slam14(1) v3_1 后端优化 数学基础 高斯牛顿非线性最小二乘

https://zhuanlan.zhihu.com/p/42383070

 

很多问题最终归结为一个最小二乘问题，如SLAM算法中的Bundle Adjustment，位姿图优化等等。求解最小二乘的方法有很多，高斯-牛顿法就是其中之一。

 

 

 

 

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/**
 * This file is part of Gauss-Newton Solver.
 *
 * Copyright (C) 2018-2020 Dongsheng Yang <ydsf16@buaa.edu.cn> (Beihang University)
 * For more information see <https://github.com/ydsf16/Gauss_Newton_solver>
 *
 * Gauss_Newton_solver is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * Gauss_Newton_solver is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with Gauss_Newton_solver. If not, see <http://www.gnu.org/licenses/>.
 */

#include <iostream>
#include <eigen3/Eigen/Core>
#include <vector>
#include <opencv2/opencv.hpp>
#include <eigen3/Eigen/Cholesky>
#include <eigen3/Eigen/QR>
#include <eigen3/Eigen/SVD>
#include <chrono>

/* 计时类 */
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
};

/*  优化方程 */
class CostFunction{
public:
        CostFunction(double* a, double* b, double* c, int max_iter, double min_step, bool is_out):
        a_(a), b_(b), c_(c), max_iter_(max_iter), min_step_(min_step), is_out_(is_out)
        {}
        
        void addObservation(double x, double y)
        {
            std::vector<double> ob;
            ob.push_back(x);
            ob.push_back(y);
            obs_.push_back(ob);
        }
        
        void calcJ_fx()
        {
            J_ .resize(obs_.size(), 3);
            fx_.resize(obs_.size(), 1);
            
            for ( size_t i = 0; i < obs_.size(); i ++)
            {
                std::vector<double>& ob = obs_.at(i);
                double& x = ob.at(0);
                double& y = ob.at(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_b()
        {
            H_ = J_.transpose() * J_;
            B_ = -J_.transpose() * fx_;
        }
        
        void calcDeltax()
        {
            deltax_ = H_.ldlt().solve(B_); 
        }
       
       void updateX()
        {
            *a_ += deltax_(0);
            *b_ += deltax_(1);
            *c_ += deltax_(2);
        }
        
        double getCost()
        {
            Eigen::MatrixXd cost= fx_.transpose() * fx_;
            return cost(0,0);
        }
        
        void solveByGaussNewton()
        {
            double sumt =0;
            bool is_conv = false;
            for( size_t i = 0; i < max_iter_; i ++)
            {
                Runtimer t;
                t.start();
                calcJ_fx();
                calcH_b();
                calcDeltax();
                double delta = deltax_.transpose() * deltax_;
                t.stop();
                if( is_out_ )
                {
                    std::cout << "Iter: " << std::left <<std::setw(3) << i << " 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) << delta << " 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;
                }
                if( delta < min_step_)
                {
                    is_conv = true;
                    break;
                }
                updateX();
            }
           
           if( is_conv  == true)
                std::cout << "\nConverged\n";
            else
                std::cout << "\nDiverged\n\n";
        }
        
        Eigen::MatrixXd fx_;
        Eigen::MatrixXd J_; // 雅克比矩阵
        Eigen::Matrix3d H_; // H矩阵
        Eigen::Vector3d B_;
        Eigen::Vector3d deltax_;
        std::vector< std::vector<double>  > obs_; // 观测
        double* a_, *b_, *c_;
        
        int max_iter_;
        double min_step_;
        bool is_out_;
};//class CostFunction




int main(int argc, char **argv) {
    
    const double aa = 0.1, bb = 0.5, cc = 2; // 实际方程的参数
    double a =0.0, b=0.0, c=0.0; // 初值
    
    /* 构造问题 */
    CostFunction cost_func(&a, &b, &c, 50, 1e-10, true);
    
    /* 制造数据 */
    const size_t N = 100; //数据个数
    cv::RNG rng(cv::getTickCount());
    for( size_t i = 0; i < N; i ++)
    {
        /* 生产带有高斯噪声的数据　*/
       double x = rng.uniform(0.0, 1.0) ;
       double y = exp(aa*x*x + bb*x + cc) + rng.gaussian(0.05);
       
       /* 添加到观测中　*/
       cost_func.addObservation(x, y);
    }
    /* 用高斯牛顿法求解 */
    cost_func.solveByGaussNewton();
    return 0;
}