From e4d0007e666d78759e2511d7055c3859d4e25186 Mon Sep 17 00:00:00 2001
From: John Kanji <jkanji@dgp.toronto.edu>
Date: Mon, 10 Feb 2020 16:24:29 -0500
Subject: [PATCH] Add Hessian Smoothing function based on Stein et al 2018

---
 mesh/hessian_smooth.m | 110 ++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 110 insertions(+)
 create mode 100644 mesh/hessian_smooth.m

diff --git a/mesh/hessian_smooth.m b/mesh/hessian_smooth.m
new file mode 100644
index 00000000..7d461278
--- /dev/null
+++ b/mesh/hessian_smooth.m
@@ -0,0 +1,110 @@
+function [U,Usteps] = hessian_smooth(V,F,varargin)
+  % HESSIAN_SMOOTH smooth a planar mesh using implicit/explicit hessian
+  % smoothing by minimizing the integrated squared hessian energy
+  % see Stein et al. 2018, http://www.cs.columbia.edu/cg/hessians/
+  %
+  % [U] = hessian_smooth(V,F)
+  % [U] = hessian_smooth(V,F,b,lambda,method,S)
+  % [U,Usteps] = hessian_smooth(V,F,'b',b,'Lambda','S',f)
+  % 
+  % Inputs:
+  %   V  #V x 2 matrix of vertex coordinates
+  %   F  #F x 3  matrix of indices of triangle corners
+  % Optional:
+  %   b  list of indices of fixed vertices
+  %   Lambda  diffusion speed parameter {0.1}
+  %   Method  method to use:
+  %     'implicit' (default)
+  %     'explicit'
+  %     'limit'
+  %   S  scalar fields to smooth (default V)
+  %   MaxIter  maximum number of iterations to solve
+  %   MaxDiff  minimum difference between consecutive iterations for
+  %            convergence
+  % Outputs:
+  %   U  smoothed function values
+  %   Usteps  list of smoothed function values for each iteration
+  %   
+  % See also: hessian_squared, laplacian_smooth
+  %
+  
+  % defaults
+  lambda = 0.1;
+  method = 'implicit';
+  b = [];
+  S = V;
+  max_iter = 1000;
+  max_diff = 1e-13;
+  % Map of parameter names to variable names
+  params_to_variables = containers.Map( ...
+    {'b','Lambda','Method','S','MaxIter','MaxDiff'}, ...
+    {'b','lambda','method','S','max_iter','max_diff'});
+  v = 1;
+  while v <= numel(varargin)
+    param_name = varargin{v};
+    if isKey(params_to_variables,param_name)
+      assert(v+1<=numel(varargin));
+      v = v+1;
+      % Trick: use feval on anonymous function to use assignin to this workspace 
+      feval(@()assignin('caller',params_to_variables(param_name),varargin{v}));
+    else
+      error('Unsupported parameter: %s',varargin{v});
+    end
+    v=v+1;
+  end
+  H = hessian_squared(V,F);
+  M = massmatrix(V,F,'barycentric');
+  I = speye(size(H));
+  
+  if strcmp(method, 'limit')
+    for d = 1:size(S,2)
+      U(:,d) = min_quad_with_fixed(0.5*H,[],b,S(b,d));
+    end
+    return
+  end
+  
+  if nargout > 1
+    Usteps = zeros([size(S) max_iter]);
+  end
+  U = S;
+  iter = 1;
+  while true
+    if nargout > 1
+      Usteps(:,:,iter) = U;
+    end
+    
+    U_prev = U;
+    switch method
+    case 'implicit'
+      Q = (M-lambda*H);
+      for d = 1:size(S,2)
+        U(:,d) = min_quad_with_fixed(Q*0.5,-U(:,d),b,S(b,d),[],[]);
+      end
+    case 'explicit'
+      Q = (I+lambda*H);
+      U = Q * U;
+      % enforce boundary
+      U(b,:) = S(b,:);
+    otherwise
+      error(['' method ' is not a supported smoothing method']);
+    end
+    
+    % Use difference from previous as stopping criterion
+    d = trace(((U-U_prev)'*M*(U-U_prev)).^2);
+    if d < max_diff
+      warning('converged...');
+      break;
+    end
+    if iter >= max_iter
+      warning('Max iterations (%d) exceeded without convergence',max_iter);
+      break;
+    end
+    iter = iter + 1;
+  end
+
+  if nargout > 1
+    Usteps = Usteps(:,:,1:iter);
+  end
+
+end
+