% PHASECONG2 - Computes edge and corner phase congruency in an image.
%
% This function calculates the PC_2 measure of phase congruency.
% This function supersedes PHASECONG
%
% There are potentially many arguments, here is the full usage:
%
% [M m or ft pc EO] = phasecong2(im, nscale, norient, minWaveLength, ...
% mult, sigmaOnf, dThetaOnSigma, k, cutOff, g)
%
% However, apart from the image, all parameters have defaults and the
% usage can be as simple as:
%
% M = phasecong2(im);
%
% Arguments:
% Default values Description
%
% nscale 4 - 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.
% 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.
% You may want to vary this up to a value of 10 or
% 20 for noisy images
% cutOff 0.5 - The fractional measure of frequency spread
% below which phase congruency values get penalized.
% g 10 - Controls the sharpness of the transition in
% the sigmoid function used to weight phase
% congruency for frequency spread.
%
% Returned values:
% M - Maximum moment of phase congruency covariance.
% This is used as a indicator of edge strength.
% m - Minimum moment of phase congruency covariance.
% This is used as a indicator of corner strength.
% or - Orientation image in integer degrees 0-180,
% positive anticlockwise.
% 0 corresponds to a vertical edge, 90 is horizontal.
% ft - *Not correctly implemented at this stage*
% A complex valued image giving the weighted mean
% phase angle at every point in the image for each
% orientation.
% pc - Cell array of phase congruency images (values between 0 and 1)
% for each orientation
% EO - A 2D cell array of complex valued convolution results
%
% EO{s,o} = convolution result for scale s and orientation o. The real part
% is the result of convolving with the even symmetric filter, the imaginary
% part is the result from convolution with the odd symmetric filter.
%
% Hence:
% abs(EO{s,o}) returns the magnitude of the convolution over the
% image at scale s and orientation o.
% angle(EO{s,o}) returns the phase angles.
%
% Notes on specifying parameters:
%
% The parameters can be specified as a full list eg.
% >> [M m or ft pc EO] = phasecong2(im, 5, 6, 3, 2.5, 0.55, 1.2, 2.0, 0.4, 10);
%
% or as a partial list with unspecified parameters taking on default values
% >> [M m or ft pc EO] = phasecong2(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:
% >> [M m or ft pc EO] = phasecong2(im, 5, 6, 3, 'cutOff', 0.3, '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
% dthetaOnSigma 1.2 norient 6
% 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, PHASESYM, GABORCONVOLVE, PLOTGABORFILTERS
% 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
%
% Peter Kovesi, "Phase Congruency Detects Corners and
% Edges". Proceedings DICTA 2003, Sydney Dec 10-12
% 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.
% May 2001 Modified to return feature type image.
% July 2003 Altered to calculate 'corner' points.
% October 2003 Speed improvements and refinements.
% July 2005 Better argument handling, changed order of return values
% August 2005 Made Octave compatible
% May 2006 Bug in checkargs fixed
% Jan 2007 Bug in setting radius to 0 for odd sized images fixed.
% April 2009 Scaling of covariance values fixed. (Negligible change to results)
% 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.
function [M, m, or, featType, PC, EO]=phasecong2(varargin)
% Get arguments and/or default values
[im, nscale, norient, minWaveLength, mult, sigmaOnf, ...
dThetaOnSigma,k, cutOff, g] = checkargs(varargin(:));
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);
totalEnergy = zero; % Total weighted phase congruency values (energy).
totalSumAn = zero; % Total filter response amplitude values.
orientation = zero; % Matrix storing orientation with greatest
% energy for each pixel.
EO = cell(nscale, norient); % Array of convolution results.
covx2 = zero; % Matrices for covariance data
covy2 = zero;
covxy = zero;
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...
for o = 1:norient % For each orientation.
% fprintf('Processing orientation %d\r',o);
angl = (o-1)*pi/norient; % Filter angle.
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.
% if o == 1 % accumulate filter info for noise compensation (nominally the same
% for all orientations, hence it is only done once)
ifftFilt = real(ifft2(filter))*sqrt(rows*cols); % Note rescaling to match power
ifftFilterArray{s} = ifftFilt; % record ifft2 of filter
% end
% 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.
% if o == 1
% 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));
% end % if o == 1
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
% Form weighting that penalizes frequency distributions that are
% particularly narrow. Calculate fractional 'width' of the frequencies
% present by taking the sum of the filter response amplitudes and dividing
% by the maximum amplitude at each point on the image.
width = sumAn_ThisOrient ./ (maxAn + epsilon) / nscale;
% Now calculate the sigmoidal weighting function for this orientation.
weight = 1.0 ./ (1 + exp( (cutOff - width)*g));
% Apply weighting to energy and then calculate phase congruency
PC{o} = weight.*Energy./sumAn_ThisOrient; % Phase congruency for this orientation
featType{o} = E+i*O;
% Build up covariance data for every point
covx = PC{o}*cos(angl);
covy = PC{o}*sin(angl);
covx2 = covx2 + covx.^2;
covy2 = covy2 + covy.^2;
covxy = covxy + covx.*covy;
end % For each orientation
fprintf(' \r');
% Edge and Corner calculations
% The following is optimised code to calculate principal vector
% of the phase congruency covariance data and to calculate
% the minimumum and maximum moments - these correspond to
% the singular values.
% First normalise covariance values by the number of orientations/2
covx2 = covx2/(norient/2);
covy2 = covy2/(norient/2);
covxy = 4*covxy/norient; % This gives us 2*covxy/(norient/2)
denom = sqrt(covxy.^2 + (covx2-covy2).^2)+epsilon;
sin2theta = covxy./denom;
cos2theta = (covx2-covy2)./denom;
or = atan2(sin2theta,cos2theta)/2; % Orientation perpendicular to edge.
or = round(or*180/pi); % Return result rounded to integer
% degrees.
neg = or < 0;
or = ~neg.*or + neg.*(or+180); % Adjust range from -90 to 90
% to 0 to 180.
M = (covy2+covx2 + denom)/2; % Maximum moment
m = (covy2+covx2 - denom)/2; % ... and minimum moment
%------------------------------------------------------------------
% 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, ...
dThetaOnSigma,k, cutOff, g] = 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 = 4; % 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.
dThetaOnSigma = 1.5; % 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.
cutOff = 0.5; % The fractional measure of frequency spread
% below which phase congruency values get penalized.
g = 10; % Controls the sharpness of the transition in
% the sigmoid function used to weight phase
% congruency for frequency spread.
% 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, dThetaOnSigma = arg{n};
elseif n == 8, k = arg{n};
elseif n == 9, cutOff = arg{n};
elseif n == 10,g = 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' ,2), 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},'dThetaOnSigma',2), dThetaOnSigma = arg{n+1};
elseif strncmpi(arg{n},'k' ,1), k = arg{n+1};
elseif strncmpi(arg{n},'cutOff' ,2), cutOff = arg{n+1};
elseif strncmpi(arg{n},'g' ,1), g = 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 cutOff < 0 || cutOff > 1
error('Cut off value must be between 0 and 1');
end