% PHASESYM - Function for computing phase symmetry on an image.
%
% This function calculates the phase symmetry of points in an image.
% This is a contrast invariant measure of symmetry.  This function can be
% used as a line and blob detector.  The greyscale 'polarity' of the lines
% that you want to find can be specified.
%
% There are potentially many arguments, here is the full usage:
%
%   [phaseSym, orientation, totalEnergy, T] = ...
%                phasesym(im, nscale, norient, minWaveLength, mult, ...
%                         sigmaOnf, k, polarity, noiseMethod)
%
% However, apart from the image, all parameters have defaults and the
% usage can be as simple as:
%
%    phaseSym = phasesym(im);
% 
% Arguments:
%              Default values      Description
%
%    nscale           5    - Number of wavelet scales, try values 3-6
%    norient          6    - Number of filter orientations.
%    minWaveLength    3    - Wavelength of smallest scale filter.
%    mult             2.1  - 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.
%    k                2.0  - No of standard deviations of the noise energy beyond
%                            the mean at which we set the noise threshold point.
%                            You may want to vary this up to a value of 10 or
%                            20 for noisy images 
%    polarity         0    - Controls 'polarity' of symmetry features to find.
%                             1 - just return 'bright' points
%                            -1 - just return 'dark' points
%                             0 - return bright and dark points.
%    noiseMethod      -1   - Parameter specifies method used to determine
%                            noise statistics. 
%                              -1 use median of smallest scale filter responses
%                              -2 use mode of smallest scale filter responses
%                               0+ use noiseMethod value as the fixed noise threshold.
%
% Return values:
%    phaseSym              - Phase symmetry image (values between 0 and 1).
%    orientation           - Orientation image. Orientation in which local
%                            symmetry energy is a maximum, in degrees
%                            (0-180), angles positive anti-clockwise. Note
%                            the orientation info is quantized by the number
%                            of orientations
%    totalEnergy           - Un-normalised raw symmetry energy which may be
%                            more to your liking.
%    T                     - Calculated noise threshold (can be useful for
%                            diagnosing noise characteristics of images).  Once you know
%                            this you can then specify fixed thresholds and save some
%                            computation time.
%
% Notes on specifying parameters:  
%
% The parameters can be specified as a full list eg.
%  >> phaseSym = phasesym(im, 5, 6, 3, 2.5, 0.55, 2.0, 0);
%
% or as a partial list with unspecified parameters taking on default values
%  >> phaseSym = phasesym(im, 5, 6, 3);
%
% or as a partial list of parameters followed by some parameters specified via a
% keyword-value pair, remaining parameters are set to defaults, for example:
%  >> phaseSym = phasesym(im, 5, 6, 3, 'polarity',-1, 'k', 2.5);
% 
% The convolutions are done via the FFT.  Many of the parameters relate to the
% specification of the filters in the frequency plane.  The values do not seem
% to be very critical and the defaults are usually fine.  You may want to
% experiment with the values of 'nscales' and 'k', the noise compensation factor.
%
% Notes on filter settings to obtain even coverage of the spectrum
% sigmaOnf       .85   mult 1.3
% sigmaOnf       .75   mult 1.6     (filter bandwidth ~1 octave)
% sigmaOnf       .65   mult 2.1  
% sigmaOnf       .55   mult 3       (filter bandwidth ~2 octaves)
%
% For maximum speed the input image should have dimensions that correspond to
% powers of 2, but the code will operate on images of arbitrary size.
%
% See Also:  PHASECONG, PHASECONG2, GABORCONVOLVE, PLOTGABORFILTERS

% References:
%     Peter Kovesi, "Symmetry and Asymmetry From Local Phase" AI'97, Tenth
%     Australian Joint Conference on Artificial Intelligence. 2 - 4 December
%     1997. http://www.cs.uwa.edu.au/pub/robvis/papers/pk/ai97.ps.gz.
%
%     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

% April 1996     Original Version written 
% August 1998    Noise compensation corrected. 
% October 1998   Noise compensation corrected.   - Again!!!
% September 1999 Modified to operate on non-square images of arbitrary size. 
% February 2001  Specialised from phasecong.m to calculate phase symmetry 
% July 2005      Better argument handling + general cleanup and speed improvements
% August 2005    Made Octave compatible.
% January 2007   Small correction and cleanup of radius calculation for odd
%                image sizes.
% May 2009       Noise compensation simplified reducing memory and
%                computation overhead.  Spread function changed to a cosine
%                eliminating parameter dThetaOnSigma and ensuring even
%                angular coverage. 
% September 2017 Changed to use FILTERGRID

% Copyright (c) 1996-2017 Peter Kovesi
% http://www.peterkovesi.com
% 
% 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.

function[phaseSym, orientation, totalEnergy, T] = phasesym(varargin)

    % Get arguments and/or default values    
    [im, nscale, norient, minWaveLength, mult, sigmaOnf, k, ...
     polarity, noiseMethod] = checkargs(varargin(:));  

    epsilon         = 1e-4;             % Used to prevent division by zero.
    [rows,cols] = size(im);
    imagefft = fft2(im);                % Fourier transform of image
    zero = zeros(rows,cols);
    
    totalEnergy = zero;                 % Matrix for accumulating weighted phase 
                                        % congruency values (energy).
    totalSumAn  = zero;                 % Matrix for accumulating filter response
                                        % amplitude values.
    orientation = zero;                 % Matrix storing orientation with greatest
                                        % energy for each pixel.

    % Pre-compute data for filter construction
    [radius, x, y] = filtergrid(rows,cols);
    theta = atan2(-y,x);              % Matrix values contain polar angle.
                                      % (note -ve y is used to give +ve
                                      % anti-clockwise angles)
                                      
    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],.4,10);   % Radius .4, 'sharpness' 10

    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

    %% The main loop...

    for o = 1:norient                     % For each orientation....
        % Construct the angular filter spread function        
        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.
        % Scale theta so that cosine spread function has the right wavelength
        % and clamp to pi.
        dtheta = min(dtheta*norient/2,pi); 
        % The spread function is cos(dtheta) between -pi and pi.  We add 1,
        % and then divide by 2 so that the value ranges 0-1
        spread = (cos(dtheta)+1)/2;              
        
        sumAn_ThisOrient  = zero;      
        Energy_ThisOrient = zero;      

        for s = 1:nscale                  % For each scale....
            filter = logGabor{s} .* spread;     % Multiply radial and angular
                                                % components to get filter.

            % Convolve image with even and odd filters returning the result in EO
            EO = ifft2(imagefft .* filter);
            An = abs(EO);                             % Amplitude of even & odd filter response.
            sumAn_ThisOrient = sumAn_ThisOrient + An; % Sum of amplitude responses.

            % At the smallest scale estimate noise characteristics from the
            % distribution of the filter amplitude responses stored in sumAn. 
            % tau is the Rayleigh parameter that is used to describe the
            % distribution.
            if s == 1 
                if noiseMethod == -1     % Use median to estimate noise statistics
                    tau = median(sumAn_ThisOrient(:))/sqrt(log(4));   
                elseif noiseMethod == -2 % Use mode to estimate noise statistics
                    tau = rayleighmode(sumAn_ThisOrient(:));
                end
            end

            % Now calculate the phase symmetry measure.
            if polarity == 0       % look for 'white' and 'black' spots
                Energy_ThisOrient = Energy_ThisOrient ...
                    + abs(real(EO)) - abs(imag(EO));
                
            elseif polarity == 1   % Just look for 'white' spots
                Energy_ThisOrient = Energy_ThisOrient ...
                    + real(EO) - abs(imag(EO));
                
            elseif polarity == -1  % Just look for 'black' spots
                Energy_ThisOrient = Energy_ThisOrient ...
                    - real(EO) - abs(imag(EO));
            end

        end                                 % ... and process the next scale        

        %% Automatically determine noise threshold
        %
        % Assuming the noise is Gaussian the response of the filters to noise will
        % form Rayleigh distribution.  We use the filter responses at the smallest
        % scale as a guide to the underlying noise level because the smallest scale
        % filters spend most of their time responding to noise, and only
        % occasionally responding to features. Either the median, or the mode, of
        % the distribution of filter responses can be used as a robust statistic to
        % estimate the distribution mean and standard deviation as these are related
        % to the median or mode by fixed constants.  The response of the larger
        % scale filters to noise can then be estimated from the smallest scale
        % filter response according to their relative bandwidths.
        %
        % This code assumes that the expected reponse to noise on the phase congruency
        % calculation is simply the sum of the expected noise responses of each of
        % the filters.  This is a simplistic overestimate, however these two
        % quantities should be related by some constant that will depend on the
        % filter bank being used.  Appropriate tuning of the parameter 'k' will
        % allow you to produce the desired output. 
        
        if noiseMethod >= 0     % We are using a fixed noise threshold
            T = noiseMethod;    % use supplied noiseMethod value as the threshold
        else
            % Estimate the effect of noise on the sum of the filter responses as
            % the sum of estimated individual responses (this is a simplistic
            % overestimate). As the estimated noise response at succesive scales
            % is scaled inversely proportional to bandwidth we have a simple
            % geometric sum.
            totalTau = tau * (1 - (1/mult)^nscale)/(1-(1/mult));
            
            % Calculate mean and std dev from tau using fixed relationship
            % between these parameters and tau. See
            % http://mathworld.wolfram.com/RayleighDistribution.html
            EstNoiseEnergyMean = totalTau*sqrt(pi/2);        % Expected mean and std
            EstNoiseEnergySigma = totalTau*sqrt((4-pi)/2);   % values of noise energy
            
            % Noise threshold, make sure it is not less than epsilon.
            T =  max(EstNoiseEnergyMean + k*EstNoiseEnergySigma, epsilon);
        end

        % Apply noise threshold,  this is effectively wavelet denoising via
        % soft thresholding.  Note 'Energy_ThisOrient' will have -ve values.
        % These will be floored out at the final normalization stage.
        Energy_ThisOrient = Energy_ThisOrient - T;
        
        % Update accumulator matrix for sumAn and totalEnergy
        totalSumAn  = totalSumAn + sumAn_ThisOrient;
        totalEnergy = totalEnergy + Energy_ThisOrient;
        
        % Update orientation matrix by finding image points where the energy in
        % this orientation is greater than in any previous orientation (the
        % change matrix) and then replacing these elements in the orientation
        % matrix with the current orientation number.
        
        if(o == 1),
            maxEnergy = Energy_ThisOrient;
        else
            change = Energy_ThisOrient > maxEnergy;
            orientation = (o - 1).*change + orientation.*(~change);
            maxEnergy = max(maxEnergy, Energy_ThisOrient);
        end
        
    end  % For each orientation
    
    % Normalize totalEnergy by the totalSumAn to obtain phase symmetry
    % totalEnergy is floored at 0 to eliminate -ve values
    phaseSym = max(totalEnergy, 0) ./ (totalSumAn + epsilon);
    
    % Convert orientation matrix values to degrees
    orientation = fix(orientation * (180 / norient));
    
    
%------------------------------------------------------------------
% CHECKARGS
%
% Function to process the arguments that have been supplied, assign
% default values as needed and perform basic checks.
    
function [im, nscale, norient, minWaveLength, mult, sigmaOnf, ...
          k, polarity, noiseMethod] = checkargs(arg)

    nargs = length(arg);
    
    if nargs < 1
        error('No image supplied as an argument');
    end    
    
    % Set up default values for all arguments and then overwrite them
    % with with any new values that may be supplied
    im              = [];
    nscale          = 5;     % Number of wavelet scales.    
    norient         = 6;     % Number of filter orientations.
    minWaveLength   = 3;     % Wavelength of smallest scale filter.    
    mult            = 2.1;   % 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.    
    k               = 2.0;   % No of standard deviations of the noise
                             % energy beyond the mean at which we set the
                             % noise threshold point. 

    polarity        = 0;     % Look for both black and white spots of symmetrry
    noiseMethod     = -1;    % Use median response of smallest scale filter
                             % to estimate noise characteristics.
    
    % Allowed argument reading states
    allnumeric   = 1;       % Numeric argument values in predefined order
    keywordvalue = 2;       % Arguments in the form of string keyword
                            % followed by numeric value
    readstate = allnumeric; % Start in the allnumeric state
    
    if readstate == allnumeric
        for n = 1:nargs
            if isa(arg{n},'char')
                readstate = keywordvalue;
                break;
            else
                if     n == 1, im            = arg{n}; 
                elseif n == 2, nscale        = arg{n};              
                elseif n == 3, norient       = arg{n};
                elseif n == 4, minWaveLength = arg{n};
                elseif n == 5, mult          = arg{n};
                elseif n == 6, sigmaOnf      = arg{n};
                elseif n == 7, k             = arg{n};              
                elseif n == 8, polarity      = arg{n};                              
                elseif n == 9,noiseMethod   = arg{n};                  
                end
            end
        end
    end

    % Code to handle parameter name - value pairs
    if readstate == keywordvalue
        while n < nargs
            
            if ~isa(arg{n},'char') || ~isa(arg{n+1}, 'double')
                error('There should be a parameter name - value pair');
            end
            
            if     strncmpi(arg{n},'im'      ,2), im =        arg{n+1};
            elseif strncmpi(arg{n},'nscale'  ,2), nscale =    arg{n+1};
            elseif strncmpi(arg{n},'norient' ,4), norient =   arg{n+1};
            elseif strncmpi(arg{n},'minWaveLength',2), minWaveLength = arg{n+1};
            elseif strncmpi(arg{n},'mult'    ,2), mult =      arg{n+1};
            elseif strncmpi(arg{n},'sigmaOnf',2), sigmaOnf =  arg{n+1};
            elseif strncmpi(arg{n},'k'       ,1), k =         arg{n+1};
            elseif strncmpi(arg{n},'polarity',2), polarity =  arg{n+1};
            elseif strncmpi(arg{n},'noiseMethod',4), noiseMethod =  arg{n+1};                
            else   error('Unrecognised parameter name');
            end

            n = n+2;
            if n == nargs
                error('Unmatched parameter name - value pair');
            end
        end
    end
    
    if isempty(im)
        error('No image argument supplied');
    end

    if ~isa(im, 'double')
        im = double(im);
    end
    
    if nscale < 1
        error('nscale must be an integer >= 1');
    end
    
    if norient < 1 
        error('norient must be an integer >= 1');
    end    

    if minWaveLength < 2
        error('It makes little sense to have a wavelength < 2');
    end          
        
    if polarity ~= -1 && polarity ~= 0 && polarity ~= 1
        error('Allowed polarity values are -1, 0 and 1')
    end
    

%%-------------------------------------------------------------------------
% RAYLEIGHMODE
%
% Computes mode of a vector/matrix of data that is assumed to come from a
% Rayleigh distribution.
%
% Usage:  rmode = rayleighmode(data, nbins)
%
% Arguments:  data  - data assumed to come from a Rayleigh distribution
%             nbins - Optional number of bins to use when forming histogram
%                     of the data to determine the mode.
%
% Mode is computed by forming a histogram of the data over 50 bins and then
% finding the maximum value in the histogram.  Mean and standard deviation
% can then be calculated from the mode as they are related by fixed
% constants.
%
% mean = mode * sqrt(pi/2)
% std dev = mode * sqrt((4-pi)/2)
% 
% See
% http://mathworld.wolfram.com/RayleighDistribution.html
% http://en.wikipedia.org/wiki/Rayleigh_distribution
%

function rmode = rayleighmode(data, nbins)
    
    if nargin == 1
        nbins = 50;  % Default number of histogram bins to use
    end

    mx = max(data(:));
    edges = 0:mx/nbins:mx;
    n = histc(data(:),edges); 
    [dum,ind] = max(n); % Find maximum and index of maximum in histogram 

    rmode = (edges(ind)+edges(ind+1))/2;

