高斯牛顿迭代（GaussNewton）代码实现

#include <cstdio>
#include <vector>
#include <iostream>
#include <opencv2/core/core.hpp>
 
using namespace std;
using namespace cv;
 
const double DELTAX = 1e-5;
const int MAXCOUNT = 100;
 
double function(const Mat &input, const Mat params)
{
    //给定函数已知x求y
    double a = params.at<double>(0,0);
    
    double b = params.at<double>(1,0);
    
    double c = params.at<double>(2,0);

    double x = input.at<double>(0,0);
 
    return exp( a*x*x + b*x + c );
}
 

double derivative(double(*function)(const Mat &input, const Mat params), const Mat &input, const Mat params, int n)
{
    // 用增加分量的方式求导数 
    Mat params1 = params.clone();
    Mat params2 = params.clone();
 
    params1.at<double>(n,0) -= DELTAX;
    params2.at<double>(n,0) += DELTAX;
 
    double y1 = function(input, params1);
    double y2 = function(input, params2);
 
    double deri = (y2 - y1) / (2*DELTAX);
 
    return deri;
}
 
void gaussNewton(double(*function)(const Mat &input, const Mat ms), const Mat &inputs, const Mat &outputs, Mat params)
{
    int num_estimates = inputs.rows;
    int num_params = params.rows;
 
    Mat r(num_estimates, 1, CV_64F);           // 残差
    Mat Jf(num_estimates, num_params, CV_64F); // 雅克比矩阵
    Mat input(1, 1, CV_64F);       
 
    double lsumR = 0;
 
    for(int i = 0; i < MAXCOUNT; i++) 
    {
        double sumR = 0;
 
        for(int j = 0; j < num_estimates; j++) 
        {
            input.at<double>(0,0) = inputs.at<double>(j,0);

            r.at<double>(j,0) = outputs.at<double>(j,0) - function(input, params);// 计算残差矩阵
 
            sumR += fabs(r.at<double>(j,0)); // 残差累加
 
            for(int k = 0; k < num_params; k++) 
            {
                Jf.at<double>(j,k) = derivative(function, input, params, k); // 根据新参数重新计算雅克比矩阵
            }
        }
 
        sumR /= num_estimates; //均残差

        if(fabs(sumR - lsumR) < 1e-8) //均残差足够小达到收敛
        {
            break;
        }
 
        Mat delta = ((Jf.t()*Jf)).inv() * Jf.t()*r;// ((Jf.t()*Jf)) 近似黑塞矩阵
        params += delta;
        lsumR = sumR;
    }
}

int main()
{
    // F = exp ( a*x*x + b*x + c )

    int num_params = 3;
    Mat params(num_params, 1, CV_64F);

     //abc参数的实际值
    params.at<double>(0,0) = 1.0; //a
    params.at<double>(1,0) = 2.0; //b
    params.at<double>(2,0) = 1.0; //c  
    
    cout<<"real("<<"a:"<< params.at<double>(0,0) <<" b:"<< params.at<double>(1,0) << " c:"<< params.at<double>(2,0) << ")"<< endl;

    int N = 100;                        
    
    double w_sigma = 1.0;               // 噪声Sigma值
    
    cv::RNG rng;                        // OpenCV随机数产生器
    
    Mat estimate_x(N, 1, CV_64F);
    Mat estimate_y(N, 1, CV_64F);
    

    for ( int i = 0; i < N; i++ )
    {
        double x = i/100.0;
        estimate_x.at<double>(i,0) = x; 
        Mat paramX(1, 1, CV_64F);
        paramX.at<double>(0,0) = x;
        estimate_y.at<double>(i,0) = function(paramX, params) + rng.gaussian ( w_sigma );
    }
    
    //abc参数的初值
    params.at<double>(0,0) = 0; //a
    params.at<double>(1,0) = 0; //b
    params.at<double>(2,0) = 0; //c

    cout<<"init("<<"a:"<< params.at<double>(0,0) <<" b:"<< params.at<double>(1,0) << " c:"<< params.at<double>(2,0) << ")"<< endl;

    gaussNewton(function, estimate_x, estimate_y, params);
 
    cout<<"result("<<"a:"<< params.at<double>(0,0) <<" b:"<< params.at<double>(1,0) << " c:"<< params.at<double>(2,0) << ")"<< endl;
 
    return 0;
}

# Project: GaussNewtonDemo
#
# All rights reserved.

cmake_minimum_required( VERSION 2.6 )
cmake_policy( SET CMP0004 OLD )

### initialize project ###########################################################################################

SET(CMAKE_BUILD_TYPE "Debug")
SET(CMAKE_CXX_FLAGS_DEBUG "$ENV{CXXFLAGS} -O0 -Wall -g2 -ggdb")
SET(CMAKE_CXX_FLAGS_RELEASE "$ENV{CXXFLAGS} -O3 -Wall")


project(GaussNewtonDemo)
find_package(Eigen3 REQUIRED)
find_package(OpenCV REQUIRED)

set(CMAKE_INSTALL_PREFIX /usr)


set(BUILD_SHARED_LIBS on)

set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -fPIC -O0")
include_directories(
        ${EIGEN3_INCLUDE_DIR}
        ${OpenCV_INCLUDE_DIR})

add_definitions( "-DPREFIX="${CMAKE_INSTALL_PREFIX}""        )

### global default options #######################################################################################

set(SOURCES 
    main.cpp
)

add_executable(GaussNewtonDemo ${SOURCES})

TARGET_LINK_LIBRARIES( GaussNewtonDemo
    ${OpenCV_LIBS} )