New analysis script - characterization of lattice geometries in real space
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Data-Analyzer/computeBondOrderParameters.m
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179
Data-Analyzer/computeBondOrderParameters.m
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function [psi_global, order_ratio, Npoints] = computeBondOrderParameters(I)
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% Analyze bond-orientational order in a 2D image, restricted to a circular region
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%
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% Input:
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% I - 2D grayscale image (double or uint8)
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%
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% Outputs:
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% psi_global - struct with fields psi2, psi4, psi6 (⟨|ψₙ|⟩ values)
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% order_ratio - scalar = ⟨|ψ₆|⟩ / ⟨|ψ₂|⟩
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% Npoints - number of detected points used
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%% Step 1: Create mask
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[Ny, Nx] = size(I);
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% Parameters for elliptical mask
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a = Nx / 2.25; % Semi-major axis (x-direction)
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b = Ny / 2.25; % Semi-minor axis (y-direction)
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theta_deg = 0; % Rotation angle in degrees
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theta = deg2rad(theta_deg); % Convert to radians
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% Meshgrid coordinates
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[Ny, Nx] = size(I);
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[X, Y] = meshgrid(1:Nx, 1:Ny);
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cx = Nx / 2;
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cy = Ny / 2;
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% Shifted coordinates
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Xc = X - cx;
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Yc = Y - cy;
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% Rotate coordinates
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Xr = cos(theta) * Xc + sin(theta) * Yc;
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Yr = -sin(theta) * Xc + cos(theta) * Yc;
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% Elliptical mask equation
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mask = (Xr / a).^2 + (Yr / b).^2 <= 1;
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% Apply mask to image
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I = double(I) .* mask;
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%% Step 2: Detect local maxima within mask
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I_smooth = imgaussfilt(I, 2);
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peaks = imregionalmax(I_smooth);
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[y, x] = find(peaks);
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Npoints = length(x);
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if Npoints < 6
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warning('Too few points (%d) to compute bond order meaningfully.', Npoints);
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psi_global = struct('psi2', NaN, 'psi4', NaN, 'psi6', NaN);
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order_ratio = NaN;
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return;
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end
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%% Step 3: Try Delaunay triangulation
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use_kNN = false;
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try
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tri = delaunay(x, y);
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catch
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warning('Delaunay triangulation failed - falling back to kNN neighbor search');
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use_kNN = true;
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end
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% 3) Find neighbors
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neighbors = cell(Npoints, 1);
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if ~use_kNN
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% Use Delaunay neighbors
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for i = 1:size(tri,1)
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for j = 1:3
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idx = tri(i,j);
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nbrs = tri(i,[1 2 3] ~= j);
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neighbors{idx} = unique([neighbors{idx}, nbrs]);
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end
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end
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else
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% Use kNN neighbors (e.g. k=6)
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k = min(6, Npoints-1);
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pts = [x, y];
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[idx, ~] = knnsearch(pts, pts, 'K', k+1); % +1 for self
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for j = 1:Npoints
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neighbors{j} = idx(j, 2:end);
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end
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end
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% 4) Compute bond order parameters
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psi2 = zeros(Npoints,1);
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psi4 = zeros(Npoints,1);
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psi6 = zeros(Npoints,1);
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for j = 1:Npoints
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nbrs = neighbors{j};
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nbrs(nbrs == j) = [];
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if isempty(nbrs)
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continue;
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end
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angles = atan2(y(nbrs) - y(j), x(nbrs) - x(j));
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psi2(j) = abs(mean(exp(1i * 2 * angles)));
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psi4(j) = abs(mean(exp(1i * 4 * angles)));
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psi6(j) = abs(mean(exp(1i * 6 * angles)));
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end
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psi_global.psi2 = mean(psi2);
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psi_global.psi4 = mean(psi4);
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psi_global.psi6 = mean(psi6);
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order_ratio = psi_global.psi6 / psi_global.psi2;
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% --- After neighbors are found and (x,y) extracted ---
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% 1) Angular distribution of all bonds pooled from all points
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all_angles = [];
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for j = 1:Npoints
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nbrs = neighbors{j};
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nbrs(nbrs == j) = [];
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if isempty(nbrs), continue; end
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angles = atan2(y(nbrs)-y(j), x(nbrs)-x(j));
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all_angles = [all_angles; angles(:)];
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end
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% Histogram of bond angles over [-pi, pi]
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nbins = 36; % 10 degree bins
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edges = linspace(-pi, pi, nbins+1);
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counts = histcounts(all_angles, edges, 'Normalization','probability');
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% Circular variance of bond angles
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R = abs(mean(exp(1i*all_angles)));
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circ_variance = 1 - R; % 0 means angles clustered; 1 means uniform
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% 2) Positional correlation function g(r) estimate
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% Compute pairwise distances
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coords = [x y];
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D = pdist(coords);
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max_r = min([Nx, Ny])/2; % max radius for g(r)
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% Compute histogram of distances
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dr = 1; % bin width in pixels (adjust)
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r_edges = 0:dr:max_r;
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[counts_r, r_bins] = histcounts(D, r_edges);
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% Normalize g(r) by ideal gas distribution (circle annulus area)
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r_centers = r_edges(1:end-1) + dr/2;
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area_annulus = pi*((r_edges(2:end)).^2 - (r_edges(1:end-1)).^2);
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density = Npoints / (Nx*Ny);
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g_r = counts_r ./ (density * area_annulus * Npoints);
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figure(1);
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clf
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set(gcf,'Position',[500 100 1250 500])
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t = tiledlayout(1, 3, 'TileSpacing', 'compact', 'Padding', 'compact'); % 1x4 grid
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nexttile
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imshow(I, []);
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hold on;
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scatter(x, y, 30, 'w', 'filled');
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hold off;
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colormap(Helper.Colormaps.plasma()); % Example colormap for original image plot
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cbar1 = colorbar;
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cbar1.Label.Interpreter = 'latex';
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xlabel('$x$ ($\mu$m)', 'Interpreter', 'latex', 'FontSize', 14);
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ylabel('$y$ ($\mu$m)', 'Interpreter', 'latex', 'FontSize', 14);
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title('$|\Psi(x,y)|^2$', 'Interpreter', 'latex', 'FontSize', 14);
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axis square;
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% --- Plot angular distribution ---
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nexttile
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polarhistogram(all_angles, nbins, 'Normalization', 'probability');
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title(sprintf('Bond angle distribution (circ. variance = %.3f)', circ_variance));
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% --- Plot g(r) ---
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nexttile
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plot(r_centers, g_r, 'LineWidth', 2);
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xlabel('r (pixels)');
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ylabel('g(r)');
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title('Radial Pair Correlation Function g(r)');
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grid on;
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axis square
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end
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63
Data-Analyzer/execution_scripts.m
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63
Data-Analyzer/execution_scripts.m
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%%
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Data = load('D:/Results - Numerics/Data_Full3D/PhaseDiagram/ImagTimePropagation/Theta0/HighN/aS_9.562000e+01_theta_000_phi_000_N_712500/Run_000/psi_gs.mat','psi','Params','Transf','Observ');
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Params = Data.Params;
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Transf = Data.Transf;
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Observ = Data.Observ;
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if isgpuarray(Data.psi)
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psi = gather(Data.psi);
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else
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psi = Data.psi;
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end
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if isgpuarray(Data.Observ.residual)
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Observ.residual = gather(Data.Observ.residual);
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else
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Observ.residual = Data.Observ.residual;
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end
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% Axes scaling and coordinates in micrometers
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x = Transf.x * Params.l0 * 1e6;
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y = Transf.y * Params.l0 * 1e6;
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z = Transf.z * Params.l0 * 1e6;
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dz = z(2)-z(1);
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% Compute probability density |psi|^2
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n = abs(psi).^2;
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nxy = squeeze(trapz(n*dz,3));
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IMG = nxy;
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extractHexaticOrder(IMG)
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%%
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Data = load('D:/Results - Numerics/Data_Full3D/PhaseDiagram/ImagTimePropagation/Theta0/HighN/aS_9.562000e+01_theta_000_phi_000_N_712500/Run_000/psi_gs.mat','psi','Params','Transf','Observ');
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% Data = load('D:/Results - Numerics/Data_Full3D/PhaseDiagram/ImagTimePropagation/Theta40/HighN/aS_9.562000e+01_theta_040_phi_000_N_508333/Run_000/psi_gs.mat','psi','Params','Transf','Observ');
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Params = Data.Params;
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Transf = Data.Transf;
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Observ = Data.Observ;
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if isgpuarray(Data.psi)
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psi = gather(Data.psi);
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else
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psi = Data.psi;
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end
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if isgpuarray(Data.Observ.residual)
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Observ.residual = gather(Data.Observ.residual);
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else
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Observ.residual = Data.Observ.residual;
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end
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% Axes scaling and coordinates in micrometers
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x = Transf.x * Params.l0 * 1e6;
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y = Transf.y * Params.l0 * 1e6;
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z = Transf.z * Params.l0 * 1e6;
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dz = z(2)-z(1);
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% Compute probability density |psi|^2
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n = abs(psi).^2;
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nxy = squeeze(trapz(n*dz,3));
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IMG = nxy;
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%
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[psi, ratio, N] = computeBondOrderParameters(IMG);
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fprintf('Points: %d\n⟨|ψ₂|⟩ = %.3f, ⟨|ψ₄|⟩ = %.3f, ⟨|ψ₆|⟩ = %.3f\n', N, psi.psi2, psi.psi4, psi.psi6);
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fprintf('(⟨|ψ₆|⟩ / ⟨|ψ₂|⟩) = %.3f\n', ratio);
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98
Data-Analyzer/extractHexaticOrder.m
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Data-Analyzer/extractHexaticOrder.m
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function extractHexaticOrder(I)
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% Input:
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% I = 2D image matrix (grayscale, double)
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% Output:
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% Plots original image + hexatic order + histogram
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%% 1) Detect dots (local maxima)
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I_smooth = imgaussfilt(I, 2);
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bw = imregionalmax(I_smooth);
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[y, x] = find(bw);
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if length(x) < 6
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error('Too few detected dots (%d) for meaningful hexatic order.', length(x));
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end
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%% 2) Compute hexatic order parameter psi6
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% Delaunay triangulation to find neighbors
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tri = delaunay(x,y);
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N = length(x);
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neighbors = cell(N,1);
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for i = 1:size(tri,1)
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neighbors{tri(i,1)} = unique([neighbors{tri(i,1)} tri(i,2) tri(i,3)]);
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neighbors{tri(i,2)} = unique([neighbors{tri(i,2)} tri(i,1) tri(i,3)]);
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neighbors{tri(i,3)} = unique([neighbors{tri(i,3)} tri(i,1) tri(i,2)]);
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end
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psi6_vals = zeros(N,1);
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for j = 1:N
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nbrs = neighbors{j};
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nbrs(nbrs == j) = [];
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if isempty(nbrs)
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psi6_vals(j) = 0;
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continue;
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end
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angles = atan2(y(nbrs)-y(j), x(nbrs)-x(j));
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psi6_vals(j) = abs(mean(exp(1i*6*angles)));
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end
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psi6_global = mean(psi6_vals);
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%% 3) Plotting
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figure('Name','Hexatic Order Analysis','NumberTitle','off', 'Units','normalized', 'Position',[0.2 0.2 0.6 0.7]);
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t = tiledlayout(2,2, 'Padding','none', 'TileSpacing','compact');
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% Top left: Original image with detected dots (square)
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ax1 = nexttile(1);
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imshow(I, []);
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hold on;
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scatter(x, y, 30, 'w', 'filled');
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hold off;
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colormap(ax1, Helper.Colormaps.plasma()); % Example colormap for original image plot
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cbar1 = colorbar(ax1);
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cbar1.Label.Interpreter = 'latex';
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xlabel(ax1, '$x$ ($\mu$m)', 'Interpreter', 'latex', 'FontSize', 14);
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ylabel(ax1, '$y$ ($\mu$m)', 'Interpreter', 'latex', 'FontSize', 14);
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title(ax1, '$|\Psi(x,y)|^2$', 'Interpreter', 'latex', 'FontSize', 14);
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axis(ax1, 'square');
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% Top right: Histogram of local |psi6| (square)
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ax2 = nexttile(2);
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histogram(psi6_vals, 20, 'FaceColor', 'b');
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xlabel(ax2, '$|\psi_6|$', 'Interpreter', 'latex', 'FontSize', 14);
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ylabel(ax2, 'Count');
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title(ax2, 'Histogram of Local Hexatic Order');
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xlim(ax2, [0 1]);
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grid(ax2, 'on');
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axis(ax2, 'square');
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% Bottom: Hexatic order magnitude on dots with bond orientation arrows (span 2 tiles)
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ax3 = nexttile([1 2]);
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scatter_handle = scatter(x, y, 50, psi6_vals, 'filled');
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colormap(ax3, 'jet'); % Different colormap for hexatic order magnitude
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cbar3 = colorbar(ax3);
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cbar3.Label.String = '|$\psi_6$|';
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cbar3.Label.Interpreter = 'latex';
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caxis(ax3, [0 1]);
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axis(ax3, 'equal');
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axis(ax3, 'tight');
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title(ax3, sprintf('Local Hexatic Order |\\psi_6|, Global = %.3f', psi6_global));
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hold(ax3, 'on');
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for j = 1:N
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nbrs = neighbors{j};
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nbrs(nbrs == j) = [];
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if isempty(nbrs)
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continue;
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end
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angles = atan2(y(nbrs)-y(j), x(nbrs)-x(j));
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psi6_complex = mean(exp(1i*6*angles));
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local_angle = angle(psi6_complex)/6;
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arrow_length = 5;
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quiver(ax3, x(j), y(j), arrow_length*cos(local_angle), arrow_length*sin(local_angle), ...
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'k', 'MaxHeadSize', 2, 'AutoScale', 'off');
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end
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hold(ax3, 'off');
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end
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