概述

本文介绍通过g2o框架，优化点和曲线的匹配（曲线拟合）。曲线的公式如下所示：

它有三个参数：a, b, lamba。

代码解析

自定义顶点

/**
 * \brief the params, a, b, and lambda for a * exp(-lambda * t) + b
 */
class VertexParams : public g2o::BaseVertex<3, Eigen::Vector3d> { // 它包含三个参数
 public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW;
  VertexParams() {}

  bool read(std::istream& /*is*/) override { return false; }

  bool write(std::ostream& /*os*/) const override { return false; }

  void setToOriginImpl() override {} // 设定参数的原始值

  void oplusImpl(const double* update) override { // 增量更新函数，调整当前参数的值
    Eigen::Vector3d::ConstMapType v(update); // 构造参数增量
    _estimate += v;
  }
};

自定义边

/**
 * \brief measurement for a point on the curve
 *
 * Here the measurement is the point which is lies on the curve.
 * The error function computes the difference between the curve
 * and the point.
 */
class EdgePointOnCurve
    : public g2o::BaseUnaryEdge<1, Eigen::Vector2d, VertexParams> { // 一条边只含有一个顶点，使用上面定义顶点的数据结构
 public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW
  EdgePointOnCurve() {}
  bool read(std::istream& /*is*/) override {
    cerr << __PRETTY_FUNCTION__ << " not implemented yet" << endl;
    return false;
  }
  bool write(std::ostream& /*os*/) const override {
    cerr << __PRETTY_FUNCTION__ << " not implemented yet" << endl;
    return false;
  }

  template <typename T>
  bool operator()(const T* params, T* error) const { // 误差函数
    const T& a = params[0];
    const T& b = params[1];
    const T& lambda = params[2];
    T fval = a * exp(-lambda * T(measurement()(0))) + b; // 计算理论值
    error[0] = fval - measurement()(1); // 和观察值进行比较
    return true;
  }

  G2O_MAKE_AUTO_AD_FUNCTIONS  // use autodiff
};

生成待拟合的点

  // generate random data
  g2o::Sampler::seedRand();
  double a = 2.;
  double b = 0.4;
  double lambda = 0.2;
  Eigen::Vector2d* points = new Eigen::Vector2d[numPoints];
  for (int i = 0; i < numPoints; ++i) {
    double x = g2o::Sampler::uniformRand(0, 10);
    double y = a * exp(-lambda * x) + b;
    // add Gaussian noise
    y += g2o::Sampler::gaussRand(0, 0.02);
    points[i].x() = x;
    points[i].y() = y;
  }

初始化求解器

  // setup the solver
  g2o::SparseOptimizer optimizer;
  optimizer.setVerbose(false);

  // allocate the solver
  g2o::OptimizationAlgorithmProperty solverProperty;
  optimizer.setAlgorithm(
      g2o::OptimizationAlgorithmFactory::instance()->construct("lm_dense",
                                                               solverProperty));

添加顶点

在本示例中，只存在一个顶点，就是待优化的参数

  // 1. add the parameter vertex
  VertexParams* params = new VertexParams();
  params->setId(0);
  params->setEstimate(
      Eigen::Vector3d(1, 1, 1));  // 参数的初始化值
  optimizer.addVertex(params);

添加边

在本示例中，每一个待拟合点，都对应一条边。也就是一个观察点就是一个观测值。

  // 2. add the points we measured to be on the curve
  for (int i = 0; i < numPoints; ++i) {
    EdgePointOnCurve* e = new EdgePointOnCurve;
    e->setInformation(Eigen::Matrix<double, 1, 1>::Identity());
    e->setVertex(0, params);
    e->setMeasurement(points[i]);
    optimizer.addEdge(e);
  }

求解器求解

  optimizer.initializeOptimization();
  optimizer.setVerbose(verbose);
  optimizer.optimize(maxIterations); // 最大的迭代次数

结果展示

  // print out the result
  cout << "Target curve" << endl;
  cout << "a * exp(-lambda * x) + b" << endl;
  cout << "Iterative least squares solution" << endl;
  cout << "a      = " << params->estimate()(0) << endl;
  cout << "b      = " << params->estimate()(1) << endl;
  cout << "lambda = " << params->estimate()(2) << endl;
  cout << endl;