% PHASECONG - Computes phase congruency on an image.
%
% Usage: [pc or ft] = phasecong(im)
%
% This function calculates the PC_2 measure of phase congruency.
% For maximum speed the input image should be square and have a
% size that is a power of 2, but the code will operate on images
% of arbitrary size.
%
%
% Returned values:
% pc - Phase congruency image (values between 0 and 1)
% or - Orientation image. Provides orientation in which
% local energy is a maximum in in degrees (0-180),
% angles positive anti-clockwise.
% ft - A complex valued image giving the weighted mean
% phase angle at every point in the image for the
% orientation having maximum energy. Use the
% function DISPFEAT to display this data.
%
% Parameters:
% im - A greyscale image to be processed.
%
% You can also specify numerous optional parameters. See the code to find
% out what they are. The convolutions are done via the FFT. Many of the
% parameters relate to the specification of the filters in the frequency
% plane. Default values for parameters are set within the file rather than
% being required as arguments because they rarely need to be changed - nor
% are they very critical. However, you may want to experiment with
% specifying/editing the values of `nscales' and `noiseCompFactor'.
%
% Note this phase congruency code is very computationally expensive and uses
% *lots* of memory.
%
%
% Example MATLAB session:
%
% >> im = imread('picci.tif');
% >> image(im); % Display the image
% >> [pc or ft] = phasecong(im);
% >> imagesc(pc), colormap(gray); % Display the phase congruency image
%
%
% To convert the phase congruency image to an edge map (with my usual parameters):
%
% >> nm = nonmaxsup(pc, or, 1.5); % Non-maxima suppression.
% The parameter 1.5 can result in edges more than 1 pixel wide but helps
% in picking up `broad' maxima.
% >> edgim = hysthresh(nm, 0.4, 0.2); % Hysteresis thresholding.
% >> edgeim = bwmorph(edgim,'skel',Inf); % Skeletonize the edgemap to fix
% % the non-maximal suppression.
% >> imagesc(edgeim), colormap(gray);
%
%
% To display the different feature types present in your image use:
%
% >> dispfeat(ft,edgim);
%
% With a small amount of editing the code can be modified to calculate
% a dimensionless measure of local symmetry in the image. The basis
% of this is that one looks for points in the image where the local
% phase is 90 or 270 degrees (the symmetric points in the cycle).
% Editing instructions are within the code.
%
% Notes on filter settings to obtain even coverage of the spectrum
% dthetaOnSigma 1.5
% sigmaOnf .85 mult 1.3
% sigmaOnf .75 mult 1.6 (bandwidth ~1 octave)
% sigmaOnf .65 mult 2.1
% sigmaOnf .55 mult 3 (bandwidth ~2 octaves)
%
% 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
% Copyright (c) 1996-2005 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.
% Original Version written April 1996
% Noise compensation corrected. August 1998
% Noise compensation corrected. October 1998 - Again!!!
% Modified to operate on non-square images of arbitrary size. September 1999
% Modified to return feature type image. May 2001
function[phaseCongruency,orientation, featType]=phasecong(im, nscale, norient, ...
minWaveLength, mult, ...
sigmaOnf, dThetaOnSigma, ...
k, cutOff)
sze = size(im);
if nargin < 2
nscale = 3; % Number of wavelet scales.
end
if nargin < 3
norient = 6; % Number of filter orientations.
end
if nargin < 4
minWaveLength = 3; % Wavelength of smallest scale filter.
end
if nargin < 5
mult = 2; % Scaling factor between successive filters.
end
if nargin < 6
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.
end
if nargin < 7
dThetaOnSigma = 1.7; % Ratio of angular interval between filter orientations
% and the standard deviation of the angular Gaussian
% function used to construct filters in the
% freq. plane.
end
if nargin < 8
k = 3.0; % No of standard deviations of the noise energy beyond the
% mean at which we set the noise threshold point.
% standard deviation to its maximum effect
% on Energy.
end
if nargin < 9
cutOff = 0.4; % The fractional measure of frequency spread
% below which phase congruency values get penalized.
end
g = 10; % Controls the sharpness of the transition in the sigmoid
% function used to weight phase congruency for frequency
% spread.
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.
imagefft = fft2(im); % Fourier transform of image
sze = size(imagefft);
rows = sze(1);
cols = sze(2);
zero = zeros(sze);
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.
estMeanE2n = [];
% Pre-compute some stuff to speed up filter construction
x = ones(rows,1) * (-cols/2 : (cols/2 - 1))/(cols/2);
y = (-rows/2 : (rows/2 - 1))' * ones(1,cols)/(rows/2);
radius = sqrt(x.^2 + y.^2); % Matrix values contain *normalised* radius from centre.
radius(round(rows/2+1),round(cols/2+1)) = 1; % Get rid of the 0 radius value in the middle
% so that taking the log of the radius will
% not cause trouble.
theta = atan2(-y,x); % Matrix values contain polar angle.
% (note -ve y is used to give +ve
% anti-clockwise angles)
sintheta = sin(theta);
costheta = cos(theta);
clear x; clear y; clear theta; % save a little memory
% The main loop...
for o = 1:norient, % For each orientation.
disp(['Processing orientation ' num2str(o)]);
angl = (o-1)*pi/norient; % Calculate filter angle.
wavelength = minWaveLength; % Initialize filter wavelength.
sumE_ThisOrient = zero; % Initialize accumulator matrices.
sumO_ThisOrient = zero;
sumAn_ThisOrient = zero;
Energy_ThisOrient = zero;
EOArray = []; % Array of complex convolution images - one for each scale.
ifftFilterArray = []; % Array of inverse FFTs of filters
% Pre-compute filter data specific to this orientation
% 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 = exp((-dtheta.^2) / (2 * thetaSigma^2)); % Calculate the angular filter component.
for s = 1:nscale, % For each scale.
% Construct the filter - first calculate the radial filter component.
fo = 1.0/wavelength; % Centre frequency of filter.
rfo = fo/0.5; % Normalised radius from centre of frequency plane
% corresponding to fo.
logGabor = exp((-(log(radius/rfo)).^2) / (2 * log(sigmaOnf)^2));
logGabor(round(rows/2+1),round(cols/2+1)) = 0; % Set the value at the center of the filter
% back to zero (undo the radius fudge).
filter = logGabor .* spread; % Multiply by the angular spread to get the filter.
filter = fftshift(filter); % Swap quadrants to move zero frequency
% to the corners.
ifftFilt = real(ifft2(filter))*sqrt(rows*cols); % Note rescaling to match power
ifftFilterArray = [ifftFilterArray ifftFilt]; % record ifft2 of filter
% Convolve image with even and odd filters returning the result in EO
EOfft = imagefft .* filter; % Do the convolution.
EO = ifft2(EOfft); % Back transform.
EOArray = [EOArray, EO]; % Record convolution result
An = abs(EO); % Amplitude of even & odd filter response.
sumAn_ThisOrient = sumAn_ThisOrient + An; % Sum of amplitude responses.
sumE_ThisOrient = sumE_ThisOrient + real(EO); % Sum of even filter convolution results.
sumO_ThisOrient = sumO_ThisOrient + imag(EO); % Sum of odd filter convolution results.
if s == 1 % Record the maximum An over all scales
maxAn = An;
else
maxAn = max(maxAn, An);
end
if s==1
EM_n = sum(sum(filter.^2)); % Record mean squared filter value at smallest
end % scale. This is used for noise estimation.
wavelength = wavelength * mult; % Finally calculate Wavelength of next filter
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,
EO = submat(EOArray,s,cols); % Extract even and odd filter
E = real(EO); O = imag(EO);
Energy_ThisOrient = Energy_ThisOrient ...
+ E.*MeanE + O.*MeanO - abs(E.*MeanO - O.*MeanE);
end
% Note: To calculate the phase symmetry measure replace the for loop above
% with the following loop. (The calculation of MeanE, MeanO, sumE_ThisOrient
% and sumO_ThisOrient can also be omitted). It is suggested that the value
% of nscale is increased (to say, 5 for a 256x256 image) and that cutOff is
% set to 0 to eliminate weighting for frequency spread.
% for s = 1:nscale,
% Energy_ThisOrient = Energy_ThisOrient ...
% + abs(real(submat(EOArray,s,cols))) - abs(imag(submat(EOArray,s,cols)));
% 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(submat(EOArray,1,cols)).^2,1,rows*cols));
meanE2n = -medianE2n/log(0.5);
estMeanE2n = [estMeanE2n meanE2n];
noisePower = meanE2n/EM_n; % Estimate of noise power.
% 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+submat(ifftFilterArray,s,cols).^2;
end
EstSumAiAj = zero;
for si = 1:(nscale-1)
for sj = (si+1):nscale
EstSumAiAj = EstSumAiAj + submat(ifftFilterArray,si,cols).*submat(ifftFilterArray,sj,cols);
end
end
EstNoiseEnergy2 = 2*noisePower*sum(sum(EstSumAn2)) + 4*noisePower*sum(sum(EstSumAiAj));
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_ThisOrient = max(Energy_ThisOrient - 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
Energy_ThisOrient = weight.*Energy_ThisOrient;
% 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;
featType = E + i*O;
else
change = Energy_ThisOrient > maxEnergy;
orientation = (o - 1).*change + orientation.*(~change);
featType = (E+i*O).*change + featType.*(~change);
maxEnergy = max(maxEnergy, Energy_ThisOrient);
end
end % For each orientation
disp('Mean Energy squared values recorded with smallest scale filter at each orientation');
disp(estMeanE2n);
% Display results
%imagesc(totalEnergy), axis image, title('total energy');
%disp('Hit any key to continue '); pause
%imagesc(totalSumAn), axis image, title('total sumAn');
%disp('Hit any key to continue '); pause
% Normalize totalEnergy by the totalSumAn to obtain phase congruency
phaseCongruency = totalEnergy ./ (totalSumAn + epsilon);
%imagesc(phaseCongruency), axis image, title('phase congruency');
% Convert orientation matrix values to degrees
orientation = orientation * (180 / norient);
featType = featType*i; % Rotate feature phase angles by 90deg so that 0
% phase corresponds to a step edge (this is a
% fudge I must have something the wrong way
% around somewhere)
%
% SUBMAT
%
% Function to extract the i'th sub-matrix 'cols' wide from a large
% matrix composed of several matricies. The large matrix is used in
% lieu of an array of matricies
function a = submat(big,i,cols)
a = big(:,((i-1)*cols+1):(i*cols));