% PHASESYMMONO - phase symmetry of an image using monogenic filters % % 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. % % This code is considerably faster than PHASESYM but you may prefer the % output from PHASESYM's oriented filters. % % There are potentially many arguments, here is the full usage: % % [phaseSym, symmetryEnergy, T] = ... % phasesymmono(im, nscale, minWaveLength, mult, ... % sigmaOnf, k, polarity, noiseMethod) % % However, apart from the image, all parameters have defaults and the % usage can be as simple as: % % phaseSym = phasesymmono(im); % % Arguments: % Default values Description % % nscale 5 - Number of wavelet scales, try values 3-6 % 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 % A value of 0 will turn off all noise compensation. % % Return values: % phaseSym - Phase symmetry image (values between 0 and 1). % symmetryEnergy - 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) % % % Notes on specifying parameters: % % The parameters can be specified as a full list eg. % >> phaseSym = phasesym(im, 5, 3, 2.5, 0.55, 2.0, 0); % % or as a partial list with unspecified parameters taking on default values % >> phaseSym = phasesym(im, 5, 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, 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: PHASESYM, PHASECONGMONO % 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 % % Michael Felsberg and Gerald Sommer, "A New Extension of Linear Signal % Processing for Estimating Local Properties and Detecting Features". DAGM % Symposium 2000, Kiel % % Michael Felsberg and Gerald Sommer. "The Monogenic Signal" IEEE % Transactions on Signal Processing, 49(12):3136-3144, December 2001 % July 2008 Code developed from phasesym where local phase information % calculated using Monogenic Filters. % April 2009 Noise compensation simplified to speedup execution. % Options to calculate noise statistics via median or mode of % smallest filter response. Option to use a fixed threshold. % Return of T for 'instrumentation' of noise detection/compensation. % Removal of orientation calculation from phasesym (not clear % how best to calculate this from monogenic filter outputs) % June 2009 Clean up % Sept 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, symmetryEnergy, T] = phasesymmono(varargin) % Get arguments and/or default values [im, nscale, minWaveLength, mult, sigmaOnf, k, ... polarity, noiseMethod] = checkargs(varargin(:)); epsilon = .0001; % Used to prevent division by zero. [rows,cols] = size(im); IM = fft2(im); % Fourier transform of image zero = zeros(rows,cols); symmetryEnergy = zero; % Matrix for accumulating weighted phase % symmetry values (energy). sumAn = zero; % Matrix for accumulating filter response % amplitude values. % Set up data for filter construction [radius, u1, u2] = filtergrid(rows,cols); % Get rid of the 0 radius value in the middle (at top left corner after % fftshifting) so that taking the log of the radius, or dividing by the % radius, will not cause trouble. radius(1,1) = 1; % Construct the monogenic filters in the frequency domain. The two % filters would normally be constructed as follows % H1 = i*u1./radius; % H2 = i*u2./radius; % However the two filters can be packed together as a complex valued % matrix, one in the real part and one in the imaginary part. Do this by % multiplying H2 by i and then adding it to H1 (note the subtraction % because i*i = -1). When the convolution is performed via the fft the % real part of the result will correspond to the convolution with H1 and % the imaginary part with H2. This allows the two convolutions to be % done as one in the frequency domain, saving time and memory. H = (i*u1 - u2)./radius; % The two monogenic filters H1 and H2 are not selective in terms of the % magnitudes of the frequencies. The code below generates bandpass % log-Gabor filters which are point-wise multiplied by IM to produce % different bandpass versions of the image before being convolved with H1 % and H2 % First construct a low-pass filter that is as large as possible, yet falls % away to zero at the boundaries. All filters are multiplied by % this to ensure no extra frequencies at the 'corners' of the FFT are % incorporated as this can upset the normalisation process when % calculating phase symmetry lp = lowpassfilter([rows,cols],.4,10); % Radius .4, 'sharpness' 10 for s = 1:nscale wavelength = minWaveLength*mult^(s-1); fo = 1.0/wavelength; % Centre frequency of filter. logGabor = exp((-(log(radius/fo)).^2) / (2 * log(sigmaOnf)^2)); logGabor = logGabor.*lp; % Apply low-pass filter logGabor(1,1) = 0; % Set the value at the 0 frequency point of the filter % back to zero (undo the radius fudge). IMF = IM.*logGabor; % Bandpassed image in the frequency domain f = real(ifft2(IMF)); % Bandpassed image in spatial domain h = ifft2(IMF.*H); % Bandpassed monogenic filtering, real part of h contains % convolution result with h1, imaginary part % contains convolution result with h2. hAmp2 = real(h).^2 + imag(h).^2; % Squared amplitude of h1 h2 filter results sumAn = sumAn + sqrt(f.^2 + hAmp2); % Magnitude of Energy. % Now calculate the phase symmetry measure. if polarity == 0 % look for 'white' and 'black' spots symmetryEnergy = symmetryEnergy + abs(f) - sqrt(hAmp2); elseif polarity == 1 % Just look for 'white' spots symmetryEnergy = symmetryEnergy + f - sqrt(hAmp2); elseif polarity == -1 % Just look for 'black' spots symmetryEnergy = symmetryEnergy - f - sqrt(hAmp2); end % 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 specify the % distribution. if s == 1 if noiseMethod == -1 % Use median to estimate noise statistics tau = median(sumAn(:))/sqrt(log(4)); elseif noiseMethod == -2 % Use mode to estimate noise statistics tau = rayleighmode(sumAn(:)); end end end % For each scale % Compensate for noise % % 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 symmetry % 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. (though the value of k seems to % be not at all critical) 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 - effectively wavelet denoising soft thresholding % and normalize symmetryEnergy by the sumAn to obtain phase symmetry. % Note the max operation is not necessary if you are after speed, it is % just 'tidy' not having -ve symmetry values phaseSym = max(symmetryEnergy-T, zero) ./ (sumAn + epsilon); %------------------------------------------------------------------ % CHECKARGS % % Function to process the arguments that have been supplied, assign % default values as needed and perform basic checks. function [im, nscale, 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. 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 symmetry noiseMethod = -1; % Use the median response of smallest scale % filter to estimate noise statistics % 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, minWaveLength = arg{n}; elseif n == 4, mult = arg{n}; elseif n == 5, sigmaOnf = arg{n}; elseif n == 6, k = arg{n}; elseif n == 7, polarity = arg{n}; elseif n == 8, 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},'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',3), 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 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;