Ceres 损失函数深度解析：从理论到工程实践

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在非线性优化问题中，异常值（outliers）会导致目标函数偏离真实的最小值。鲁棒核函数通过降低异常值的权重，使得优化过程对异常值不敏感。数学上，鲁棒核函数作用于残差上，形式如下：

$$
\rho(s) = \rho\left(|f(x)|^2\right)
$$

其中，(f(x)) 是残差函数，(\rho) 是鲁棒核函数。Ceres Solver 通过修改残差的平方项来抑制异常值的影响。

以下是几种常用的损失函数及其特性：

损失函数	公式	适用场景
HuberLoss	(\rho(s) = \begin{cases} s & \text{if} s \leq 1 \ 2\sqrt{s} – 1 & \text{otherwise} \end{cases})	适度抑制异常值
CauchyLoss	(\rho(s) = \log(1 + s))	强异常值抑制
SoftLOneLoss	(\rho(s) = 2(\sqrt{1 + s} – 1))	平滑过渡，适合中等异常值

下面是一个完整的 Ceres 示例，展示如何在 Bundle Adjustment 中使用不同的损失函数：

#include <ceres/ceres.h>
#include <gflags/gflags.h>

DEFINE_string(loss_function, "huber", "Type of loss function (huber, cauchy, softlone)");
DEFINE_double(scale, 1.0, "Scale parameter for the loss function");

struct BundleAdjustmentCostFunctor {BundleAdjustmentCostFunctor(const Eigen::Vector2d& observed) : observed_(observed) {}

    template <typename T>
    bool operator()(const T* const camera, const T* const point, T* residuals) const {
        // Project 3D point to 2D image plane
        // ... (omitted for brevity)
        return true;
    }

    static ceres::CostFunction* Create(const Eigen::Vector2d& observed) {
        return new ceres::AutoDiffCostFunction<BundleAdjustmentCostFunctor, 2, 9, 3>(new BundleAdjustmentCostFunctor(observed));
    }

    Eigen::Vector2d observed_;
};

int main(int argc, char** argv) {google::ParseCommandLineFlags(&argc, &argv, true);

    ceres::Problem problem;
    ceres::LossFunction* loss_function = nullptr;

    if (FLAGS_loss_function == "huber") {loss_function = new ceres::HuberLoss(FLAGS_scale);
    } else if (FLAGS_loss_function == "cauchy") {loss_function = new ceres::CauchyLoss(FLAGS_scale);
    } else if (FLAGS_loss_function == "softlone") {loss_function = new ceres::SoftLOneLoss(FLAGS_scale);
    }

    // Add residuals to the problem
    // ... (omitted for brevity)

    ceres::Solver::Options options;
    options.minimizer_progress_to_stdout = true;
    ceres::Solver::Summary summary;
    ceres::Solve(options, &problem, &summary);

    std::cout << summary.FullReport() << std::endl;
    return 0;
}