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Resampling volume or image with affine matrix

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Resampling volume or image with affine matrix

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13 Aug 2008 (Updated )

Use affine matrix to convert 3D vol or 2D img to orthogonal one with corresponding affine matrix.

affine(old_img, old_M, new_elem_size, verbose, bg, method)
%  Using 2D or 3D affine matrix to rotate, translate, scale, reflect and
%  shear a 2D image or 3D volume. 2D image is represented by a 2D matrix,
%  3D volume is represented by a 3D matrix, and data type can be real 
%  integer or floating-point.
%
%  You may notice that MATLAB has a function called 'imtransform.m' for
%  2D spatial transformation. However, keep in mind that 'imtransform.m'
%  assumes y for the 1st dimension, and x for the 2nd dimension. They are
%  equivalent otherwise.
%
%  In addition, if you adjust the 'new_elem_size' parameter, this 'affine.m'
%  is equivalent to 'interp2.m' for 2D image, and equivalent to 'interp3.m'
%  for 3D volume.
%
%  Usage: [new_img new_M] = ...
%	affine(old_img, old_M, [new_elem_size], [verbose], [bg], [method]);
%
%  old_img  -	original 2D image or 3D volume. We assume x for the 1st
%		dimension, y for the 2nd dimension, and z for the 3rd
%		dimension.
%
%  old_M  -	a 3x3 2D affine matrix for 2D image, or a 4x4 3D affine
%		matrix for 3D volume. We assume x for the 1st dimension,
%		y for the 2nd dimension, and z for the 3rd dimension.
%
%  new_elem_size (optional)  -  size of voxel along x y z direction for 
%		a transformed 3D volume, or size of pixel along x y for
%		a transformed 2D image. We assume x for the 1st dimension
%		y for the 2nd dimension, and z for the 3rd dimension.
%		'new_elem_size' is 1 if it is default or empty.
%
%		You can increase its value to decrease the resampling rate,
%		and make the 2D image or 3D volume more coarse. It works
%		just like 'interp3'.
%
%  verbose (optional) - 1, 0
%		1:  show transforming progress in percentage
%		2:  progress will not be displayed
%		'verbose' is 1 if it is default or empty.
%
%  bg (optional)  -	background voxel intensity in any extra corner that
%		is caused by the interpolation. 0 in most cases. If it is
%		default or empty, 'bg' will be the average of two corner
%		voxel intensities in original data.
%
%  method (optional)  -	1, 2, or 3
%		1:  for Trilinear interpolation
%		2:  for Nearest Neighbor interpolation
%		3:  for Fischer's Bresenham interpolation
%		'method' is 1 if it is default or empty.
%
%  new_img  -	transformed 2D image or 3D volume
%
%  new_M  -	transformed affine matrix
%
%  Example 1 (3D rotation):
%	load mri.mat;   old_img = double(squeeze(D));
%	old_M = [0.88 0.5 3 -90; -0.5 0.88 3 -126; 0 0 2 -72; 0 0 0 1];
%	new_img = affine(old_img, old_M, 2);
%	[x y z] = meshgrid(1:128,1:128,1:27);
%	sz = size(new_img);
%	[x1 y1 z1] = meshgrid(1:sz(2),1:sz(1),1:sz(3));
%	figure; slice(x, y, z, old_img, 64, 64, 13.5);
%	shading flat; colormap(map); view(-66, 66);
%	figure; slice(x1, y1, z1, new_img, sz(1)/2, sz(2)/2, sz(3)/2);
%	shading flat; colormap(map); view(-66, 66);
%
%  Example 2 (2D interpolation):
%	load mri.mat;   old_img=D(:,:,1,13)';
%	old_M = [1 0 0; 0 1 0; 0 0 1];
%	new_img = affine(old_img, old_M, [.2 .4]);
%	figure; image(old_img); colormap(map);
%	figure; image(new_img); colormap(map);
%
%  This program is inspired by:
%  SPM5 Software from Wellcome Trust Centre for Neuroimaging
%	http://www.fil.ion.ucl.ac.uk/spm/software
%  Fischer, J., A. del Rio (2004). A Fast Method for Applying Rigid
%	Transformations to Volume Data, WSCG2004 Conference.
%	http://wscg.zcu.cz/wscg2004/Papers_2004_Short/M19.pdf
%  
%  - Jimmy Shen (jimmy@rotman-baycrest.on.ca)
%
function [new_img, new_M] = affine(old_img, old_M, new_elem_size, verbose, bg, method)

   if ~exist('old_img','var') | ~exist('old_M','var')
      error('Usage: [new_img new_M] = affine(old_img, old_M, [new_elem_size], [verbose], [bg], [method]);');
   end

   if ndims(old_img) == 3
      if ~isequal(size(old_M),[4 4])
         error('old_M should be a 4x4 affine matrix for 3D volume.');
      end
   elseif ndims(old_img) == 2
      if ~isequal(size(old_M),[3 3])
         error('old_M should be a 3x3 affine matrix for 2D image.');
      end
   else
      error('old_img should be either 2D image or 3D volume.');
   end

   if ~exist('new_elem_size','var') | isempty(new_elem_size)
      new_elem_size = [1 1 1];
   elseif length(new_elem_size) < 2
      new_elem_size = new_elem_size(1)*ones(1,3);
   elseif length(new_elem_size) < 3
      new_elem_size = [new_elem_size(:); 1]';
   end

   if ~exist('method','var') | isempty(method)
      method = 1;
   elseif ~exist('bresenham_line3d.m','file') & method == 3
      error([char(10) char(10) 'Please download 3D Bresenham''s line generation program from:' char(10) char(10) 'http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=21057' char(10) char(10) 'to test Fischer''s Bresenham interpolation method.' char(10) char(10)]);
   end

   %  Make compatible to MATLAB earlier than version 7 (R14), which
   %  can only perform arithmetic on double data type
   %
   old_img = double(old_img);
   old_dim = size(old_img);

   if ~exist('bg','var') | isempty(bg)
      bg = mean([old_img(1) old_img(end)]);
   end

   if ~exist('verbose','var') | isempty(verbose)
      verbose = 1;
   end

   if ndims(old_img) == 2
      old_dim(3) = 1;
      old_M = old_M(:, [1 2 3 3]);
      old_M = old_M([1 2 3 3], :);
      old_M(3,:) = [0 0 1 0];
      old_M(:,3) = [0 0 1 0]';
   end

   %  Vertices of img in voxel
   %
   XYZvox = [	1		1		1
		1		1		old_dim(3)
		1		old_dim(2)	1
		1		old_dim(2)	old_dim(3)
		old_dim(1)	1		1
		old_dim(1)	1		old_dim(3)
		old_dim(1)	old_dim(2)	1
		old_dim(1)	old_dim(2)	old_dim(3)   ]';

   old_R = old_M(1:3,1:3);
   old_T = old_M(1:3,4);

   %  Vertices of img in millimeter
   %
   XYZmm = old_R*(XYZvox-1) + repmat(old_T, [1, 8]);

   %  Make scale of new_M according to new_elem_size
   %
   new_M = diag([new_elem_size 1]);

   %  Make translation so minimum vertex is moved to [1,1,1]
   %
   new_M(1:3,4) = round( min(XYZmm,[],2) );

   %  New dimensions will be the maximum vertices in XYZ direction (dim_vox)
   %  i.e. compute   dim_vox   via   dim_mm = R*(dim_vox-1)+T
   %  where, dim_mm = round(max(XYZmm,[],2));
   %
   new_dim = ceil(new_M(1:3,1:3) \ ( round(max(XYZmm,[],2))-new_M(1:3,4) )+1)';

   %  Initialize new_img with new_dim
   %
   new_img = zeros(new_dim(1:3));

   %  Mask out any changes from Z axis of transformed volume, since we
   %  will traverse it voxel by voxel below. We will only apply unit
   %  increment of mask_Z(3,4) to simulate the cursor movement
   %
   %  i.e. we will use   mask_Z * new_XYZvox   to replace   new_XYZvox
   %
   mask_Z = diag(ones(1,4));
   mask_Z(3,3) = 0;

   %  It will be easier to do the interpolation if we invert the process
   %  by not traversing the original volume. Instead, we traverse the
   %  transformed volume, and backproject each voxel in the transformed 
   %  volume back into the original volume. If the backprojected voxel
   %  in original volume is within its boundary, the intensity of that
   %  voxel can be used by the cursor location in the transformed volume.
   %
   %  First, we traverse along Z axis of transformed volume voxel by voxel
   %
   for z = 1:new_dim(3)

      if verbose & ~mod(z,10)
         fprintf('%.2f percent is done.\n', 100*z/new_dim(3));
      end

      %  We need to find out the mapping from voxel in the transformed
      %  volume (new_XYZvox) to voxel in the original volume (old_XYZvox)
      %
      %  The following equation works, because they all equal to XYZmm:
      %  new_R*(new_XYZvox-1) + new_T  ==  old_R*(old_XYZvox-1) + old_T
      %
      %  We can use modified new_M1 & old_M1 to substitute new_M & old_M
      %      new_M1 * new_XYZvox       ==       old_M1 * old_XYZvox
      %
      %  where: M1 = M;   M1(:,4) = M(:,4) - sum(M(:,1:3),2);
      %  and:             M(:,4) == [T; 1] == sum(M1,2)
      %
      %  Therefore:   old_XYZvox = old_M1 \ new_M1 * new_XYZvox;
      %
      %  Since we are traverse Z axis, and   new_XYZvox   is replaced
      %  by   mask_Z * new_XYZvox, the above formula can be rewritten
      %  as:    old_XYZvox = old_M1 \ new_M1 * mask_Z * new_XYZvox;
      %
      %  i.e. we find the mapping from new_XYZvox to old_XYZvox:
      %  M = old_M1 \ new_M1 * mask_Z;
      %
      %  First, compute modified old_M1 & new_M1
      %
      old_M1 = old_M;   old_M1(:,4) = old_M(:,4) - sum(old_M(:,1:3),2);
      new_M1 = new_M;   new_M1(:,4) = new_M(:,4) - sum(new_M(:,1:3),2);

      %  Then, apply unit increment of mask_Z(3,4) to simulate the
      %  cursor movement
      %
      mask_Z(3,4) = z;

      %  Here is the mapping from new_XYZvox to old_XYZvox
      %
      M = old_M1 \ new_M1 * mask_Z;

      switch method
      case 1
         new_img(:,:,z) = trilinear(old_img, new_dim, old_dim, M, bg);
      case 2
         new_img(:,:,z) = nearest_neighbor(old_img, new_dim, old_dim, M, bg);
      case 3
         new_img(:,:,z) = bresenham(old_img, new_dim, old_dim, M, bg);
      end

   end;			% for z

   if ndims(old_img) == 2
      new_M(3,:) = [];
      new_M(:,3) = [];
   end

   return;					% affine


%--------------------------------------------------------------------
function img_slice = trilinear(img, dim1, dim2, M, bg)

   img_slice = zeros(dim1(1:2));
   TINY = 5e-2;					% tolerance

   %  Dimension of transformed 3D volume
   %
   xdim1 = dim1(1);
   ydim1 = dim1(2);

   %  Dimension of original 3D volume
   %
   xdim2 = dim2(1);
   ydim2 = dim2(2);
   zdim2 = dim2(3);

   %  initialize new_Y accumulation
   %
   Y2X = 0;
   Y2Y = 0;
   Y2Z = 0;

   for y = 1:ydim1

      %  increment of new_Y accumulation
      %
      Y2X = Y2X + M(1,2);		% new_Y to old_X
      Y2Y = Y2Y + M(2,2);		% new_Y to old_Y
      Y2Z = Y2Z + M(3,2);		% new_Y to old_Z

      %  backproject new_Y accumulation and translation to old_XYZ
      %
      old_X = Y2X + M(1,4);
      old_Y = Y2Y + M(2,4);
      old_Z = Y2Z + M(3,4);

      for x = 1:xdim1

         %  accumulate the increment of new_X, and apply it
         %  to the backprojected old_XYZ
         %
         old_X = M(1,1) + old_X  ;
         old_Y = M(2,1) + old_Y  ;
         old_Z = M(3,1) + old_Z  ;

         %  within boundary of original image
         %
         if (	old_X > 1-TINY & old_X < xdim2+TINY & ...
		old_Y > 1-TINY & old_Y < ydim2+TINY & ...
		old_Z > 1-TINY & old_Z < zdim2+TINY	)

            %  Calculate distance of old_XYZ to its neighbors for
            %  weighted intensity average
            %
            dx = old_X - floor(old_X);
            dy = old_Y - floor(old_Y);
            dz = old_Z - floor(old_Z);

            x000 = floor(old_X);
            x100 = x000 + 1;

            if floor(old_X) < 1
               x000 = 1;
               x100 = x000;
            elseif floor(old_X) > xdim2-1
               x000 = xdim2;
               x100 = x000;
            end

            x010 = x000;
            x001 = x000;
            x011 = x000;

            x110 = x100;
            x101 = x100;
            x111 = x100;

            y000 = floor(old_Y);
            y010 = y000 + 1;

            if floor(old_Y) < 1
               y000 = 1;
               y100 = y000;
            elseif floor(old_Y) > ydim2-1
               y000 = ydim2;
               y010 = y000;
            end

            y100 = y000;
            y001 = y000;
            y101 = y000;

            y110 = y010;
            y011 = y010;
            y111 = y010;

            z000 = floor(old_Z);
            z001 = z000 + 1;

            if floor(old_Z) < 1
               z000 = 1;
               z001 = z000;
            elseif floor(old_Z) > zdim2-1
               z000 = zdim2;
               z001 = z000;
            end

            z100 = z000;
            z010 = z000;
            z110 = z000;

            z101 = z001;
            z011 = z001;
            z111 = z001;

            x010 = x000;
            x001 = x000;
            x011 = x000;

            x110 = x100;
            x101 = x100;
            x111 = x100;

            v000 = double(img(x000, y000, z000));
            v010 = double(img(x010, y010, z010));
            v001 = double(img(x001, y001, z001));
            v011 = double(img(x011, y011, z011));

            v100 = double(img(x100, y100, z100));
            v110 = double(img(x110, y110, z110));
            v101 = double(img(x101, y101, z101));
            v111 = double(img(x111, y111, z111));

            img_slice(x,y) = v000*(1-dx)*(1-dy)*(1-dz) + ...
               v010*(1-dx)*dy*(1-dz) + ...
               v001*(1-dx)*(1-dy)*dz + ...
               v011*(1-dx)*dy*dz + ...
               v100*dx*(1-dy)*(1-dz) + ...
               v110*dx*dy*(1-dz) + ...
               v101*dx*(1-dy)*dz + ...
               v111*dx*dy*dz;

         else
            img_slice(x,y) = bg;

         end	% if boundary

      end	% for x
   end		% for y

   return;					% trilinear


%--------------------------------------------------------------------
function img_slice = nearest_neighbor(img, dim1, dim2, M, bg)

   img_slice = zeros(dim1(1:2));

   %  Dimension of transformed 3D volume
   %
   xdim1 = dim1(1);
   ydim1 = dim1(2);

   %  Dimension of original 3D volume
   %
   xdim2 = dim2(1);
   ydim2 = dim2(2);
   zdim2 = dim2(3);

   %  initialize new_Y accumulation
   %
   Y2X = 0;
   Y2Y = 0;
   Y2Z = 0;

   for y = 1:ydim1

      %  increment of new_Y accumulation
      %
      Y2X = Y2X + M(1,2);		% new_Y to old_X
      Y2Y = Y2Y + M(2,2);		% new_Y to old_Y
      Y2Z = Y2Z + M(3,2);		% new_Y to old_Z

      %  backproject new_Y accumulation and translation to old_XYZ
      %
      old_X = Y2X + M(1,4);
      old_Y = Y2Y + M(2,4);
      old_Z = Y2Z + M(3,4);

      for x = 1:xdim1

         %  accumulate the increment of new_X and apply it
         %  to the backprojected old_XYZ
         %
         old_X = M(1,1) + old_X  ;
         old_Y = M(2,1) + old_Y  ;
         old_Z = M(3,1) + old_Z  ;

         xi = round(old_X);
         yi = round(old_Y);
         zi = round(old_Z);

         %  within boundary of original image
         %
         if (	xi >= 1 & xi <= xdim2 & ...
		yi >= 1 & yi <= ydim2 & ...
		zi >= 1 & zi <= zdim2	)

            img_slice(x,y) = img(xi,yi,zi);

         else
            img_slice(x,y) = bg;

         end	% if boundary

      end	% for x
   end		% for y

   return;					% nearest_neighbor


%--------------------------------------------------------------------
function img_slice = bresenham(img, dim1, dim2, M, bg)

   img_slice = zeros(dim1(1:2));

   %  Dimension of transformed 3D volume
   %
   xdim1 = dim1(1);
   ydim1 = dim1(2);

   %  Dimension of original 3D volume
   %
   xdim2 = dim2(1);
   ydim2 = dim2(2);
   zdim2 = dim2(3);

   for y = 1:ydim1

      start_old_XYZ = round(M*[0     y 0 1]');
      end_old_XYZ   = round(M*[xdim1 y 0 1]');

      [X Y Z] = bresenham_line3d(start_old_XYZ, end_old_XYZ);

      %  line error correction
      %
%      del = end_old_XYZ - start_old_XYZ;
 %     del_dom = max(del);
  %    idx_dom = find(del==del_dom);
   %   idx_dom = idx_dom(1);
    %  idx_other = [1 2 3];
     % idx_other(idx_dom) = [];
      %del_x1 = del(idx_other(1));
%      del_x2 = del(idx_other(2));
 %     line_slope = sqrt((del_x1/del_dom)^2 + (del_x2/del_dom)^2 + 1);
  %    line_error = line_slope - 1;
% line error correction removed because it is too slow

      for x = 1:xdim1

         %  rescale ratio
         %
         i = round(x * length(X) / xdim1);

         if i < 1
            i = 1;
         elseif i > length(X)
            i = length(X);
         end

         xi = X(i);
         yi = Y(i);
         zi = Z(i);

         %  within boundary of the old XYZ space
         %
         if (	xi >= 1 & xi <= xdim2 & ...
		yi >= 1 & yi <= ydim2 & ...
		zi >= 1 & zi <= zdim2	)

            img_slice(x,y) = img(xi,yi,zi);

%            if line_error > 1
 %              x = x + 1;

%               if x <= xdim1
 %                 img_slice(x,y) = img(xi,yi,zi);
  %                line_error = line_slope - 1;
   %            end
    %        end		% if line_error
% line error correction removed because it is too slow

         else
            img_slice(x,y) = bg;

         end	% if boundary

      end	% for x
   end		% for y

   return;					% bresenham

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