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+utils/find_shift_fast_1D.m

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+% FIND_SHIFT_FAST_1D uses cross-correlation to find shift between 1D patterns o1 and
+% o2, if the patterns are 2D, perform the search along the axis `ax`
+%
+% shift = find_shift_fast_1D(o1, o2,  ax, sigma)
+%
+% Inputs:
+%   **o1            - aligned array 1D/2D (will be aligned along 1st axis)
+%   **o2            - template for alignment 1D or 2D 
+%   **ax            - perform search along this axis 
+%  *optional*
+%   **sigma         - filtering intensity [0-1 range],  sigma <= 0 no filtering, recommended sigma < 0.05 
+%   **padding       - pading [in pixels] the provided array by zeros, prevent circular boundary condition in FFT, default = 0 
+% *returns*
+%   ++shift         - (vector) displacement of the 1D/2D arrays  
+
+%*-----------------------------------------------------------------------*
+%|                                                                       |
+%|  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 
+% 
+
+
+
+function shift = find_shift_fast_1D(o1, o2, ax, sigma, padding)
+
+    if nargin < 3
+        ax = 2; 
+    end
+    if nargin < 4
+        sigma = 0;
+    end
+    if nargin < 5
+        padding = 0; 
+    else
+        padding = ceil(padding/2)*2;
+    end
+    
+    max_shift = size(o1,ax)/3;  % avoid too large corrections !!!
+    
+    Ndims = ndims(o1);
+   
+    if ax ~= 1 
+        error('FIXME: Not tested axis')
+    end
+    
+    %% symmetrize before spectral filtering !!
+    o1 = cat(ax, o1, flipud(o1)); 
+    o2 = cat(ax, o2, flipud(o2)); 
+    Npix = size(o1);
+
+    shape = ones(1,Ndims);
+    shape(ax) = Npix(ax);
+
+    if sigma > 0
+        %% high pass filter 
+        o1 = fft(o1, [],ax);
+        o2 = fft(o2, [],ax);
+        x = reshape((-Npix(ax)/2+1:Npix(ax)/2)/Npix(ax), shape);
+        spectral_filter = fftshift(exp(1./(-(x.^2)/(sigma^2))));
+        spectral_filter(floor(end/2+[-3:3])) = 0; %% remove some strange artefacts 
+        o1 = bsxfun(@times, o1, spectral_filter);
+        o2 = bsxfun(@times, o2, spectral_filter);
+        o1 = ifft(o1, [],ax);
+        o2 = ifft(o2, [],ax);
+    
+    end
+
+    % remove symetrization  !!
+    o1 = o1(1:end/2,:);
+    o2 = o2(1:end/2,:);
+    o1 = padarray(o1, padding/2, 'both');
+    o2 = padarray(o2, padding/2, 'both');
+    
+    
+    Npix = size(o1);
+    shape = ones(1,Ndims);
+    shape(ax) = Npix(ax);
+
+    %% remove edge issues (after symetrized filtering )
+    spatial_filter = reshape(tukeywin(prod(shape)), shape);
+    o1 = bsxfun(@times, o1, spatial_filter);
+    o2 = bsxfun(@times, o2, spatial_filter);
+    
+    o1 = fft(o1, [],ax);
+    o2 = fft(o2, [],ax);
+
+    %% cross-correlation 
+    xcorrmat = abs(ifft(o1.*conj(o2),[],ax)); 
+    %% 1D fftshift 
+    xcorrmat = circshift(xcorrmat, floor(Npix(ax)/2), ax); 
+    if ax == 2; error('FIXME: Not tested axis'); end 
+    
+    % choose only optimim withing reduced range 
+    xcorrmat([1:ceil(end/2-max_shift), ceil(end/2+max_shift):end],:) = 0; 
+    
+    %% take only small region around maximum 
+    WIN = 10; 
+    kernel_size = [1,1];
+    kernel_size(ax) = WIN; 
+    mask = conv2(single(bsxfun(@eq, xcorrmat, max(xcorrmat,[],ax))), ones(kernel_size), 'same'); 
+    xcorrmat(~mask) = nan; 
+    xcorrmat = max(0, bsxfun(@minus, xcorrmat, min(xcorrmat,[],ax)));
+    xcorrmat(~mask) = 0; 
+    xcorrmat = bsxfun(@times, xcorrmat, 1./max(xcorrmat,[],ax)).^4;
+    
+    
+    %% find center of mass 
+    MASS = sum(xcorrmat,ax);
+    grid = reshape(1:Npix(ax),shape); 
+    shift = sum(bsxfun(@times, xcorrmat, grid),ax) ./ MASS - floor(Npix(ax)/2)-1;
+    
+end