function [FSIM, FSIMc] = FeatureSIM(imageRef, imageDis)
% ========================================================================
% FSIM Index with automatic downsampling, Version 1.0
% Copyright(c) 2010 Lin ZHANG, Lei Zhang, Xuanqin Mou and David Zhang
% All Rights Reserved.
%
% ----------------------------------------------------------------------
% Permission to use, copy, or modify this software and its documentation
% for educational and research purposes only and without fee is here
% granted, provided that this copyright notice and the original authors'
% names appear on all copies and supporting documentation. This program
% shall not be used, rewritten, or adapted as the basis of a commercial
% software or hardware product without first obtaining permission of the
% authors. The authors make no representations about the suitability of
% this software for any purpose. It is provided "as is" without express
% or implied warranty.
%----------------------------------------------------------------------
%
% This is an implementation of the algorithm for calculating the
% Feature SIMilarity (FSIM) index between two images.
%
% Please refer to the following paper
%
% Lin Zhang, Lei Zhang, Xuanqin Mou, and David Zhang,"FSIM: a feature similarity
% index for image qualtiy assessment", IEEE Transactions on Image Processing, vol. 20, no. 8, pp. 2378-2386, 2011.
%
%----------------------------------------------------------------------
%
%Input : (1) imageRef: the first image being compared
% (2) imageDis: the second image being compared
%
%Output: (1) FSIM: is the similarty score calculated using FSIM algorithm. FSIM
% only considers the luminance component of images. For colorful images,
% they will be converted to the grayscale at first.
% (2) FSIMc: is the similarity score calculated using FSIMc algorithm. FSIMc
% considers both the grayscale and the color information.
%Note: For grayscale images, the returned FSIM and FSIMc are the same.
%
%-----------------------------------------------------------------------
%
%Usage:
%Given 2 test images img1 and img2. For gray-scale images, their dynamic range should be 0-255.
%For colorful images, the dynamic range of each color channel should be 0-255.
%
%[FSIM, FSIMc] = FeatureSIM(img1, img2);
%-----------------------------------------------------------------------
[rows, cols] = size(imageRef(:,:,1));
I1 = ones(rows, cols);
I2 = ones(rows, cols);
Q1 = ones(rows, cols);
Q2 = ones(rows, cols);
if ndims(imageRef) == 3 %images are colorful
Y1 = 0.299 * double(imageRef(:,:,1)) + 0.587 * double(imageRef(:,:,2)) + 0.114 * double(imageRef(:,:,3));
Y2 = 0.299 * double(imageDis(:,:,1)) + 0.587 * double(imageDis(:,:,2)) + 0.114 * double(imageDis(:,:,3));
I1 = 0.596 * double(imageRef(:,:,1)) - 0.274 * double(imageRef(:,:,2)) - 0.322 * double(imageRef(:,:,3));
I2 = 0.596 * double(imageDis(:,:,1)) - 0.274 * double(imageDis(:,:,2)) - 0.322 * double(imageDis(:,:,3));
Q1 = 0.211 * double(imageRef(:,:,1)) - 0.523 * double(imageRef(:,:,2)) + 0.312 * double(imageRef(:,:,3));
Q2 = 0.211 * double(imageDis(:,:,1)) - 0.523 * double(imageDis(:,:,2)) + 0.312 * double(imageDis(:,:,3));
else %images are grayscale
Y1 = imageRef;
Y2 = imageDis;
end
Y1 = double(Y1);
Y2 = double(Y2);
%%%%%%%%%%%%%%%%%%%%%%%%%
% Downsample the image
%%%%%%%%%%%%%%%%%%%%%%%%%
minDimension = min(rows,cols);
F = max(1,round(minDimension / 256));
aveKernel = fspecial('average',F);
aveI1 = conv2(I1, aveKernel,'same');
aveI2 = conv2(I2, aveKernel,'same');
I1 = aveI1(1:F:rows,1:F:cols);
I2 = aveI2(1:F:rows,1:F:cols);
aveQ1 = conv2(Q1, aveKernel,'same');
aveQ2 = conv2(Q2, aveKernel,'same');
Q1 = aveQ1(1:F:rows,1:F:cols);
Q2 = aveQ2(1:F:rows,1:F:cols);
aveY1 = conv2(Y1, aveKernel,'same');
aveY2 = conv2(Y2, aveKernel,'same');
Y1 = aveY1(1:F:rows,1:F:cols);
Y2 = aveY2(1:F:rows,1:F:cols);
%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculate the phase congruency maps
%%%%%%%%%%%%%%%%%%%%%%%%%
PC1 = phasecong2(Y1);
PC2 = phasecong2(Y2);
%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculate the gradient map
%%%%%%%%%%%%%%%%%%%%%%%%%
dx = [3 0 -3; 10 0 -10; 3 0 -3]/16;
dy = [3 10 3; 0 0 0; -3 -10 -3]/16;
IxY1 = conv2(Y1, dx, 'same');
IyY1 = conv2(Y1, dy, 'same');
gradientMap1 = sqrt(IxY1.^2 + IyY1.^2);
IxY2 = conv2(Y2, dx, 'same');
IyY2 = conv2(Y2, dy, 'same');
gradientMap2 = sqrt(IxY2.^2 + IyY2.^2);
%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculate the FSIM
%%%%%%%%%%%%%%%%%%%%%%%%%
T1 = 0.85; %fixed
T2 = 160; %fixed
PCSimMatrix = (2 * PC1 .* PC2 + T1) ./ (PC1.^2 + PC2.^2 + T1);
gradientSimMatrix = (2*gradientMap1.*gradientMap2 + T2) ./(gradientMap1.^2 + gradientMap2.^2 + T2);
PCm = max(PC1, PC2);
SimMatrix = gradientSimMatrix .* PCSimMatrix .* PCm;
FSIM = sum(sum(SimMatrix)) / sum(sum(PCm));
%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculate the FSIMc
%%%%%%%%%%%%%%%%%%%%%%%%%
T3 = 200;
T4 = 200;
ISimMatrix = (2 * I1 .* I2 + T3) ./ (I1.^2 + I2.^2 + T3);
QSimMatrix = (2 * Q1 .* Q2 + T4) ./ (Q1.^2 + Q2.^2 + T4);
lambda = 0.03;
SimMatrixC = gradientSimMatrix .* PCSimMatrix .* real((ISimMatrix .* QSimMatrix) .^ lambda) .* PCm;
FSIMc = sum(sum(SimMatrixC)) / sum(sum(PCm));
return;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [ResultPC]=phasecong2(im)
% ========================================================================
% Copyright (c) 1996-2009 Peter Kovesi
% School of Computer Science & Software Engineering
% The University of Western Australia
% http://www.csse.uwa.edu.au/
%
% Permission is hereby granted, free of charge, to any person obtaining a copy
% of this software and associated documentation files (the "Software"), to deal
% in the Software without restriction, subject to the following conditions:
%
% The above copyright notice and this permission notice shall be included in all
% copies or substantial portions of the Software.
%
% The software is provided "as is", without warranty of any kind.
% References:
%
% Peter Kovesi, "Image Features From Phase Congruency". Videre: A
% Journal of Computer Vision Research. MIT Press. Volume 1, Number 3,
% Summer 1999 http://mitpress.mit.edu/e-journals/Videre/001/v13.html
nscale = 4; % Number of wavelet scales.
norient = 4; % Number of filter orientations.
minWaveLength = 6; % Wavelength of smallest scale filter.
mult = 2; % Scaling factor between successive filters.
sigmaOnf = 0.55; % Ratio of the standard deviation of the
% Gaussian describing the log Gabor filter's
% transfer function in the frequency domain
% to the filter center frequency.
dThetaOnSigma = 1.2; % Ratio of angular interval between filter orientations
% and the standard deviation of the angular Gaussian
% function used to construct filters in the
% freq. plane.
k = 2.0; % No of standard deviations of the noise
% energy beyond the mean at which we set the
% noise threshold point.
% below which phase congruency values get
% penalized.
epsilon = .0001; % Used to prevent division by zero.
thetaSigma = pi/norient/dThetaOnSigma; % Calculate the standard deviation of the
% angular Gaussian function used to
% construct filters in the freq. plane.
[rows,cols] = size(im);
imagefft = fft2(im); % Fourier transform of image
zero = zeros(rows,cols);
EO = cell(nscale, norient); % Array of convolution results.
estMeanE2n = [];
ifftFilterArray = cell(1,nscale); % Array of inverse FFTs of filters
% Pre-compute some stuff to speed up filter construction
% Set up X and Y matrices with ranges normalised to +/- 0.5
% The following code adjusts things appropriately for odd and even values
% of rows and columns.
if mod(cols,2)
xrange = [-(cols-1)/2:(cols-1)/2]/(cols-1);
else
xrange = [-cols/2:(cols/2-1)]/cols;
end
if mod(rows,2)
yrange = [-(rows-1)/2:(rows-1)/2]/(rows-1);
else
yrange = [-rows/2:(rows/2-1)]/rows;
end
[x,y] = meshgrid(xrange, yrange);
radius = sqrt(x.^2 + y.^2); % Matrix values contain *normalised* radius from centre.
theta = atan2(-y,x); % Matrix values contain polar angle.
% (note -ve y is used to give +ve
% anti-clockwise angles)
radius = ifftshift(radius); % Quadrant shift radius and theta so that filters
theta = ifftshift(theta); % are constructed with 0 frequency at the corners.
radius(1,1) = 1; % Get rid of the 0 radius value at the 0
% frequency point (now at top-left corner)
% so that taking the log of the radius will
% not cause trouble.
sintheta = sin(theta);
costheta = cos(theta);
clear x; clear y; clear theta; % save a little memory
% Filters are constructed in terms of two components.
% 1) The radial component, which controls the frequency band that the filter
% responds to
% 2) The angular component, which controls the orientation that the filter
% responds to.
% The two components are multiplied together to construct the overall filter.
% Construct the radial filter components...
% First construct a low-pass filter that is as large as possible, yet falls
% away to zero at the boundaries. All log Gabor filters are multiplied by
% this to ensure no extra frequencies at the 'corners' of the FFT are
% incorporated as this seems to upset the normalisation process when
% calculating phase congrunecy.
lp = lowpassfilter([rows,cols],.45,15); % Radius .45, 'sharpness' 15
logGabor = cell(1,nscale);
for s = 1:nscale
wavelength = minWaveLength*mult^(s-1);
fo = 1.0/wavelength; % Centre frequency of filter.
logGabor{s} = exp((-(log(radius/fo)).^2) / (2 * log(sigmaOnf)^2));
logGabor{s} = logGabor{s}.*lp; % Apply low-pass filter
logGabor{s}(1,1) = 0; % Set the value at the 0 frequency point of the filter
% back to zero (undo the radius fudge).
end
% Then construct the angular filter components...
spread = cell(1,norient);
for o = 1:norient
angl = (o-1)*pi/norient; % Filter angle.
% For each point in the filter matrix calculate the angular distance from
% the specified filter orientation. To overcome the angular wrap-around
% problem sine difference and cosine difference values are first computed
% and then the atan2 function is used to determine angular distance.
ds = sintheta * cos(angl) - costheta * sin(angl); % Difference in sine.
dc = costheta * cos(angl) + sintheta * sin(angl); % Difference in cosine.
dtheta = abs(atan2(ds,dc)); % Absolute angular distance.
spread{o} = exp((-dtheta.^2) / (2 * thetaSigma^2)); % Calculate the
% angular filter component.
end
% The main loop...
EnergyAll(rows,cols) = 0;
AnAll(rows,cols) = 0;
for o = 1:norient % For each orientation.
sumE_ThisOrient = zero; % Initialize accumulator matrices.
sumO_ThisOrient = zero;
sumAn_ThisOrient = zero;
Energy = zero;
for s = 1:nscale, % For each scale.
filter = logGabor{s} .* spread{o}; % Multiply radial and angular
% components to get the filter.
ifftFilt = real(ifft2(filter))*sqrt(rows*cols); % Note rescaling to match power
ifftFilterArray{s} = ifftFilt; % record ifft2 of filter
% Convolve image with even and odd filters returning the result in EO
EO{s,o} = ifft2(imagefft .* filter);
An = abs(EO{s,o}); % Amplitude of even & odd filter response.
sumAn_ThisOrient = sumAn_ThisOrient + An; % Sum of amplitude responses.
sumE_ThisOrient = sumE_ThisOrient + real(EO{s,o}); % Sum of even filter convolution results.
sumO_ThisOrient = sumO_ThisOrient + imag(EO{s,o}); % Sum of odd filter convolution results.
if s==1 % Record mean squared filter value at smallest
EM_n = sum(sum(filter.^2)); % scale. This is used for noise estimation.
maxAn = An; % Record the maximum An over all scales.
else
maxAn = max(maxAn, An);
end
end % ... and process the next scale
% Get weighted mean filter response vector, this gives the weighted mean
% phase angle.
XEnergy = sqrt(sumE_ThisOrient.^2 + sumO_ThisOrient.^2) + epsilon;
MeanE = sumE_ThisOrient ./ XEnergy;
MeanO = sumO_ThisOrient ./ XEnergy;
% Now calculate An(cos(phase_deviation) - | sin(phase_deviation)) | by
% using dot and cross products between the weighted mean filter response
% vector and the individual filter response vectors at each scale. This
% quantity is phase congruency multiplied by An, which we call energy.
for s = 1:nscale,
E = real(EO{s,o}); O = imag(EO{s,o}); % Extract even and odd
% convolution results.
Energy = Energy + E.*MeanE + O.*MeanO - abs(E.*MeanO - O.*MeanE);
end
% Compensate for noise
% We estimate the noise power from the energy squared response at the
% smallest scale. If the noise is Gaussian the energy squared will have a
% Chi-squared 2DOF pdf. We calculate the median energy squared response
% as this is a robust statistic. From this we estimate the mean.
% The estimate of noise power is obtained by dividing the mean squared
% energy value by the mean squared filter value
medianE2n = median(reshape(abs(EO{1,o}).^2,1,rows*cols));
meanE2n = -medianE2n/log(0.5);
estMeanE2n(o) = meanE2n;
noisePower = meanE2n/EM_n; % Estimate of noise power.
% Now estimate the total energy^2 due to noise
% Estimate for sum(An^2) + sum(Ai.*Aj.*(cphi.*cphj + sphi.*sphj))
EstSumAn2 = zero;
for s = 1:nscale
EstSumAn2 = EstSumAn2 + ifftFilterArray{s}.^2;
end
EstSumAiAj = zero;
for si = 1:(nscale-1)
for sj = (si+1):nscale
EstSumAiAj = EstSumAiAj + ifftFilterArray{si}.*ifftFilterArray{sj};
end
end
sumEstSumAn2 = sum(sum(EstSumAn2));
sumEstSumAiAj = sum(sum(EstSumAiAj));
EstNoiseEnergy2 = 2*noisePower*sumEstSumAn2 + 4*noisePower*sumEstSumAiAj;
tau = sqrt(EstNoiseEnergy2/2); % Rayleigh parameter
EstNoiseEnergy = tau*sqrt(pi/2); % Expected value of noise energy
EstNoiseEnergySigma = sqrt( (2-pi/2)*tau^2 );
T = EstNoiseEnergy + k*EstNoiseEnergySigma; % Noise threshold
% The estimated noise effect calculated above is only valid for the PC_1 measure.
% The PC_2 measure does not lend itself readily to the same analysis. However
% empirically it seems that the noise effect is overestimated roughly by a factor
% of 1.7 for the filter parameters used here.
T = T/1.7; % Empirical rescaling of the estimated noise effect to
% suit the PC_2 phase congruency measure
Energy = max(Energy - T, zero); % Apply noise threshold
EnergyAll = EnergyAll + Energy;
AnAll = AnAll + sumAn_ThisOrient;
end % For each orientation
ResultPC = EnergyAll ./ AnAll;
return;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% LOWPASSFILTER - Constructs a low-pass butterworth filter.
%
% usage: f = lowpassfilter(sze, cutoff, n)
%
% where: sze is a two element vector specifying the size of filter
% to construct [rows cols].
% cutoff is the cutoff frequency of the filter 0 - 0.5
% n is the order of the filter, the higher n is the sharper
% the transition is. (n must be an integer >= 1).
% Note that n is doubled so that it is always an even integer.
%
% 1
% f = --------------------
% 2n
% 1.0 + (w/cutoff)
%
% The frequency origin of the returned filter is at the corners.
%
% See also: HIGHPASSFILTER, HIGHBOOSTFILTER, BANDPASSFILTER
%
% Copyright (c) 1999 Peter Kovesi
% School of Computer Science & Software Engineering
% The University of Western Australia
% http://www.csse.uwa.edu.au/
%
% Permission is hereby granted, free of charge, to any person obtaining a copy
% of this software and associated documentation files (the "Software"), to deal
% in the Software without restriction, subject to the following conditions:
%
% The above copyright notice and this permission notice shall be included in
% all copies or substantial portions of the Software.
%
% The Software is provided "as is", without warranty of any kind.
% October 1999
% August 2005 - Fixed up frequency ranges for odd and even sized filters
% (previous code was a bit approximate)
function f = lowpassfilter(sze, cutoff, n)
if cutoff < 0 || cutoff > 0.5
error('cutoff frequency must be between 0 and 0.5');
end
if rem(n,1) ~= 0 || n < 1
error('n must be an integer >= 1');
end
if length(sze) == 1
rows = sze; cols = sze;
else
rows = sze(1); cols = sze(2);
end
% Set up X and Y matrices with ranges normalised to +/- 0.5
% The following code adjusts things appropriately for odd and even values
% of rows and columns.
if mod(cols,2)
xrange = [-(cols-1)/2:(cols-1)/2]/(cols-1);
else
xrange = [-cols/2:(cols/2-1)]/cols;
end
if mod(rows,2)
yrange = [-(rows-1)/2:(rows-1)/2]/(rows-1);
else
yrange = [-rows/2:(rows/2-1)]/rows;
end
[x,y] = meshgrid(xrange, yrange);
radius = sqrt(x.^2 + y.^2); % A matrix with every pixel = radius relative to centre.
f = ifftshift( 1 ./ (1.0 + (radius ./ cutoff).^(2*n)) ); % The filter
return;