% 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