function [Xd,lam,U] = denoise_kpca(Xo,A1,A2,A3,A4,A5)% Reconstruct the data mapped on the first principal components% % >> Xd = denoise_kpca(X, kernel, kernel_par);% % Denoising can be done by moving the point in inputspace so that% its corresponding map to feature space is optimized. This means% that the data point in feature space is as close as possible with% its corresponding reconstructed points using the principal% components. If the principal components are to be calculated on% the same data X as the one one wants to denoise, use the command:% % >> Xd         = denoise_kpca(X, kernel, kernel_par);% >> [Xd,lam,U] = denoise_kpca(X, kernel, kernel_par, [], type, nb);% % When one wants to denoise data 'Xt' other than the data used to obtain the principal components:% % >> Xd          = denoise_kpca(X, kernel, kernel_par, Xt);% >> [Xd, lam, U] = denoise_kpca(X, kernel, kernel_par, Xt, type, nb);% %% Full syntax% % >> [Xd, lam, U] = denoise_kpca(X, kernel, kernel_par, Xt);% >> [Xd, lam, U] = denoise_kpca(X, kernel, kernel_par, Xt, type);% %       Outputs    %         Xd         : N x d (Nt x d) matrix with denoised data X (Xt)%         lam(*)     : nb x 1 vector with eigenvalues of principal components%         U(*)       : N x nb (Nt x d) matrix with principal eigenvectors%       Inputs    %         X          : N x d matrix with data points used for finding the principal components%         kernel     : Kernel type (e.g. 'RBF_kernel')%         kernel_par : Kernel parameter (bandwidth in the case of the 'RBF_kernel')%         Xt(*)      : Nt x d matrix with the points to denoise (if not specified, X is denoised instead)%         type(*)    : 'eig'(*), 'svd', 'eigs', 'eign'%         nb(*)      : Number of principal components used in approximation% % >> Xd  = denoise_kpca(X, U, lam, kernel, kernel_par, Xt);% %       Outputs    %         Xd         : N x d (Nt x d) matrix with denoised data X (Xt)%       Inputs    %         X          : N x d matrix with data points used for finding the principal components%         U          : N x nb (Nt x d) matrix with principal eigenvectors%         lam        : nb x 1 vector with eigenvalues of principal components%         kernel     : Kernel type (e.g. 'RBF_kernel')%         kernel_par : Kernel parameter (bandwidth in the case of the 'RBF_kernel')%         Xt(*)      : Nt x d matrix with the points to denoise (if not specified, X is denoised instead)% % See also:%   kpca, kernel_matrix, RBF_kernel% Copyright (c) 2002,  KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlabif ~exist('fminunc'),  error('This function needs the optimization function ''fminunc''.');end    if isstr(A1),  kernel = A1;  par = A2;  eval('if isempty(Xt), Xt = A3; end','Xt = Xo;');  eval('etype = A4;','etype = ''svd'';');  eval('nb = A5;','nb = ''inf'';');  [lam,U] = kpca(Xo,kernel,par);  [lam,U] = kpca(Xo,kernel,par,[], etype, nb);else  U = A1;  lam = A2;  kernel = A3;  par = A4;  eval('Xt = A5;','Xt = Xo;');endwarning off[nb,d] = size(Xt);for n=1:nb,  x = Xt(n,:);  %dist_phi(x,x,kernel,par,U,lam,Xo)  Xd(n,:) = fminunc(@dist_phi,x,[],x,kernel,par,U,lam,Xo);endwarning onfunction d = dist_phi(x,xor,kernel,par,U,lam,Xo)% the distance in feature space between x and the subspace spanned% by U,lamk = kernel_matrix(Xo,kernel,par,[x;xor]);betas = k(:,1)'*U;alphas = k(:,2)'*U;d = feval(kernel,x,x,par)-2*betas*alphas';