我们先用一个简单的例子来说明如何求解最小二乘问题，然后展示如何手写高斯牛顿法和优化库求解此问题
#include <iostream>
#include <chrono>
#include <opencv2/opencv.hpp>
#include <Eigen/Core>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
int main(int argc, char **argv) {
double ar = 1.0, br = 2.0, cr = 1.0; // 真实参数值
double ae = 2.0, be = -1.0, ce = 5.0; // 估计参数值
int N = 100; // 数据点
double w_sigma = 1.0; // 噪声Sigma值
double inv_sigma = 1.0 / w_sigma;
cv::RNG rng; // OpenCV随机数产生器
vector<double> x_data, y_data; // 数据
for (int i = 0; i < N; i++) {
double x = i / 100.0;
x_data.push_back(x);
y_data.push_back(exp(ar * x * x + br * x + cr) + rng.gaussian(w_sigma * w_sigma));
}
// 开始Gauss-Newton迭代
int iterations = 100; // 迭代次数
double cost = 0, lastCost = 0; // 本次迭代的cost和上一次迭代的cost
chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
for (int iter = 0; iter < iterations; iter++) {
Matrix3d H = Matrix3d::Zero(); // Hessian = J^T W^{-1} J in Gauss-Newton
Vector3d b = Vector3d::Zero(); // bias
cost = 0;
for (int i = 0; i < N; i++) {
double xi = x_data[i], yi = y_data[i]; // 第i个数据点
double error = yi - exp(ae * xi * xi + be * xi + ce);
Vector3d J; // 雅可比矩阵
J[0] = -xi * xi * exp(ae * xi * xi + be * xi + ce); // de/da
J[1] = -xi * exp(ae * xi * xi + be * xi + ce); // de/db
J[2] = -exp(ae * xi * xi + be * xi + ce); // de/dc
H += inv_sigma * inv_sigma * J * J.transpose();
b += -inv_sigma * inv_sigma * error * J;
cost += error * error;
}
// 求解线性方程 Hx=b
Vector3d dx = H.ldlt().solve(b);
if (isnan(dx[0])) {
cout << "result is nan!" << endl;
break;
}
if (iter > 0 && cost >= lastCost) {
cout << "cost: " << cost << ">= last cost: " << lastCost << ", break." << endl;
break;
}
ae += dx[0];
be += dx[1];
ce += dx[2];
lastCost = cost;
cout << "total cost: " << cost << ", \t\tupdate: " << dx.transpose() <<
"\t\testimated params: " << ae << "," << be << "," << ce << endl;
}
chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
chrono::duration<double> time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
cout << "solve time cost = " << time_used.count() << " seconds. " << endl;
cout << "estimated abc = " << ae << ", " << be << ", " << ce << endl;
return 0;
}
其结果如下，经过多次迭代，曲线拟合趋于收敛的状态！
#include <iostream>
#include <g2o/core/g2o_core_api.h>
#include <g2o/core/base_vertex.h>
#include <g2o/core/base_unary_edge.h>
#include <g2o/core/block_solver.h>
#include <g2o/core/optimization_algorithm_levenberg.h>
#include <g2o/core/optimization_algorithm_gauss_newton.h>
#include <g2o/core/optimization_algorithm_dogleg.h>
#include <g2o/solvers/dense/linear_solver_dense.h>
#include <Eigen/Core>
#include <opencv2/core/core.hpp>
#include <cmath>
#include <chrono>
using namespace std;
// 曲线模型的顶点，模板参数：优化变量维度和数据类型
class CurveFittingVertex : public g2o::BaseVertex<3, Eigen::Vector3d> {
public:
EIGEN_MAKE_ALIGNED_OPERATOR_NEW
// 重置
virtual void setToOriginImpl() override {
_estimate << 0, 0, 0;
}
// 更新
virtual void oplusImpl(const double *update) override {
_estimate += Eigen::Vector3d(update);
}
// 存盘和读盘：留空
virtual bool read(istream &in) {}
virtual bool write(ostream &out) const {}
};
// 误差模型 模板参数：观测值维度，类型，连接顶点类型
class CurveFittingEdge : public g2o::BaseUnaryEdge<1, double, CurveFittingVertex> {
public:
EIGEN_MAKE_ALIGNED_OPERATOR_NEW
CurveFittingEdge(double x) : BaseUnaryEdge(), _x(x) {}
// 计算曲线模型误差
virtual void computeError() override {
const CurveFittingVertex *v = static_cast<const CurveFittingVertex *> (_vertices[0]);
const Eigen::Vector3d abc = v->estimate();
_error(0, 0) = _measurement - std::exp(abc(0, 0) * _x * _x + abc(1, 0) * _x + abc(2, 0));
}
// 计算雅可比矩阵
virtual void linearizeOplus() override {
const CurveFittingVertex *v = static_cast<const CurveFittingVertex *> (_vertices[0]);
const Eigen::Vector3d abc = v->estimate();
double y = exp(abc[0] * _x * _x + abc[1] * _x + abc[2]);
_jacobianOplusXi[0] = -_x * _x * y;
_jacobianOplusXi[1] = -_x * y;
_jacobianOplusXi[2] = -y;
}
virtual bool read(istream &in) {}
virtual bool write(ostream &out) const {}
public:
double _x; // x 值， y 值为 _measurement
};
int main(int argc, char **argv) {
double ar = 1.0, br = 2.0, cr = 1.0; // 真实参数值
double ae = 2.0, be = -1.0, ce = 5.0; // 估计参数值
int N = 100; // 数据点
double w_sigma = 1.0; // 噪声Sigma值
double inv_sigma = 1.0 / w_sigma;
cv::RNG rng; // OpenCV随机数产生器
vector<double> x_data, y_data; // 数据
for (int i = 0; i < N; i++) {
double x = i / 100.0;
x_data.push_back(x);
y_data.push_back(exp(ar * x * x + br * x + cr) + rng.gaussian(w_sigma * w_sigma));
}
// 构建图优化，先设定g2o
typedef g2o::BlockSolver<g2o::BlockSolverTraits<3, 1>> BlockSolverType; // 每个误差项优化变量维度为3，误差值维度为1
typedef g2o::LinearSolverDense<BlockSolverType::PoseMatrixType> LinearSolverType; // 线性求解器类型
// 梯度下降方法，可以从GN, LM, DogLeg 中选
auto solver = new g2o::OptimizationAlgorithmGaussNewton(
g2o::make_unique<BlockSolverType>(g2o::make_unique<LinearSolverType>()));
g2o::SparseOptimizer optimizer; // 图模型
optimizer.setAlgorithm(solver); // 设置求解器
optimizer.setVerbose(true); // 打开调试输出
// 往图中增加顶点
CurveFittingVertex *v = new CurveFittingVertex();
v->setEstimate(Eigen::Vector3d(ae, be, ce));
v->setId(0);
optimizer.addVertex(v);
// 往图中增加边
for (int i = 0; i < N; i++) {
CurveFittingEdge *edge = new CurveFittingEdge(x_data[i]);
edge->setId(i);
edge->setVertex(0, v); // 设置连接的顶点
edge->setMeasurement(y_data[i]); // 观测数值
edge->setInformation(Eigen::Matrix<double, 1, 1>::Identity() * 1 / (w_sigma * w_sigma)); // 信息矩阵：协方差矩阵之逆
optimizer.addEdge(edge);
}
// 执行优化
cout << "start optimization" << endl;
chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
optimizer.initializeOptimization();
optimizer.optimize(10);
chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
chrono::duration<double> time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
cout << "solve time cost = " << time_used.count() << " seconds. " << endl;
// 输出优化值
Eigen::Vector3d abc_estimate = v->estimate();
cout << "estimated model: " << abc_estimate.transpose() << endl;
return 0;
}