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2026-09-18 00:09:10 +09:00 · 2026-08-07 15:56:42 +09:00
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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Unwrapping phase based on Ghiglia and Romero (1994) based on weighted and unweighted least-square method
+% URL: https://doi.org/10.1364/JOSAA.11.000107
+% Inputs:
+%   * psi: wrapped phase from -pi to pi
+%   * weight: weight of the phase (optional, default: all ones)
+% Output:
+%   * phi: unwrapped phase from the weighted (or unweighted) least-square phase unwrapping
+% Author: Muhammad F. Kasim (University of Oxford, 2016)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+function phi = phase_unwrap(psi, weight)
+    if (nargin < 2) % unweighted phase unwrap
+        % get the wrapped differences of the wrapped values
+        dx = [zeros([size(psi,1),1]), wrapToPi(diff(psi, 1, 2)), zeros([size(psi,1),1])];
+        dy = [zeros([1,size(psi,2)]); wrapToPi(diff(psi, 1, 1)); zeros([1,size(psi,2)])];
+        rho = diff(dx, 1, 2) + diff(dy, 1, 1);
+        
+        % get the result by solving the poisson equation
+        phi = solvePoisson(rho);
+        
+    else % weighted phase unwrap
+        % check if the weight has the same size as psi
+        if (~all(size(weight) == size(psi)))
+            error('Argument error: Size of the weight must be the same as size of the wrapped phase');
+        end
+        
+        % vector b in the paper (eq 15) is dx and dy
+        dx = [wrapToPi(diff(psi, 1, 2)), zeros([size(psi,1),1])];
+        dy = [wrapToPi(diff(psi, 1, 1)); zeros([1,size(psi,2)])];
+        
+        % multiply the vector b by weight square (W^T * W)
+        WW = weight .* weight;
+        WWdx = WW .* dx;
+        WWdy = WW .* dy;
+        
+        % applying A^T to WWdx and WWdy is like obtaining rho in the unweighted case
+        WWdx2 = [zeros([size(psi,1),1]), WWdx];
+        WWdy2 = [zeros([1,size(psi,2)]); WWdy];
+        rk = diff(WWdx2, 1, 2) + diff(WWdy2, 1, 1);
+        normR0 = norm(rk(:));
+        
+        % start the iteration
+        eps = 1e-6;
+        k = 0;
+        phi = zeros(size(psi));
+        while (~all(rk == 0))
+            zk = solvePoisson(rk);
+            k = k + 1;
+            if (k == 1) pk = zk;
+            else 
+                betak = sum(sum(rk .* zk)) / sum(sum(rkprev .* zkprev));
+                pk = zk + betak * pk;
+            end
+            
+            % save the current value as the previous values
+            rkprev = rk;
+            zkprev = zk;
+            
+            % perform one scalar and two vectors update
+            Qpk = applyQ(pk, WW);
+            alphak = sum(sum(rk .* zk)) / sum(sum(pk .* Qpk));
+            phi = phi + alphak * pk;
+            rk = rk - alphak * Qpk;
+            
+            % check the stopping conditions
+            if ((k >= numel(psi)) || (norm(rk(:)) < eps * normR0)) break; end;
+        end
+    end
+end
+
+function phi = solvePoisson(rho)
+    % solve the poisson equation using dct
+    dctRho = dct2(rho);
+    [N, M] = size(rho);
+    [I, J] = meshgrid([0:M-1], [0:N-1]);
+    dctPhi = dctRho ./ 2 ./ (cos(pi*I/M) + cos(pi*J/N) - 2);
+    dctPhi(1,1) = 0; % handling the inf/nan value
+    
+    % now invert to get the result
+    phi = idct2(dctPhi);
+    
+end
+
+% apply the transformation (A^T)(W^T)(W)(A) to 2D matrix
+function Qp = applyQ(p, WW)
+    % apply (A)
+    dx = [diff(p, 1, 2), zeros([size(p,1),1])];
+    dy = [diff(p, 1, 1); zeros([1,size(p,2)])];
+    
+    % apply (W^T)(W)
+    WWdx = WW .* dx;
+    WWdy = WW .* dy;
+    
+    % apply (A^T)
+    WWdx2 = [zeros([size(p,1),1]), WWdx];
+    WWdy2 = [zeros([1,size(p,2)]); WWdy];
+    Qp = diff(WWdx2,1,2) + diff(WWdy2,1,1);
+end