function x = rbfpreimg( model, options, init_point  )% RBFPREIMG RBF pre-image by Schoelkopf's fixed-point algorithm.%% Synopsis:%  x = rbfpreimg( model )%  x = rbfpreimg( model, options  )%  x = rbfpreimg( model, options, init_point  )%% Description:%  x = rbfpreimg( model ) it is an implementation of the %   Schoelkopf's fixed-point algorithm to solve the pre-image %   problem for kernel expansion wiht RBF kernel [Schol98a]. %   The kernel expansion is given in the input structure model.%%  x = rbfpreimg( model, options  ) use structure options to %   set up control parameters: %    tmax ... maximal number of iterations.%    eps ... minimal change in the norm of the optimized vector. %%  x = rbfpreimg( model, options, init_point  ) use to set up%    starting point of the optimization otherwise it is seleceted %    randomly.%% Input:%  model [struct] Kernel expansion:%   .Alpha [nsv x 1] Coefficients of the kernel expansion.%   .sv.X [dim x nsv] Vectors of defining the expansion.%   .options.ker [string] Must be equl to 'rbf'.%   .options.arg [1x1] Argument of the RBF kernel.% %  options [struct] Control parameters:%   .tmax [1x1] Maximal number of iterations (default 1e6).%   .eps [1x1] Minimal change of the optimized vector x.%%  init_point [dim x 1] Initial point of optimization.%  % Output:%  x [dim x 1] Pre-image of the RBF kernel expansion.%  % See also %  RBFPREIMG2, RBFPREIMG3, 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:% 4-may-2004, VF% 1-July-2003, VF% 30-June-2003, VF% input arguments%-------------------------------if nargin < 2, options = []; else options = c2s( options); endif ~isfield(options,'tmax'), options.tmax = 1e6; endif ~isfield(options,'eps'), options.eps = 1e-12; end[dim,nsv] = size(model.sv.X);% select initial point %-----------------------if nargin == 3,  x = init_point;else  x = model.sv.X*(2*rand(nsv,1)-ones(nsv,1));endold_x = x;step=0;change = inf;% main loop%----------------------while step < options.tmax & change > options.eps,    step = step + 1;      k = kernel(model.sv.X, x, model.options.ker, model.options.arg );   ka = k.*model.Alpha(:);   den=sum(ka);   if den > 0,     x = model.sv.X*ka/den;   else     % reinit     x = model.sv.X*(2*rand(nsv,1)-ones(nsv,1));   end   change = sum((old_x-x).^2);      old_x = x;end return;