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+utils/prop_free_nonparallel_plane.m

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+% PROP_FREE_NONPARALLEL_PLANE  Near field propagation into a surface nonparallel with beam
+%
+%    [U, H, h_tilted] = prop_free_nonparallel_plane(U, z, lambda, pixel_size, ax=1)
+%     returns the propagated wavefield
+%    Inputs: 
+%       **U           stack of images 
+%       **z             propagation distance along horizontal axis (vector)
+%       **lambda        wavelenght [m]
+%       **pixel_size    pixel size [m]
+%       **ax            (optional)  axis along which is the tilted plane 
+%    *returns*
+%       ++U           propagated stack of images 
+%
+%
+%     Example for tilted plane propagation:
+%     Npix = size(img);
+%     grid = ((-Npix(2)/2+1):Npix(2)/2)*pixel_size * tand(90-grazing_angle); 
+%    [img_propag, H, h_tilted] = prop_free_tilted_plane(img, grid, lambda, pixel_size)
+% 
+%
+    
+%*-----------------------------------------------------------------------*
+%|                                                                       |
+%|  Except where otherwise noted, this work is licensed under a          |
+%|  Creative Commons Attribution-NonCommercial-ShareAlike 4.0            |
+%|  International (CC BY-NC-SA 4.0) license.                             |
+%|                                                                       |
+%|  Copyright (c) 2017 by Paul Scherrer Institute (http://www.psi.ch)    |
+%|                                                                       |
+%|       Author: CXS group, PSI                                          |
+%*-----------------------------------------------------------------------*
+% You may use this code with the following provisions:
+%
+% If the code is fully or partially redistributed, or rewritten in another
+%   computing language this notice should be included in the redistribution.
+%
+% If this code, or subfunctions or parts of it, is used for research in a 
+%   publication or if it is fully or partially rewritten for another 
+%   computing language the authors and institution should be acknowledged 
+%   in written form in the publication: “Data processing was carried out 
+%   using the “cSAXS matlab package” developed by the CXS group,
+%   Paul Scherrer Institut, Switzerland.” 
+%   Variations on the latter text can be incorporated upon discussion with 
+%   the CXS group if needed to more specifically reflect the use of the package 
+%   for the published work.
+%
+% A publication that focuses on describing features, or parameters, that
+%    are already existing in the code should be first discussed with the
+%    authors.
+%   
+% This code and subroutines are part of a continuous development, they 
+%    are provided “as they are” without guarantees or liability on part
+%    of PSI or the authors. It is the user responsibility to ensure its 
+%    proper use and the correctness of the results.
+
+function [U, H, h_tilted] = prop_free_nonparallel_plane(U, z, lambda, pixel_size, ax)
+    %%  nearfield propagator into tilted plane 
+    
+    if all(z == 0)
+        H = 1;
+        h = 1; 
+        U = U;
+        return
+    end
+    if nargin  < 5
+        ax = 1;
+    end
+
+    %% standard 1D ASM for vertical axis 
+    Np = size(U);
+    if ax == 1 
+       Np = Np([2,1]);
+       pixel_size = pixel_size(min(end,[2,1])); 
+    end 
+    
+    assert(length(z) == Np(2), 'Shift vector z has wrong length')
+    
+    type = ones(1,'like',U);
+    xgrid = type * (-fix(Np(1)/2):ceil(Np(1)/2)-1);
+    k = 2 * pi / lambda(1);
+    extent = Np .* pixel_size;
+    kx =  2 * pi .*xgrid / extent(1) ;
+    H = exp(1i*z(:)' .* (kx'.^2) / (2*k));
+        
+    H = fftshift(H, 1);  % 1D ASM propagator 
+    
+    %% convolution ASM for horizontal axis
+    % fft transform to get real space convolution kernel
+    h = ifftshift(fft(H,[],1),1)/(Np(1));
+        
+    h = utils.crop_pad(h, [Np(2), Np(2)]);
+
+    % apply shift on the convolution propagator for the axis along the beam 
+    h_tilted = utils.imshift_fft_ax(h, (1:Np(2))-Np(2)/2-1,1);
+    
+    % adjustnemnts to make the propagated image consistent with the original one 
+    h_tilted = fliplr(rot90(h_tilted));
+    H = fliplr(H); 
+
+    %%%%%%%%%%%%%% propagate the image  %%%%%%%%%%%%%%%%%%%%%
+    if ax == 1
+        % propagate horizontal axis
+        U = ifft(H.' .* fft(U,[],2),[],2);
+        % propagate vertical axis !! use matrix multiplication !! in realspace
+        % to describe the convolution that is needed for propagation to the
+        % tilted axis ->   h_tilted * U
+        U = utils.mtimes_stack(h_tilted,U);  % convolution applied by matrix multiplication, make it for entire image stack in parallel 
+    else
+        % propagate horizontal axis
+        U = ifft(H .* fft(U,[],1),[],1);
+        % propagate vertical axis !! use matrix multiplication !! in realspace
+        % to describe the convolution that is needed for propagation to the
+        % tilted axis 
+        U = utils.mtimes_stack(U,h_tilted);
+         
+        
+    end
+    
+end