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% Call function without arguments for instructions on how to use it
% Filename: $RCSfile: stxm_online.m,v $
%
% $Revision: 1.16 $ $Date: 2011/04/04 17:03:48 $
% $Author: $
% $Tag: $
%
% Description:
% plot a STXM scan
%
% Note:
% Call without arguments for a brief help text.
%
% Dependencies:
% - image_read
%
% history:
%
% April 4th 2011:
% do not normalize the dark field since this is problematic for SAXS with a
% beam stop
%
% September 29th 2010:
% include changes by Martin Dierolf and Joan Vila in the standard version
% of stxm_online
%
% December 10th 2008:
% add bug-fixes and suggestions from Martin Dierolf:
% DirPerLine parameter could not be set via the command line,
% BurstMode flag was always active, is now coupled to dir_per_line,
% new Parameter ZeroOrderR
%
% September 5th 2008:
% use compile_x12sa_filename,
% plot as 2x2 sub figures
%
% June 14th 2008: 1st documented version based on work
%*-----------------------------------------------------------------------*
%|                                                                       |
%|  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 [varargout] = stxm_online(first_scan_number, Ny, varargin)
import beamline.pilatus_valid_pixel_roi
import io.image_read
import plotting.image_show
import utils.compile_x12sa_filename
import utils.find_files
% set default values
% Pilatus 2M
detector_number = 1;
% single directory or directory per line format
dir_per_line = 1;
% figure number for display
fig_no = 2;
% number of points along a scan line, 0 for automatic determination from
% the first line
Nx = 0;
% size of the regio of interest
roi_dim = 128;
% automatic determination of the center position
cen_x = 0;
cen_y = 0;
% dark field integration starting radius
dark_field_r = 20;
% radius of excluded area around center
zero_order_r = 0;
% calculate the first moment rather than a Fourier transform to get the
% differential phase contrast
first_moment = 1;
% use additionally differentiation of the integrated phase
integrated_phase = 1;
% do not update the plot every line to save some time
update_interval = 3;
% save resulting figure
figure_dir = '~/Data10/analysis/online/stxm/figures/';
% save the resulting data
data_dir = '~/Data10/analysis/online/stxm/data/';
% valid pixel mask
filename_valid_mask = '~/Data10/analysis/data/pilatus_valid_mask.mat';
phase = [];
gx = [];
gy = [];
full_screen_position_integrated_phase = [ 5 525 1201 420];
print_a4_position_integrated_phase = [ 5 525 743 420 ];
full_screen_position_standard = [ 5 109 1201 836];
print_a4_position_standard = [ 5 109 743 836 ];
% check minimum number of input arguments
if (nargin < 2)
fprintf('Usage:\n')
fprintf('[trans,dpcx,dpcy,df]=%s(<(first) scan number>, <no. of scan lines> [[,<name>,<value>] ...]);\n',...
mfilename);
fprintf('The optional <name>,<value> pairs are:\n');
fprintf('''DetectorNumber'',<1-Pilatus 2M, 2-Pilatus 300k, 3-Pilatus 100k>\n');
fprintf('''Nx'',<no. of points per line> default is %d (0 means automatic determination from first scan line)\n',Nx);
fprintf('''ROIdim'',<no. of points> region of interest used for data analysis, default is %d\n',roi_dim);
fprintf('''CenX'',<point> 0 means automatic determination, default is %d\n',cen_x);
fprintf('''CenY'',<point> 0 means automatic determination, default is %d\n',cen_y);
fprintf('''DarkFieldR'',<min. radius> dark field integration starts at this radius, default is %.0f\n',dark_field_r);
fprintf('''FigNo'',<integer value> figure number for data display, default is %d\n',fig_no);
fprintf('''DirPerLine'',<0-no,1-yes> separate directory for each scan line, default is %d\n',dir_per_line);
fprintf('''ZeroOrderR'', <min. radius> pixel values inside this radius are set to zero, default is %d\n', zero_order_r);
fprintf('''FirstMoment'',<0-no,1-yes> calculate the first moment rather than a Fourier transform to get the differential phase contrast, default is %d\n',first_moment);
fprintf('''IntegratedPhase'',<0-no,1-yes> differentiate additionally the sum signal and re-differentiate it, default is %d\n',integrated_phase);
fprintf('''UpdateInterval'',<integer N> update the plot each Nth line, default is %d\n',update_interval);
fprintf('''FigureDir'',''directory'' save the resulting plot in eps, jpeg and Matlab fig format, '''' for no saving, default is %s\n',figure_dir);
fprintf('''DataDir'',''directory'' save the resulting data as Matlab file, '''' for no saving, default is %s\n',data_dir);
fprintf('''FilenameValidMask'',<path and filename> Matlab file with the valid pixel indices ind_valid, [] for no valid pixel mask,\n');
fprintf(' default is %s\n',filename_valid_mask);
fprintf('Additional <name>,<value> pairs recognized by compile_x12sa_filename and by image_read can be specified. Please call them for an overview\n');
fprintf('\n');
error('At least the (first) scan number and the number of scan lines have to be specified as input parameter.');
end
% accept cell array with name/value pairs as well
no_of_in_arg = nargin;
if (nargin == 3)
if (isempty(varargin))
% ignore empty cell array
no_of_in_arg = no_of_in_arg -1;
else
if (iscell(varargin{1}))
% use a filled one given as first and only variable parameter
varargin = varargin{1};
no_of_in_arg = 2 + length(varargin);
end
end
end
% check number of input arguments
if (rem(no_of_in_arg,2) ~= 0)
error('The optional parameters have to be specified as ''name'',''value'' pairs');
end
% parse the variable input arguments
vararg = cell(0,0);
for ind = 1:2:length(varargin)
name = varargin{ind};
value = varargin{ind+1};
switch name
case 'DetectorNumber'
detector_number = value;
case 'Nx'
Nx = value;
case 'ROIdim'
roi_dim = value;
case 'CenX'
cen_x = value;
case 'CenY'
cen_y = value;
case 'DarkFieldR'
dark_field_r = value;
case 'FigNo'
fig_no = value;
case 'DirPerLine'
dir_per_line = value;
case 'ZeroOrderR'
zero_order_r = value;
case 'FirstMoment'
first_moment = value;
case 'IntegratedPhase'
integrated_phase = value;
case 'UpdateInterval'
update_interval = value;
case 'FilenameValidMask'
filename_valid_mask = value;
otherwise
vararg{end+1} = name; %#ok<AGROW>
vararg{end+1} = value; %#ok<AGROW>
end
end
% pass some parameters to image_show
vararg(11:(end+10)) = vararg;
vararg{ 1} = 'RetryReadSleep';
vararg{ 2} = 5.0;
vararg{ 3} = 'RetryReadMax';
vararg{ 4} = 5;
vararg{ 5} = 'ErrorIfNotFound';
vararg{ 6} = 0;
% vararg{ 7} = 'BurstMode';
% if (dir_per_line)
% vararg{ 8} = 1;
% else
% vararg{ 8} = 0;
% end
vararg{7} = 'UnhandledParError';
vararg{8} = 0;
vararg{9} = 'DetectorNumber';
vararg{10} = detector_number;
% region of interest index in each dimension
roi_rel_ind = -round(0.5*roi_dim):(round(0.5*roi_dim)-1);
% load the indices of valid pixels
if ((~isempty(filename_valid_mask)) && (exist(filename_valid_mask,'file')))
fprintf('loading the valid pixel mask %s\n',filename_valid_mask);
load(filename_valid_mask);
end
% wait for the data to be available
scan_no_check = first_scan_number;
if ((dir_per_line) && (Ny > 1))
scan_no_check = scan_no_check +1;
end
filename_mask = compile_x12sa_filename(scan_no_check,0,'DetectorNumber',detector_number);
[~, fnames] = find_files(filename_mask);
data_available = (~isempty(fnames));
if (~data_available)
fprintf('Waiting for %s to become available.\n',filename_mask);
while (~data_available);
pause(1);
[~, fnames] = find_files(filename_mask);
data_available = (~isempty(fnames));
end
end
% check that number of points per line determination will be possible
% determine number of points per line
if (Nx <= 0)
if (~dir_per_line)
error('The number of points per line can only automatically be determined if separate scan directories are used for each line.');
end
vararg_remain = vararg;
vararg_remain(3:(end+2)) = vararg_remain;
vararg_remain{1} = 'SubExpWildcard';
vararg_remain{2} = 1;
[fmask,vararg_remain] = ...
compile_x12sa_filename(first_scan_number,0,vararg_remain); %#ok<NASGU>
Nx = length(dir(fmask));
if (Nx < 1)
error('No matching files found for %s',fmask);
end
end
fprintf('%d lines with %d points per line in\n',Ny,Nx);
if (integrated_phase)
figure(fig_no +1);
hold off;
clf;
% print as layed out on the screen, i.e., preserve aspect ratio
set(gcf,'PaperPositionMode','auto');
% paper size
set(gcf,'PaperType','A4');
% background color
set(gcf,'Color','white');
% resize and position
set(gcf,'Position',full_screen_position_integrated_phase);
colormap(bone(256));
end
figure(fig_no);
hold off;
clf;
% print as layed out on the screen, i.e., preserve aspect ratio
set(gcf,'PaperPositionMode','auto');
% paper size
set(gcf,'PaperType','A4');
% background color
set(gcf,'Color','white');
% resize and position
set(gcf,'Position',full_screen_position_standard);
colormap(bone(256));
% STXM display loop
point_no = 0;
scan_number = first_scan_number;
frame = [];
for ii=Ny:-1:1
sub_exp_no = 0;
for jj=Nx:-1:1
if (dir_per_line)
vararg_remain = vararg;
vararg_remain(3:(end+2)) = vararg_remain;
vararg_remain{1} = 'SubExpNo';
vararg_remain{2} = sub_exp_no;
[filename,vararg_remain] = ...
compile_x12sa_filename(scan_number,0,vararg_remain);
else
[filename,vararg_remain] = ...
compile_x12sa_filename(scan_number,point_no,vararg);
end
last_frame = frame;
[frame,vararg_remain] = image_read(filename,vararg_remain);
if (isempty(frame.data))
fprintf('%s not found, repeating the previous frame\n',filename);
frame = last_frame;
end
if (~isempty(vararg_remain))
vararg_remain
error('There are unhandled parameters.');
end
if (point_no == 0)
trans = zeros(Ny,Nx);
dpcx = trans;
dpcy = trans;
df = trans;
if ((cen_x <= 0) || (cen_y <= 0))
[cx, cy] = find_center(frame.data);
fprintf('Beam cemter guess (x,y) = (%d,%d)\n',cx,cy);
if (cen_x <= 0)
cen_x = cx;
end
if (cen_y <= 0)
cen_y = cy;
end
end
roi_x_ind = cen_x + roi_rel_ind;
if ((roi_x_ind(1) < 1) || (roi_x_ind(end) > size(frame.data,2)))
error('Region of interest out of range in x\n');
end
roi_y_ind = cen_y + roi_rel_ind;
if ((roi_y_ind(1) < 1) || (roi_y_ind(end) > size(frame.data,1)))
error('Region of interest out of range in y\n');
end
[yy,xx] = meshgrid(roi_rel_ind,roi_rel_ind);
[~, rho] = cart2pol(xx,yy);
ind_df = find((rho > dark_field_r) & (rho < roi_rel_ind(end)));
if (~isempty(filename_valid_mask))
% in case of less than full detector readout cut out the right part of
% the valid pixel mask
valid_mask = pilatus_valid_pixel_roi(valid_mask,'RoiSize',size(frame.data));
else
% if the valid pixel mask is not used specify all pixels to
% be valid
valid_mask.indices = 1:(size(frame.data,1)*size(frame.data,2));
end
% calculate the indices of the valid and invalid pixels within
% the region of interest
frame_valid = zeros(size(frame.data));
frame_valid(valid_mask.indices) = 1;
frame_valid = frame_valid(roi_y_ind,roi_x_ind);
ind_invalid = find(frame_valid == 0);
% ind_valid = find(frame_valid ~= 0);
ind_df = setdiff(ind_df,ind_invalid);
end
% cut out the region of interest
frame_roi = frame.data(roi_y_ind,roi_x_ind);
frame_roi(ind_invalid) = 0;
% set central part of detector frame to zero, if specified
if (zero_order_r> 0)
frame_roi(rho<zero_order_r) = 0; %min(frame_roi(:));
end
% data analysis for the current point
if (first_moment)
[tr,px,py] = stxm_pt2(frame_roi);
else
[tr,px,py] = stxm_pt(frame_roi);
end
trans(ii,jj) = tr;
dpcx(ii,jj) = px;
dpcy(ii,jj) = py;
% df(ii,jj) = sum(frame_roi(ind_df)) / sum(sum(frame_roi(ind_valid)));
df(ii,jj) = sum(frame_roi(ind_df));
point_no = point_no +1;
sub_exp_no = sub_exp_no +1;
end
% plot linewise each update_interval-th line
if (Nx > 1) && (Ny > 1)
if ((ii == Ny) || (rem(ii,update_interval) == 1) || (ii == 1))
if(gcf ~= fig_no)
figure(fig_no);
end
iv = 2;
ih = 2;
colormap(bone(256));
subplot(iv,ih,1);
imagesc(trans);
axis xy; axis equal; axis tight;
colorbar;
axis_min = min(min(trans(trans ~= 0)));
if (isnan(axis_min))
axis_min = 0;
end
axis_max = max(max(trans(trans ~= 0)));
if (isnan(axis_max))
axis_max = 0;
end
caxis([(axis_min-.0001) (axis_max+.0001)]);
title_str = [ 'transmission #' num2str(first_scan_number,'%d') ];
if (dir_per_line)
title_str = [ title_str '-' num2str(first_scan_number+Ny-1,'%d') ]; %#ok<AGROW>
end
title_str = sprintf('%s (detector %d)',title_str,detector_number);
title(title_str);
subplot(iv,ih,2);
imagesc(df);
axis xy; axis equal; axis tight;
colorbar;
axis_min = min(min(df(df ~= 0)));
if (isnan(axis_min))
axis_min = 0;
end
axis_max = max(max(df(df~=0)));
if (isnan(axis_max))
axis_max = 0;
end
caxis([(axis_min-.0001) (axis_max+.0001)]);
title('dark field');
subplot(iv,ih,3);
imagesc(dpcx);
axis xy; axis equal; axis tight;
colorbar;
axis_min = min(min(dpcx(dpcx ~= 0)));
if (isnan(axis_min))
axis_min = 0;
end
axis_max = max(max(dpcx(dpcx~=0)));
if (isnan(axis_max))
axis_max = 0;
end
caxis([(axis_min-.0001) (axis_max+.0001)]);
title('DPC x');
subplot(iv,ih,4);
imagesc(dpcy);
axis xy; axis equal; axis tight;
colorbar;
axis_min = min(min(dpcy(dpcy ~= 0)));
if (isnan(axis_min))
axis_min = 0;
end
axis_max = max(max(dpcy(dpcy~=0)));
if (isnan(axis_max))
axis_max = 0;
end
caxis([(axis_min-.0001) (axis_max+.0001)]);
title('DPC y');
drawnow;
end
end
if (dir_per_line)
scan_number = scan_number +1;
end
end
% store return arguments
if nargout > 0
varargout{1} = trans;
end
if nargout > 1
varargout{2} = dpcx;
end
if nargout > 2
varargout{3} = dpcy;
end
if nargout > 3
varargout{4} = df;
end
if nargout > 4
varargout{5} = phase;
end
if nargout > 5
varargout{6} = gx;
end
if nargout > 6
varargout{7} = gy;
end
if (integrated_phase)
% calculate the integrated phase from the differential phase contrast
% in horizontal and vertical direction
phase = phase_from_dpc(dpcx,dpcy, 'fourier');
% calculate the 1D differential phase contrast from the integrated
% phase
[gx, gy] = gradient(phase);
figure(fig_no +1);
iv = 1;
ih = 3;
colormap(bone(256));
subplot(iv,ih,1);
imagesc(phase);
axis xy; axis equal; axis tight;
colorbar;
axis_min = min(phase(phase ~= 0));
if (isnan(axis_min))
axis_min = 0;
end
axis_max = max(phase(phase~=0));
if (isnan(axis_max))
axis_max = 0;
end
caxis([(axis_min-.0001) (axis_max+.0001)]);
title_str = [ 'integrated phase #' num2str(first_scan_number,'%d') ];
if (dir_per_line)
title_str = [ title_str '-' num2str(first_scan_number+Ny-1,'%d') ];
end
title_str = sprintf('%s (detector %d)',title_str,detector_number);
title(title_str);
subplot(iv,ih,2);
imagesc(gx);
axis xy; axis equal; axis tight;
colorbar;
axis_min = min(gx(gx ~= 0));
if (isnan(axis_min))
axis_min = 0;
end
axis_max = max(gx(gx ~= 0));
if (isnan(axis_max))
axis_max = 0;
end
caxis([(axis_min-.0001) (axis_max+.0001)]);
title('DPC x from integrated phase');
subplot(iv,ih,3);
imagesc(gy);
axis xy; axis equal; axis tight;
colorbar;
axis_min = min(gy(gy ~= 0));
if (isnan(axis_min))
axis_min = 0;
end
axis_max = max(gy(gy ~= 0));
if (isnan(axis_max))
axis_max = 0;
end
caxis([(axis_min-.0001) (axis_max+.0001)]);
title('DPC y from integrated phase');
drawnow;
end
% file name for saving
filename = sprintf('stxm_scans_%d_%05d-%05d',detector_number,...
first_scan_number,first_scan_number+Ny-1);
% save figures
if (~isempty(figure_dir))
figure(fig_no);
% create output directories and write the plot in different formats
if (~exist(figure_dir,'dir'))
mkdir(figure_dir)
end
if ((figure_dir(end) ~= '/') && (figure_dir(end) ~= '\'))
figure_dir = [ figure_dir '/' ];
end
fprintf('output directory for figures is %s\n',figure_dir);
% resize to a smaller width as print layout
set(gcf,'Position',print_a4_position_standard);
subdir = [ figure_dir 'jpg/' ];
if (~exist(subdir,'dir'))
mkdir(subdir);
end
fprintf('saving %s.jpg\n',filename);
print('-djpeg','-r300',[subdir filename '.jpg'] );
subdir = [ figure_dir 'eps/' ];
if (~exist(subdir,'dir'))
mkdir(subdir);
end
fprintf('saving %s.eps\n',filename);
print('-depsc','-r1200',[subdir filename '.eps'] );
% resize to full screen
set(gcf,'Position',full_screen_position_standard);
subdir = [ figure_dir 'fig/' ];
if (~exist(subdir,'dir'))
mkdir(subdir);
end
fprintf('saving %s.fig\n',filename);
hgsave([subdir filename '.fig']);
if (integrated_phase)
figure(fig_no +1);
subdir = [ figure_dir 'jpg/' ];
if (~exist(subdir,'dir'))
mkdir(subdir);
end
% resize to a smaller width as print layout
set(gcf,'Position',print_a4_position_integrated_phase);
fprintf('saving %s_integrated_phase.jpg\n',filename);
print('-djpeg','-r300',[subdir filename '_integrated_phase.jpg'] );
subdir = [ figure_dir 'eps/' ];
if (~exist(subdir,'dir'))
mkdir(subdir);
end
fprintf('saving %s_integrated_phase.eps\n',filename);
print('-depsc','-r1200',[subdir filename '_integrated_phase.eps'] );
% resize to a smaller width as print layout
set(gcf,'Position',full_screen_position_integrated_phase);
subdir = [ figure_dir 'fig/' ];
if (~exist(subdir,'dir'))
mkdir(subdir);
end
fprintf('saving %s_integrated_phase.fig\n',filename);
hgsave([subdir filename '_integrated_phase.fig']);
end
end
% save resulting data
if (~isempty(data_dir))
if ((data_dir(end) ~= '/') && (data_dir(end) ~= '\'))
data_dir = [ data_dir '/' ];
end
% create output directory
if (~exist(data_dir,'dir'))
mkdir(data_dir)
end
% save data
fprintf('saving %s.mat\n',[data_dir filename]);
if (integrated_phase)
save([data_dir filename],'trans','dpcx','dpcy','df', 'phase', 'gx', 'gy');
else
save([data_dir filename],'trans','dpcx','dpcy','df');
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [cx, cy] = find_center(f)
f = medfilt2(f,[5 5]);
[~, cx] = max(sum(f,1));
[~, cy] = max(sum(f,2));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [tr, px, py] = stxm_pt(a)
persistent c1 c2 s1 s2 sz
if (isempty(sz)) || (any(sz ~= size(a)))
sz = size(a);
c1 = -cos(2*pi*(0:sz(1)-1)/sz(1));
s1 = sin(2*pi*(0:sz(1)-1)/sz(1));
c2 = -cos(2*pi*(0:sz(2)-1)/sz(2));
s2 = sin(2*pi*(0:sz(2)-1)/sz(2));
end
a1 = sum(a,1);
a2 = sum(a,2)';
tr = sum(a1);
px = atan2(sum(a1.*c1), sum(a1.*s1));
py = atan2(sum(a2.*c2), sum(a2.*s2));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [tr, px, py ] = stxm_pt2(a)
persistent x y sz
if (isempty(sz)) || (any(sz ~= size(a)))
sz = size(a);
% masking out the invalid pixels is done by setting the
% corresponding intensities to zero before calling this function
[y,x] = ndgrid((0:sz(1)-1)-sz(1)/2, (0:sz(1)-1)-sz(1)/2);
% x2 = x.^2;
end
tr = sum(sum(a));
px = sum(sum(a.*x))/tr;
py = sum(sum(a.*y))/tr;
% p2 = (sum(a1.*x2)/tr + sum(a2.*x2)/tr - px^2 - py^2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function p = phase_from_dpc(dpcx,dpcy,varargin)
%
% Integrates the phase from a combination of x and y gradients.
% phase_from_dpc(dpcx,dpcy,'fourier') uses the Fourier method (default),
% phase_from_dpc(dpcx,dpcy,'finitdiff') uses a finite difference method.
if nargin > 2
method = varargin{1};
else
%method = 'fourier';
method = 'finitediff';
end
px = -dpcy;
py = -dpcx;
sz = size(px);
switch lower(method)
case 'fourier'
f = zeros(2*sz);
f(1:sz(1),1:sz(2)) = px + 1i*py;
f(1:sz(1),sz(2)+1:end) = fliplr(px + 1i*py);
f(sz(1)+1:end,1:sz(2)) = flipud(px + 1i*py);
f(sz(1)+1:end,sz(2)+1:end) = rot90(px + 1i*py,2);
[x1,x2] = ndgrid(-sz(1):(sz(1)-1),-sz(2):(sz(2)-1));
q1 = pi*fftshift(x1)/sz(1);
q2 = pi*fftshift(x2)/sz(2);
qc = q2 - 1i*q1;
inv_qc = 1./qc;
inv_qc(1,1) = 0;
nf = ifftn(fftn(f).*inv_qc);
p = real(nf(1:sz(1),1:sz(2)));
case 'finitediff'
ggx = pgradient(dpcx);
[~, ggy] = pgradient(dpcy);
f = .25*(ggx + ggy);
ta = zeros(sz);
for i = 1:10000
ta = ta + (pdel2(ta) - f);
% Zero boundary conditions
%ta(1,:) = 0;
%ta(:,1) = 0;
%ta(end,:) = 0;
%ta(:,end) = 0;
% Zero normal gradient boundary condition
ta(1,:) = ta(2,:);
ta(:,1) = ta(:,2);
ta(end,:) = ta(end-1,:);
ta(:,end) = ta(:,end-1);
if mod(i,1000)==0
figure(1); imagesc(real(ta)); colormap(bone(256)); colorbar; drawnow;
end
p = ta;
end
end