Eigen C++非線性最小二乘法代碼實現

最近在研究中需要使用LM等優化算法，對特定的東西進行優化。理論懂了一些些，論文也看了不少， 但是實際上手了兩天卻發現，自己根本寫不出來，很多東西一知半解，現在還是要靜下心，急不得啊

參考資料

• 非線性優化（一）：優化方法
• 非線性優化（二）：徒手實現LM算法
• 卡爾曼濾波家族
• 卡爾曼濾波：從入門到精通

#include <iostream>
#include <eigen3/Eigen/Core>
#include <eigen3/Eigen/Dense>
#include <opencv2/opencv.hpp>
#include <eigen3/Eigen/Cholesky>
#include <chrono>

/* 計時類 */
class Runtimer{
public:
    inline void start()
    {
        t_s_  = std::chrono::steady_clock::now();
    }
    
    inline void stop()
    {
        t_e_ = std::chrono::steady_clock::now();
    }
    
    inline double duration()
    {
        return std::chrono::duration_cast<std::chrono::duration<double>>(t_e_ - t_s_).count() * 1000.0;
    }
    
private:
    std::chrono::steady_clock::time_point t_s_; //start time ponit
    std::chrono::steady_clock::time_point t_e_; //stop time point
};

/*  優化方程 */
class LevenbergMarquardt{
public:
    LevenbergMarquardt(double* a, double* b, double* c):
    a_(a), b_(b), c_(c)
    {
        epsilon_1_ = 1e-6;
        epsilon_2_ = 1e-6;
        max_iter_ = 50; 	// 迭代次數
        is_out_ = true;
    }
    
    void setParameters(double epsilon_1, double epsilon_2, int max_iter, bool is_out)
    {
        epsilon_1_ = epsilon_1;
        epsilon_2_ = epsilon_2;
        max_iter_ = max_iter;
        is_out_ = is_out;
    }
    
    void addObservation(const double& x, const double& y)
    {
        obs_.push_back(Eigen::Vector2d(x, y));
    }
    
    void calcJ_fx() // 計算 Jacobi 矩陣
    {
        J_ .resize(obs_.size(), 3);
        fx_.resize(obs_.size(), 1);
        
        for ( size_t i = 0; i < obs_.size(); i ++)
        {
            const Eigen::Vector2d& ob = obs_.at(i);
            const double& x = ob(0);
            const double& y = ob(1);
            double j1 = -x*x*exp(*a_ * x*x + *b_*x + *c_);
            double j2 = -x*exp(*a_ * x*x + *b_*x + *c_);
            double j3 = -exp(*a_ * x*x + *b_*x + *c_);
            J_(i, 0 ) = j1;
            J_(i, 1) = j2;
            J_(i, 2) = j3;
            fx_(i, 0) = y - exp( *a_ *x*x + *b_*x +*c_);
        }
    }
    
    void calcH_g() // 計算 Hassian 矩陣
    {
        H_ = J_.transpose() * J_;
        g_ = -J_.transpose() * fx_;
    }
        
    double getCost()  // 獲得代價函數
    {
        Eigen::MatrixXd cost= fx_.transpose() * fx_;
        return cost(0,0);
    }
    
    double F(double a, double b, double c)
    {
        Eigen::MatrixXd fx;
        fx.resize(obs_.size(), 1);
        
        for ( size_t i = 0; i < obs_.size(); i ++)
        {
            const Eigen::Vector2d& ob = obs_.at(i);
            const double& x = ob(0);
            const double& y = ob(1);
            fx(i, 0) = y - exp( a *x*x + b*x +c);
        }
        Eigen::MatrixXd F = 0.5 * fx.transpose() * fx; //MatrixXd 動態矩陣大小可以通過 resize()改變大小
        return F(0,0);
    }
    
    double L0_L( Eigen::Vector3d& h)
    {
           Eigen::MatrixXd L = -h.transpose() * J_.transpose() * fx_ - 0.5 * h.transpose() * J_.transpose() * J_ * h;
           return L(0,0);
    }

    void solve()
    {
        int k = 0;
        double nu = 2.0;
        calcJ_fx();
        calcH_g();
        bool found = ( g_.lpNorm<Eigen::Infinity>() < epsilon_1_ );
        
        std::vector<double> A;
        A.push_back( H_(0, 0) );
        A.push_back( H_(1, 1) );
        A.push_back( H_(2,2) );
        auto max_p = std::max_element(A.begin(), A.end());
        double mu = *max_p;
        
        double sumt =0;

        while ( !found && k < max_iter_)
        {
            Runtimer t;
            t.start();
            
            k = k +1;
            Eigen::Matrix3d G = H_ + mu * Eigen::Matrix3d::Identity();
            Eigen::Vector3d h = G.ldlt().solve(g_);
            
            if( h.norm() <= epsilon_2_ * ( sqrt(*a_**a_ + *b_**b_ + *c_**c_ ) +epsilon_2_ ) )
                found = true;
            else
            {
                double na = *a_ + h(0);
                double nb = *b_ + h(1);
                double nc = *c_ + h(2);
                
                double rho =( F(*a_, *b_, *c_) - F(na, nb, nc) )  / L0_L(h);

                if( rho > 0)
                {
                    *a_ = na;
                    *b_ = nb;
                    *c_ = nc;
                    calcJ_fx();
                    calcH_g();
                                      
                    found = ( g_.lpNorm<Eigen::Infinity>() < epsilon_1_ );
                    mu = mu * std::max<double>(0.33, 1 - std::pow(2*rho -1, 3));
                    nu = 2.0;
                }
                else
                {
                    mu = mu * nu; 
                    nu = 2*nu;
                }// if rho > 0
            }// if step is too small
            
            t.stop();
            if( is_out_ )
            {
                std::cout << "Iter: " << std::left <<std::setw(3) << k << " Result: "<< std::left <<std::setw(10)  << *a_ << " " << std::left <<std::setw(10)  << *b_ << " " << std::left <<std::setw(10) << *c_ << 
                " step: " << std::left <<std::setw(14) << h.norm() << " cost: "<< std::left <<std::setw(14)  << getCost() << " time: " << std::left <<std::setw(14) << t.duration()  <<
                " total_time: "<< std::left <<std::setw(14) << (sumt += t.duration()) << std::endl;
            }   
        } // while
        
        if( found  == true)
            std::cout << "\nConverged\n\n";
        else
            std::cout << "\nDiverged\n\n";
        
    }//function 
     
    Eigen::MatrixXd fx_; 
    Eigen::MatrixXd J_; // 雅克比矩陣
    Eigen::Matrix3d H_; // H矩陣
    Eigen::Vector3d g_;
    
    std::vector< Eigen::Vector2d> obs_; // 觀測
   
    /* 要求的三個參數 */
    double* a_, *b_, *c_;
    
    /* parameters */
    double epsilon_1_, epsilon_2_;
    int max_iter_;
    bool is_out_;
};//class LevenbergMarquardt


int main(int argc, char **argv) {
    const double aa = 0.1, bb = 0.5, cc = 2; // 實際方程的參數
    double a =0.0, b=0.0, c=0.0; // 初值
    
    /* 構造問題 */
    LevenbergMarquardt lm(&a, &b, &c);
    lm.setParameters(1e-10, 1e-10, 100, true);
    
    /* 製造數據 */
    const size_t N = 100; //數據個數
    cv::RNG rng(cv::getTickCount());  // 隨機數產生工具
    for( size_t i = 0; i < N; i ++)
    {
        /* 生產帶有高斯噪聲的數據　*/
        double x = rng.uniform(0.0, 1.0) ;
        double y = exp(aa*x*x + bb*x + cc) + rng.gaussian(0.05);
        
        /* 添加到觀測中　*/
        lm.addObservation(x, y);
    }
    /* 用LM法求解 */
    lm.solve();
    
    return 0;
}