--- /dev/null
+## Copyright (C) 2010 Alex Opie <lx_op@orcon.net.nz>
+##
+##
+## This program is free software; you can redistribute it and/or modify it
+## under the terms of the GNU General Public License as published by
+## the Free Software Foundation; either version 3 of the License, or (at
+## your option) any later version.
+##
+## This program is distributed in the hope that it will be useful, but
+## WITHOUT ANY WARRANTY; without even the implied warranty of
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+## General Public License for more details.
+##
+## You should have received a copy of the GNU General Public License
+## along with this program; see the file COPYING. If not, see
+## <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @defun @var{recon} = iradon (@var{proj}, @var{theta}, @var{interp}, @
+## @var{filter}, @var{scaling}, @var{output_size})
+##
+## Performs filtered back-projection on the projections in @var{proj}
+## to reconstruct an approximation of the original image.
+##
+## @var{proj} should be a matrix whose columns are projections of an
+## image (or slice). Each element of @var{theta} is used as the angle
+## (in degrees) that the corresponding column of @var{proj} was
+## projected at. If @var{theta} is omitted, it is assumed that
+## projections were taken at evenly spaced angles between 0 and 180 degrees.
+## @var{theta} can also be a scalar, in which case it is taken as the
+## angle between projections if more than one projection is provided.
+##
+## @var{interp} determines the type of interpolation that is used
+## in the back-projection. It must be one of the types accepted by
+## @command{interp1}, and defaults to 'Linear' if it is omitted.
+##
+## @var{filter} and @var{scaling} determine the type of rho filter
+## to apply. See the help for @command{rho_filter} for their use.
+##
+## @var{output_size} sets the edge length of the output image (it
+## is always square). This argument does not scale the image. If it
+## is omitted, the length is taken to be
+## @group
+## 2 * floor (size (proj, 1) / (2 * sqrt (2))).
+## @end group
+##
+## If @var{proj} was obtained using @command{radon}, there is no
+## guarantee that the reconstructed image will be exactly the same
+## size as the original.
+##
+## @defunx [@var{recon}, @var{filt}] = iradon (...)
+##
+## This form also returns the filter frequency response in the vector
+## @var{filt}.
+## @end defun
+##
+## Performs filtered back-projection in order to reconstruct an
+## image based on its projections.
+##
+## Filtered back-projection is the most common means of reconstructing
+## images from CT scans. It is a two step process: First, each of
+## the projections is filtered with a `rho filter', so named due
+## to its frequency domain definition, which is simply |rho|, where
+## rho is the radial axis in a polar coordinate system. Second,
+## the filtered projections are each `smeared' across the image
+## space. This is the back-projection part.
+##
+## Usage example:
+## @example
+## P = phantom ();
+## projections = radon (P, 1:179);
+## reconstruction = iradon (filtered_projections, 1:179, 'Spline', 'Hann');
+## figure, imshow (reconstruction, [])
+## @end example
+
+function [recon, filt] = iradon (proj, theta, interp, filter, scaling, output_size)
+
+ if (nargin == 0)
+ error ("No projections provided to iradon");
+ endif
+
+ if (nargin < 6)
+ output_size = 2 * floor (size (proj, 1) / (2 * sqrt (2)));
+ endif
+ if (nargin < 5) || (length (scaling) == 0)
+ scaling = 1;
+ endif
+ if (nargin < 4) || (length (filter) == 0)
+ filter = "Ram-Lak";
+ endif
+ if (nargin < 3) || (length (interp) == 0)
+ interp = "linear";
+ endif
+ if (nargin < 2) || (length (theta) == 0)
+ theta = 180 * (0:1:size (proj, 2) - 1) / size (proj, 2);
+ endif
+
+ if (isscalar (theta)) && (size (proj, 2) != 1)
+ theta = (0:size (proj, 2) - 1) * theta;
+ endif
+
+ if (length (theta) != size (proj, 2))
+ error ("iradon: Number of projections does not match number of angles")
+ endif
+ if (!isscalar (scaling))
+ error ("iradon: Frequency scaling value must be a scalar");
+ endif
+ if (!length (find (strcmpi (interp, {'nearest', 'linear', 'spline', \
+ 'pchip', 'cubic'}))))
+ error ("iradon: Invalid interpolation method specified");
+ endif
+
+ ## Convert angles to radians
+ theta *= pi / 180;
+
+ ## First, filter the projections
+ [filtered, filt] = rho_filter (proj, filter, scaling);
+
+ ## Next, back-project
+ recon = back_project (filtered, theta, interp, output_size);
+
+endfunction
+
+
+function recon = back_project (proj, theta, interpolation, dim)
+ ## Make an empty image
+ recon = zeros (dim, dim);
+
+ ## Zero pad the projections if the requested image
+ ## has a diagonal longer than the projections
+ diagonal = ceil (dim * sqrt (2)) + 1;
+ if (size (proj, 1) < diagonal)
+ diff = 2 * ceil ((diagonal - size (proj, 1)) / 2);
+ proj = padarray (proj, diff / 2);
+ endif
+
+ ## Create the x & y values for each pixel
+ centre = floor ((dim + 1) / 2);
+ x = (0:dim - 1) - centre + 1;
+ x = repmat (x, dim, 1);
+
+ y = (dim - 1: -1 : 0)' - centre;
+ y = repmat (y, 1, dim);
+
+ ## s axis for projections, needed by interp1
+ s = (0:size (proj, 1) - 1) - floor (size (proj, 1) / 2);
+
+ ## Sum each projection's contribution
+ for i = 1:length (theta)
+ s_dash = (x * cos (theta (i)) + y * sin (theta (i)));
+ interpolated = interp1 (s, proj (:, i), s_dash (:), ["*", interpolation]);
+ recon += reshape (interpolated, dim, dim);
+ endfor
+
+ ## Scale the reconstructed values to their original size
+ recon *= pi / (2 * length (theta));
+
+endfunction
+
+%!demo
+%! P = phantom ();
+%! figure, imshow (P, []), title ("Original image")
+%! projections = radon (P, 0:179);
+%! reconstruction = iradon (projections, 0:179, 'Spline', 'Hann');
+%! figure, imshow (reconstruction, []), title ("Reconstructed image")
+
+% $Log$
+% 2010-30-03 lxop
+% First submitted to Octave-Forge
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff