function X = Adj_SeparateAngles(R,deep,is_real)
% Adj_SeparateAngles -- Adjoint of the frequency angular partioning 
% Usage:
%   X = Adj_SeparateAngles(R,deep)
% Inputs:
%   R         Array of smoothly localized Fourier samples 
%   deep      Depth of the angular partition; number of angular
%             wedges per cardinal point (North,South,East West) is
%             equal to 2^deep
% Outputs
%   X         Squared array ---interpreted as a smoothly windowed FT. 
% See Also
%   SeparateAngles, AtA, AtA_Toeplitz
%
% By Emmanuel Candes, 2003-2004

  if nargin < 3,
    is_real = 0;
  end
  
  nn  = size(R);
  n = 4*nn(3);
  n2 = n/2;	
  boxlen = n/2^deep;
  boxcnt = 2^deep;	
  
  X = zeros(n);
  XX = zeros(n);
  
  [ix,w] = DetailMeyerWindow([n2/4 n2/2],3);
  lx = reverse(-ix);
  
  m      =  0:(boxcnt - 1);
  ym     =  (m-boxcnt/2).*boxlen + boxlen/2;
  slope  =  -ym./n2;		 
  
  % Columns
  for r = 1:(n2/2),	    
    k = lx(r); t = n2 + k + 1;
    shift  =  ym + slope.*(k+n2);
    alpha =  -k./n2;
    w = MakeSineWindow(boxlen,alpha);	
    
    y = squeeze(R(1,:,r,:)).';
    y =  Adj_Evaluate_FT(y(:),shift,boxlen,0,w);
    
    X(:,t)  = y;
    
    rsym = n2/2+1-r; t = n2 - k + 1; 
    if (is_real)
      X(:,t) = conj(y);
    else
      shift = -shift + 1;
      y = squeeze(R(2,:,rsym,:)).';
      X(:,t) = Adj_Evaluate_FT(y(:),shift,boxlen,0,w);
    end
  end
  
  X = fft_mid0(X)./sqrt(n);
  
  % Rows
  for r = 1:(n2/2),	      
    k = lx(r);  t = n2 + k + 1;
    shift  =  ym + slope.*(k+n2);
    alpha =  -k./n2;
    w = MakeSineWindow(boxlen,alpha);	
    
    y = squeeze(R(3,:,r,:)).';
    y = Adj_Evaluate_FT(y(:),shift,boxlen,0,w);
    
    XX(:,t)  = y; 
    
    rsym = n2/2+1-r; t = n2 - k + 1; 
    if (is_real)
      XX(:,t) = conj(y);
    else
		shift = -shift + 1;
		y = squeeze(R(4,:,rsym,:)).';
      XX(:,t) = Adj_Evaluate_FT(y(:),shift,boxlen,0,w);
    end
    
  end	
  
  XX = fft_mid0(XX).'/sqrt(n);
  X = X + XX; %summation