function model = minball(X,options)% MINBALL Minimal enclosing ball in kernel feature space. %% Synopsis:%  model = minball(X)%  model = minball(X,options)%% Description:%  It computes center and radius of the minimal ball%  enclosing data X mapped into a feature space induced %  by a given kernel. The problem leads to a QP problem which is %  solve by 'quadprog' of the MATLAB Optimization toolbox.% % Input:%  X [dim x num_data] Input data.%  options [struct] Control parameters:%   .ker [string] Kernel identifier (default 'linear'). See 'help kernel'.%   .arg [1 x nargs] Kernel arguments.%   .eps [1x1] Multipliers less then eps are set to zero (default 1e-12).%   .mu [1x1] Regularization constant given to diagonal of the %     kernel matrix (default 1e-12).%% Output:%  model [struct] Center of the ball in the kernel feature space:%   .sv.X [dim x nsv] Data determining the center.%   .Alpha [nsv x 1] Data weights.%   .r [1x1] Radius of the minimal enclosing ball.%   .b [1x1] Squared norm of the center equal to Alpha'*K*Alpha.%   .options [struct] Copy of used options.%% Example:%  data = load('riply_trn');%  options = struct('ker','linear','arg',1);%  model = minball(data.X,options);%  [Ax,Ay] = meshgrid(linspace(-5,5,100),linspace(-5,5,100));%  dist = kdist([Ax(:)';Ay(:)'],model);%  figure; hold on; %  ppatterns(data.X); ppatterns(model.sv.X,'ro',12);%  contour( Ax, Ay, reshape(dist,100,100),[model.r model.r]);%% See also %  KDIST.%% 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:% 25-aug-2004, VF, added model.fun = 'kdist' and .diag_add changed to .mu % 16-may-2004, VF% 15-jun-2002, VF% process input arguments%-----------------------------------------if nargin < 2, options = []; else options=c2s(options); endif ~isfield(options,'ker'), options.ker = 'linear'; endif ~isfield(options,'arg'), options.arg = 1; endif ~isfield(options,'eps'), options.eps = 1e-12; endif ~isfield(options,'mu'), options.mu = 1e-12; end[dim,num_data] = size(X);% kernel matrix with regularization%-----------------------------------------K = kernel( X, options.ker, options.arg )+...    eye(num_data,num_data)*options.mu;% set up QP problem%-----------------------------------------f = -diag(K);H=2*K;Aeq = ones(1,num_data);beq = 1;LB = zeros(num_data,1);UB = inf*ones(num_data,1);% optimization%----------------------------qp_options=optimset('Display','off');model.Alpha=quadprog(H,f,[],[],Aeq,beq,LB,UB,zeros(num_data,1),qp_options);% take non-zero Alpha's%---------------------inx= find(model.Alpha > options.eps);model.Alpha = model.Alpha(inx);% compute radius%---------------------K = K(inx,inx);model.b = model.Alpha'*K*model.Alpha;model.r = sum( sqrt( diag(K) - 2*K*model.Alpha + model.b ))/length(inx);% setup model%---------------------model.sv.X= X(:,inx);model.nsv = length(inx);model.options=options;model.fun = 'kdist';return;% EOF