% GABORCONVOLVE - function for convolving image with log-Gabor filters
%
% Usage: [EO, BP] = gaborconvolve(im, nscale, norient, minWaveLength, mult, ...
% sigmaOnf, dThetaOnSigma, Lnorm, feedback)
%
% Arguments:
% The convolutions are done via the FFT. Many of the parameters relate
% to the specification of the filters in the frequency plane.
%
% Variable Suggested Description
% name value
% ----------------------------------------------------------
% im Image to be convolved.
% nscale = 4; Number of wavelet scales.
% norient = 6; Number of filter orientations.
% minWaveLength = 3; Wavelength of smallest scale filter.
% mult = 1.7; Scaling factor between successive filters.
% sigmaOnf = 0.65; 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 = 1.3; Ratio of angular interval between filter
% orientations and the standard deviation of
% the angular Gaussian function used to
% construct filters in the freq. plane.
% Lnorm 0 Optional integer indicating what norm the
% filters should be normalized to. A value of 1
% will produce filters with the same L1 norm, 2
% will produce filters with matching L2
% norm. the default value of 0 results in no
% normalization (the filters have unit height
% Gaussian transfer functions on a log frequency
% scale)
% feedback 0/1 Optional parameter. If set to 1 a message
% indicating which orientation is being
% processed is printed on the screen.
%
% Returns:
%
% EO - 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.
%
% BP - Cell array of bandpass images corresponding to each scale s.
%
%
% Notes on filter settings to obtain even coverage of the spectrum energy
% dThetaOnSigma 1.2 - 1.3
% sigmaOnf .90 mult 1.15
% sigmaOnf .85 mult 1.2
% sigmaOnf .75 mult 1.4 (bandwidth ~1 octave)
% sigmaOnf .65 mult 1.7
% sigmaOnf .55 mult 2.2 (bandwidth ~2 octaves)
%
% The determination of mult given sigmaOnf is entirely empirical. What I do is
% plot out the sum of the squared filter amplitudes in the frequency domain and
% see how even the coverage of the spectrum is. If there are concentric 'gaps'
% in the spectrum one needs to reduce mult and/or reduce sigmaOnf (which
% increases filter bandwidth)
%
% If there are 'gaps' radiating outwards then one needs to reduce dthetaOnSigma
% (increasing angular bandwidth of the filters)
%
% For details of log-Gabor filters see:
% D. J. Field, "Relations Between the Statistics of Natural Images and the
% Response Properties of Cortical Cells", Journal of The Optical Society of
% America A, Vol 4, No. 12, December 1987. pp 2379-2394
% Copyright (c) 2001-2010 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.
% May 2001 - Original version
% April 2010 - Reworked to tidy things up. Return of bandpass images added.
% March 2013 - Restored use of dThetaOnSigma to control angular spread of filters.
function [EO, BP] = gaborconvolve(im, nscale, norient, minWaveLength, mult, ...
sigmaOnf, dThetaOnSigma, Lnorm, feedback)
if ndims(im) == 3
warning('Colour image supplied: Converting to greyscale');
im = rgb2gray(im);
end
if ~exist('Lnorm','var'), Lnorm = 0; end
if ~exist('feedback','var'), feedback = 0; end
if ~isa(im,'double'), im = double(im); end
[rows cols] = size(im);
imagefft = fft2(im); % Fourier transform of image
EO = cell(nscale, norient); % Pre-allocate cell array
BP = cell(nscale,1);
% 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.
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. This keeps the overall norm of each filter not too dissimilar.
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).
% Compute bandpass image for each scale
if Lnorm == 2 % Normalize filters to have same L2 norm
L = sqrt(sum(logGabor{s}(:).^2));
elseif Lnorm == 1 % Normalize to have same L1
L = sum(sum(abs(real(ifft2(logGabor{s})))));
elseif Lnorm == 0 % No normalization
L = 1;
else
error('Lnorm must be 0, 1 or 2');
end
logGabor{s} = logGabor{s}./L;
BP{s} = ifft2(imagefft .* logGabor{s});
end
% The main loop...
for o = 1:norient, % For each orientation.
if feedback
fprintf('Processing orientation %d \r', o);
end
angl = (o-1)*pi/norient; % Calculate filter angle.
wavelength = minWaveLength; % Initialize filter wavelength.
% 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.
% Calculate the standard deviation of the angular Gaussian
% function used to construct filters in the freq. plane.
thetaSigma = pi/norient/dThetaOnSigma;
spread = exp((-dtheta.^2) / (2 * thetaSigma^2));
for s = 1:nscale, % For each scale.
filter = logGabor{s} .* spread; % Multiply by the angular spread to get the filter
if Lnorm == 2 % Normalize filters to have the same L2 norm ** why sqrt 2 **????
L = sqrt(sum(real(filter(:)).^2 + imag(filter(:)).^2 ))/sqrt(2);
elseif Lnorm == 1 % Normalize to have same L1
L = sum(sum(abs(real(ifft2(filter)))));
elseif Lnorm == 0 % No normalization
L = 1;
end
filter = filter./L;
% Do the convolution, back transform, and save the result in EO
EO{s,o} = ifft2(imagefft .* filter);
wavelength = wavelength * mult; % Finally calculate Wavelength of next filter
end % ... and process the next scale
end % For each orientation
if feedback, fprintf(' \r'); end