function x = rbfpreimg3(model,nn)% RBFPREIMG3 RBF pre-image problem by Kwok-Tsang's algorithm.%% Synopsis:%  x = rbfpreimg3(model)%  x = rbfpreimg3(model,nn)%% Description:%  x = rbfpreimg3(model) is an implementation of the algorithm %    by [Kwok03] to solve the pre-image problem for kernel expansion %    with RBF kernel. The kernel expansion is given in the input %    structure model.%%  x = rbfpreimg3(model,nn) use to set number of nearest %    neighbours (default 10).%% Input:%  model [struct] RBF kernel expansion:%   .Alpha [nsv x 1] Weights.%   .sv.X [dim x nsv] Vectors.%   .options.arg [1x1] RBF kernel width.%%  nn [1x1] Number of nearest neighbours.%  % Output:%  x [dim x 1] Pre-image of the RBF kernel expansion.%  % See also %  RBFPREIMG1, RBFPREIMG2, RSRBF, KPCAREC.%% About: Statistical Pattern Recognition Toolbox% (C) 1999-2003, Written by Vojtech Franc and Vaclav Hlavac% <a href="http://www.cvut.cz">Czech Technical University Prague</a>% <a href="http://www.feld.cvut.cz">Faculty of Electrical Engineering</a>% <a href="http://cmp.felk.cvut.cz">Center for Machine Perception</a>% Modifications:% 17-may-2004, VF% 21-Feb-2004, VF% 17-February-2004, Petr Posik[dim, num_data] = size(model.sv.X);% default number of used nearest neighbours if nargin < 2, nn = min([num_data, 10]); endK = kernel(model.sv.X, 'rbf', model.options.arg );Const2 = model.Alpha(:)'*K*model.Alpha(:);df2 = 1 + Const2 - 2*K*model.Alpha(:);d2 = -2*model.options.arg^2 * log( 1 - 0.5*df2);% select nn neighbours[dummy, inx] = sort( df2 );X = model.sv.X(:,inx(1:nn));df2 = df2(inx(1:nn));d2 = d2(inx(1:nn));H = eye(nn,nn) - 1/nn * ones(nn,nn);[U,L,V] = svd(X*H);r = rank(L);Z = L*V';d02 = sum(Z.^2)';z = -0.5*pinv(Z')*(d2-d02);x = U*z + sum(X,2)/nn;return;