeature = rerange([cA,cH;cV,cD]);
[cA,cH,cV,cD] = dwt2(imf,'dmey');
fFeature = rerange([cA,cH;cV,cD]);
otherwise
error('Please specify a feature extraction method among ''gradient'', ''edge'', ''dct'', ''wavelet'', or ''none'' (raw pixels)!');
end
% Sliding window
[m,n] = size(aFeature);
w = floor(w/2);
fmi_map = ones(m-2*w, n-2*w);
for p = w+1:m-w
for q = w+1:n-w
aSub = aFeature(p-w:p+w, q-w:q+w);
bSub = bFeature(p-w:p+w, q-w:q+w);
fSub = fFeature(p-w:p+w, q-w:q+w);
l = round((2*w+1).^2);
if aSub == fSub
fmi_af = 1;
else
aMax = max(aSub(:));
aMin = min(aSub(:));
if aMax == aMin
aSub = ones(2*w+1);
else
aSub = (aSub - aMin)/(aMax - aMin);
end
fMax = max(fSub(:));
fMin = min(fSub(:));
if fMax == fMin
fSub = ones(2*w+1);
else
fSub = (fSub - fMin)/(fMax - fMin);
end
% Normalize the feature images to get the marginal PDFs
aPdf = aSub(:)./(sum(sum(aSub)));
fPdf = fSub(:)./(sum(sum(fSub)));
% PDF to CDF tranformation
aCdf = zeros(size(aPdf));
fCdf = zeros(size(fPdf));
aCdf(1) = aPdf(1);
fCdf(1) = fPdf(1);
for i = 2:l
aCdf(i) = aPdf(i)+aCdf(i-1);
fCdf(i) = fPdf(i)+fCdf(i-1);
end
% Pearson correlation between marginal PDFs
aTemp = aPdf - mean(aPdf);
fTemp = fPdf - mean(fPdf);
if sum(aTemp.*fTemp) == 0
c = 0;
else
c = sum(aTemp.*fTemp)/sqrt(sum(aTemp.*aTemp)*sum(fTemp.*fTemp));
end
% Population standard deviations
e_aPdf = 0; e2_aPdf = 0; e_fPdf = 0; e2_fPdf = 0;
for i = 1:l
e_aPdf = e_aPdf + i.*aPdf(i); % Expected value of aPdf
e2_aPdf = e2_aPdf + (i.^2).*aPdf(i); % 2nd-order moment of aPdf
e_fPdf = e_fPdf + i.*fPdf(i); % Expected value of fPdf
e2_fPdf = e2_fPdf + (i.^2).*fPdf(i); % 2nd-order moment of fPdf
end
aSd = sqrt(e2_aPdf - e_aPdf.^2); % Population standard deviation of aPdf
fSd = sqrt(e2_fPdf - e_fPdf.^2); % Population standard deviation of fPdf
jointEntropy = 0;
% Joint PDF calculation using simplified Nelsen method with joint entropy calculation
if c >= 0
if c == 0 || aSd == 0 || fSd == 0
phi = 0;
else
covUp = 0;
for i = 1:l
for j = 1:l
% Frechet's upper bound bivariate distributions between f & a
covUp = covUp + 0.5*(fCdf(i)+aCdf(j)-abs(fCdf(i)-aCdf(j))) - fCdf(i).*aCdf(j);
end
end
corrUp = covUp/(fSd*aSd); % Upper correlation between f & a
phi = c/corrUp;
end
jpdfUp = 0.5*(fCdf(1)+aCdf(1)-abs(fCdf(1)-aCdf(1)));
jpdf = phi.*jpdfUp + (1-phi).*fPdf(1).*aPdf(1);
if jpdf~=0
jointEntropy = real(-jpdf.*log2(jpdf)); % Joint entropy
end
% 1-D boundaries
for i=2:l
jpdfUp = 0.5*(fCdf(i)+aCdf(1)-abs(fCdf(i)-aCdf(1))) - ...
0.5*(fCdf(i-1)+aCdf(1)-abs(fCdf(i-1)-aCdf(1)));
jpdf = phi.*jpdfUp + (1-phi).*fPdf(i).*aPdf(1);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
for j=2:l
jpdfUp = 0.5*(fCdf(1)+aCdf(j)-abs(fCdf(1)-aCdf(j))) - ...
0.5*(fCdf(1)+aCdf(j-1)-abs(fCdf(1)-aCdf(j-1)));
jpdf = phi.*jpdfUp + (1-phi).*fPdf(1).*aPdf(j);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
% 2-D walls
for i=2:l
for j=2:l
jpdfUp = 0.5*(fCdf(i)+aCdf(j)-abs(fCdf(i)-aCdf(j))) - ...
0.5*(fCdf(i-1)+aCdf(j)-abs(fCdf(i-1)-aCdf(j))) - ...
0.5*(fCdf(i)+aCdf(j-1)-abs(fCdf(i)-aCdf(j-1))) + ...
0.5*(fCdf(i-1)+aCdf(j-1)-abs(fCdf(i-1)-aCdf(j-1)));
jpdf = phi.*jpdfUp + (1-phi).*fPdf(i).*aPdf(j);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
end
end
if c < 0
if aSd == 0 || fSd == 0
theta = 0;
else
covLo = 0;
for i = 1:l
for j = 1:l
% Frechet's lower bound bivariate distributions between f & a
covLo = covLo + 0.5*(fCdf(i)+aCdf(j)-1+abs(fCdf(i)+aCdf(j)-1)) - fCdf(i).*aCdf(j);
end
end
corrLo = covLo/(fSd*aSd); % Lower correlation between f & a
theta = c/corrLo;
end
jpdfLo = 0.5*(fCdf(1)+aCdf(1)-1+abs(fCdf(1)+aCdf(1)-1));
jpdf = theta.*jpdfLo + (1-theta).*fPdf(1).*aPdf(1);
if jpdf~=0
jointEntropy = real(-jpdf.*log2(jpdf)); % Joint entropy
end
% 1-D boundaries
for i=2:l
jpdfLo = 0.5*(fCdf(i)+aCdf(1)-1+abs(fCdf(i)+aCdf(1)-1)) - ...
0.5*(fCdf(i-1)+aCdf(1)-1+abs(fCdf(i-1)+aCdf(1)-1));
jpdf = theta.*jpdfLo + (1-theta).*fPdf(i).*aPdf(1);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
for j=2:l
jpdfLo = 0.5*(fCdf(1)+aCdf(j)-1+abs(fCdf(1)+aCdf(j)-1)) - ...
0.5*(fCdf(1)+aCdf(j-1)-1+abs(fCdf(1)+aCdf(j-1)-1));
jpdf = theta.*jpdfLo + (1-theta).*fPdf(1).*aPdf(j);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
% 2-D walls
for i=2:l
for j=2:l
jpdfLo = 0.5*(fCdf(i)+aCdf(j)-1+abs(fCdf(i)+aCdf(j)-1)) - ...
0.5*(fCdf(i-1)+aCdf(j)-1+abs(fCdf(i-1)+aCdf(j)-1)) - ...
0.5*(fCdf(i)+aCdf(j-1)-1+abs(fCdf(i)+aCdf(j-1)-1)) + ...
0.5*(fCdf(i-1)+aCdf(j-1)-1+abs(fCdf(i-1)+aCdf(j-1)-1));
jpdf = theta.*jpdfLo + (1-theta).*fPdf(i).*aPdf(j);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
end
end
% Marginal entropies
index = find(aPdf~=0);
aEntropy = sum(-aPdf(index).*log2(aPdf(index)));
index = find(fPdf~=0);
fEntropy = sum(-fPdf(index).*log2(fPdf(index)));
% Mutual information between a & f
mi = aEntropy + fEntropy - jointEntropy;
% Overall normalized mutual information
if mi == 0
fmi_af = 0;
else
fmi_af = 2.*mi./(aEntropy+fEntropy);
end
end
if bSub == fSub
fmi_bf = 1;
else
bMax = max(bSub(:));
bMin = min(bSub(:));
if bMax == bMin
bSub = ones(2*w+1);
else
bSub = (bSub - bMin)/(bMax - bMin);
end
fMax = max(fSub(:));
fMin = min(fSub(:));
if fMax == fMin
fSub = ones(2*w+1);
else
fSub = (fSub - fMin)/(fMax - fMin);
end
% PDF of the gradients of the images
bPdf = bSub(:)./(sum(sum(bSub)));
fPdf = fSub(:)./(sum(sum(fSub)));
l = length(bPdf);
% PDF to CDF tranformation
bCdf = zeros(size(bPdf));
fCdf = zeros(size(fPdf));
bCdf(1) = bPdf(1);
fCdf(1) = fPdf(1);
for i=2:l
bCdf(i) = bPdf(i)+bCdf(i-1);
fCdf(i) = fPdf(i)+fCdf(i-1);
end
% Pearson correlation between marginal PDFs
bTemp = bPdf - mean(bPdf);
fTemp = fPdf - mean(fPdf);
if sum(bTemp.*fTemp) == 0
c = 0;
else
c = sum(bTemp.*fTemp)/sqrt(sum(bTemp.*bTemp)*sum(fTemp.*fTemp));
end
% Population standard deviations
e_bPdf = 0; e2_bPdf = 0; e_fPdf = 0; e2_fPdf = 0;
for i = 1:l
e_bPdf = e_bPdf + i.*bPdf(i); % Expected value of the aPdf
e2_bPdf = e2_bPdf + (i.^2).*bPdf(i); % 2nd-order moment of the aPdf
e_fPdf = e_fPdf + i.*fPdf(i); % Expected value of the bPdf
e2_fPdf = e2_fPdf + (i.^2).*fPdf(i); % 2nd-order moment of the bPdf
end
bSd = sqrt(e2_bPdf - e_bPdf.^2); % Population standard deviation of the intensity
fSd = sqrt(e2_fPdf - e_fPdf.^2); % Population standard deviation of the intensity
jointEntropy = 0;
% JPDF calculation using simplified Nelsen method with joint entropy calculation
if c >= 0
if c == 0 || bSd == 0 || fSd == 0
phi = 0;
else
covUp = 0;
for i = 1:l
for j = 1:l
% Frechet's upper bound bivariate distributions between f & b
covUp = covUp + 0.5*(fCdf(i)+bCdf(j)-abs(fCdf(i)-bCdf(j))) - fCdf(i).*bCdf(j);
end
end
corrUp = covUp/(fSd*bSd); % Upper correlation between f & b
phi = c/corrUp;
end
jpdfUp = 0.5*(fCdf(1)+bCdf(1)-abs(fCdf(1)-bCdf(1)));
jpdf = phi.*jpdfUp + (1-phi).*fPdf(1).*bPdf(1);
if jpdf~=0
jointEntropy = real(-jpdf.*log2(jpdf)); % Joint entropy
end
% 1-D boundaries
for i=2:l
jpdfUp = 0.5*(fCdf(i)+bCdf(1)-abs(fCdf(i)-bCdf(1))) - ...
0.5*(fCdf(i-1)+bCdf(1)-abs(fCdf(i-1)-bCdf(1)));
jpdf = phi.*jpdfUp + (1-phi).*fPdf(i).*bPdf(1);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
for j=2:l
jpdfUp = 0.5*(fCdf(1)+bCdf(j)-abs(fCdf(1)-bCdf(j))) - ...
0.5*(fCdf(1)+bCdf(j-1)-abs(fCdf(1)-bCdf(j-1)));
jpdf = phi.*jpdfUp + (1-phi).*fPdf(1).*bPdf(j);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
% 2-D walls
for i=2:l
for j=2:l
jpdfUp = 0.5*(fCdf(i)+bCdf(j)-abs(fCdf(i)-bCdf(j))) - ...
0.5*(fCdf(i-1)+bCdf(j)-abs(fCdf(i-1)-bCdf(j))) - ...
0.5*(fCdf(i)+bCdf(j-1)-abs(fCdf(i)-bCdf(j-1))) + ...
0.5*(fCdf(i-1)+bCdf(j-1)-abs(fCdf(i-1)-bCdf(j-1)));
jpdf = phi.*jpdfUp + (1-phi).*fPdf(i).*bPdf(j);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
end
end
if c < 0
if bSd == 0 || fSd == 0
theta = 0;
else
covLo = 0;
for i = 1:l
for j = 1:l
% Frechet's lower bound bivariate distributions between f & b
covLo = covLo + 0.5*(fCdf(i)+bCdf(j)-1+abs(fCdf(i)+bCdf(j)-1)) - fCdf(i).*bCdf(j);
end
end
corrLo = covLo/(fSd*bSd); % Lower correlation between f & b
theta = c/corrLo;
end
jpdfLo = 0.5*(fCdf(1)+bCdf(1)-1+abs(fCdf(1)+bCdf(1)-1));
jpdf = theta.*jpdfLo + (1-theta).*fPdf(1).*bPdf(1);
if jpdf~=0
jointEntropy = real(-jpdf.*log2(jpdf)); % Joint entropy
end
% 1-D boundaries
for i=2:l
jpdfLo = 0.5*(fCdf(i)+bCdf(1)-1+abs(fCdf(i)+bCdf(1)-1)) - ...
0.5*(fCdf(i-1)+bCdf(1)-1+abs(fCdf(i-1)+bCdf(1)-1));
jpdf = theta.*jpdfLo + (1-theta).*fPdf(i).*bPdf(1);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
for j=2:l
jpdfLo = 0.5*(fCdf(1)+bCdf(j)-1+abs(fCdf(1)+bCdf(j)-1)) - ...
0.5*(fCdf(1)+bCdf(j-1)-1+abs(fCdf(1)+bCdf(j-1)-1));
jpdf = theta.*jpdfLo + (1-theta).*fPdf(1).*bPdf(j);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
% 2-D walls
for i=2:l
for j=2:l
jpdfLo = 0.5*(fCdf(i)+bCdf(j)-1+abs(fCdf(i)+bCdf(j)-1)) - ...
0.5*(fCdf(i-1)+bCdf(j)-1+abs(fCdf(i-1)+bCdf(j)-1)) - ...
0.5*(fCdf(i)+bCdf(j-1)-1+abs(fCdf(i)+bCdf(j-1)-1)) + ...
0.5*(fCdf(i-1)+bCdf(j-1)-1+abs(fCdf(i-1)+bCdf(j-1)-1));
jpdf = theta.*jpdfLo + (1-theta).*fPdf(i).*bPdf(j);
if jpdf~=0
jointEntropy = jointEntropy + real(-jpdf.*log2(jpdf)); % Joint entropy
end
end
end
end
% Marginal entropies
index = find(bPdf~=0);
bEntropy = sum(-bPdf(index).*log2(bPdf(index)));
index = find(fPdf~=0);
fEntropy = sum(-fPdf(index).*log2(fPdf(index)));
% Mutual information between b & f
mi = bEntropy + fEntropy - jointEntropy;
% Overall normalized mutual information
if mi == 0
fmi_bf = 0;
else
fmi_bf = 2.*mi./(bEntropy+fEntropy);
end
end
fmi_map(p-w,q-w) = (fmi_af + fmi_bf)./2;
end
end
nfmi = mean2(fmi_map);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function imNorm = rerange(im)
% RERANGE changes the range of images into interval [0,1]
im = double(im);
[m,n] = size(im);
imMax = max(im(:));
imMin = min(im(:));
if imMax == imMin
imNorm = ones(m,n);
else
imNorm = (im - imMin)/(imMax - imMin);
end