function [eigvector, eigvalue] = LDA(X,gnd,options)
% LDA: Linear Discriminant Analysis 
%
%       [eigvector, eigvalue] = LDA(X, gnd, options)
% 
%             Input:
%               X       - Data matrix. Each row vector of fea is a data point.
%               gnd   - Colunm vector of the label information for each
%                       data point. 
%               options - Struct value in Matlab. The fields in options
%                         that can be set:
%
%                            Regu  -  1: regularized solution, 
%                                        a* = argmax (a'X'WXa)/(a'X'DXa+alpha*I) 
%                                     0: solve the sinularity problem by SVD 
%                                     Default: 1 
%
%                            alpha -  The regularization parameter. Valid
%                                     when Regu==1. Default value is 0.1. 
%
%                            ReguType  -  'Ridge': Tikhonov regularization
%                                         'Custom': User provided
%                                                   regularization matrix
%                                          Default: 'Ridge' 
%                        regularizerR  -   (nFea x nFea) regularization
%                                          matrix which should be provided
%                                          if ReguType is 'Custom'. nFea is
%                                          the feature number of data
%                                          matrix
%                        Fisherface     -  1: Fisherface approach
%                                             PCARatio = nSmp - nClass
%                                          Default: 0
%
%                            PCARatio     -  The percentage of principal
%                                            component kept in the PCA
%                                            step. The percentage is
%                                            calculated based on the
%                                            eigenvalue. Default is 1
%                                            (100%, all the non-zero
%                                            eigenvalues will be kept.
%                                            If PCARatio > 1, the PCA step
%                                            will keep exactly PCARatio principle
%                                            components (does not exceed the
%                                            exact number of non-zero components).  
%                           
%
%             Output:
%               eigvector - Each column is an embedding function, for a new
%                           data point (row vector) x,  y = x*eigvector
%                           will be the embedding result of x.
%               eigvalue  - The sorted eigvalue of LDA eigen-problem. 
% 
%
%
%    Examples:
%       
%       fea = rand(50,70);
%       gnd = [ones(10,1);ones(15,1)*2;ones(10,1)*3;ones(15,1)*4];
%       options = [];
%       options.Fisherface = 1;
%       [eigvector, eigvalue] = LDA(fea, gnd, options);
%       Y = fea*eigvector;
% 
%
% See also LPP, constructW, LGE
%
%
%
%Reference:
%
%   P. N. Belhumeur, J. P. Hespanha, and D. J. Kriegman, 揈igenfaces
%   vs. fisherfaces: recognition using class specific linear
%   projection,