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+% INVERT_DVF simple iterative method for estimation of the inverse deformation
+% field 
+% Chen, Mingli, et al. "A simple fixed‐point approach to invert a deformation field a." Medical physics 35.1 (2008): 81-88.
+% 
+% [deform_tensors,inv_deform_tensors] = invert_DVF(deform_tensors, Npix_vol)
+% 
+%    **deform_tensors    cell of 3D arrays containing forward deformation DVF       
+%    **Npix_vol          pixel size of the recosntructed volume      
+% Outputs: 
+%    ++deform_tensors_linear        calculate forward DVF 
+%    ++inv_deform_tensors_linear    calculate inverse DVF 
+
+%*-----------------------------------------------------------------------*
+%|                                                                       |
+%|  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) 2018 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 [deform_tensors,inv_deform_tensors] = invert_DVF(deform_tensors, Npix_vol)
+
+
+   Niter = 10; 
+   %% find invert transformation 
+   N = length(deform_tensors); 
+    for block = 1:N
+        % init guess 
+        for ax = 1:3
+            inv_deform_tensors{block}{ax} = -deform_tensors{block}{ax}; 
+        end
+        for i = 1:Niter
+            for ax = 1:3
+                scale = size(deform_tensors{block}{ax}, ax) / Npix_vol(ax) ; % calculate the deformation in deform_tensors grid 
+                inv_deform_tensors{block}{ax} =  inv_deform_tensors{block}{ax}*0.5 + 0.5*utils.interp3_gpu(-deform_tensors{block}{ax}, ...
+                                    scale*inv_deform_tensors{block}{1},scale*inv_deform_tensors{block}{2},scale*inv_deform_tensors{block}{3});
+            end
+        end
+        % in my code I assume that deform_tensors and inv_deform_tensors
+        % have the same direction (in the astra the direction is swapped)
+        for ax = 1:3
+            inv_deform_tensors{block}{ax} = -gather(inv_deform_tensors{block}{ax}); 
+        end
+    end
+ 
+end