% 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
% Copyright (c) 1996-2009 Peter Kovesi
% School of Computer Science & Software Engineering
% The University of Western Australia
% pk at csse uwa edu au
% 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[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.
% Pre-compute some stuff to speed up filter construction
%
% Set up u1 and u2 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
[u1,u2] = meshgrid(xrange, yrange);
u1 = ifftshift(u1); % Quadrant shift to put 0 frequency at the corners
u2 = ifftshift(u2);
radius = sqrt(u1.^2 + u2.^2); % Matrix values contain frequency
% values as a radius from centre
% (but quadrant shifted)
% 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;