%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Name: M_step.m
% Input: x - a nxd matrix (nx3 if using RGB)
%        Q - vector of values from the complete data log-likelihood function
%        h - a nxk matrix, the expectation of the hidden variable z given the data set and distribution params
%        S - cluster covariance matrices
%        k - the number of clusters
%        flag - flag to use improved EM to avoid singular covariance matrix
% Output: S - cluster covariance matrices
%         m - cluster means
%         pi - mixing coefficients
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [S, m, pi] = M_step(x, h, S, k, flag)

  % get size of data
  [num_data, dim] = size(x);
  eps = 1e-15;
  lambda = 1e-3; % value for improved version of EM

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  % update mixing coefficients
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  pi = zeros(k, 1);
  for i = 1:num_data
    row = h(i, :);
    maxValue = max(row);
    maxIdx = find(row == maxValue);
    pi(maxIdx) = pi(maxIdx) + 1;
  end
  pi = pi ./ num_data;
  

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  % TODO: update cluster means
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  % TODO: Calculate the covariance matrix estimate
  %       further modifications will need to be made when doing 2(d)
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


end