phasecong.m

hopfield neural network for binary image recognition

  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));