ransac算法拟合平面

### 使用RANSAC算法进行平面拟合

#### 平面拟合原理概述
RANSAC（Random Sample Consensus）是一种迭代方法，旨在从一组观测数据中估计数学模型的参数，特别适用于存在大量异常值的情况。对于平面拟合任务而言，目标是从三维空间中的点集中找出最能代表这些点的一个平面。

具体来说，在每次迭代过程中，随机选取三个不共线的点来定义一个假设的平面，并计算其余所有点到此平面上的距离。如果某个点到该平面的距离小于设定的阈值，则认为这个点属于内点集(inliers)。经过多轮这样的操作之后，拥有最多内点的那个平面即被认为是最优解[^1]。

#### Python 实现示例
下面展示了一个使用Python和NumPy库实现简单版RANSAC来进行平面拟合的例子：

import numpy as np

def fit_plane_ransac(points, threshold=0.01, max_iterations=1000):
    best_inliers = None
    best_model = None
    
    n_points = points.shape[0]

    for _ in range(max_iterations):
        # Randomly select three non-collinear points to form a plane.
        indices = np.random.choice(n_points, size=3, replace=False)
        
        p1, p2, p3 = points[indices]
        
        # Check collinearity by computing the volume of parallelepiped formed by vectors (p2- p1
        v2 = p3 - p1
        
        cross_product = np.cross(v1, v2)

        if np.linalg.norm(cross_product) < 1e-6:
            continue
            
        normal_vector = cross_product / np.linalg.norm(cross_product)
        d = -np.dot(normal_vector, p1)

        model_parameters = (*normal_vector, d)

        distances = abs(np.dot(points, normal_vector) + d)
        current_inliers = points[(distances <= threshold)]

        if best_inliers is None or len(current_inliers) > len(best_inliers):
            best_inliers = current_inliers.copy()
            best_model = model_parameters

    return best_model, best_inliers


if __name__ == "__main__":
    from mpl_toolkits.mplot3d import Axes3D
    import matplotlib.pyplot as plt

    fig = plt.figure(figsize=(8, 6))
    ax = fig.add_subplot(111, projection='3d')

    # Generate some sample data with noise and outliers.
    X, Y = np.meshgrid(range(-5, 5), range(-5, 5))
    Z_clean = 2 * X + 3 * Y + 7
    Z_noisy = Z_clean + np.random.normal(scale=0.5, size=X.shape)

    outlier_indices = np.random.randint(low=0, high=len(Z_noisy.ravel()), size=int(len(Z_noisy.ravel())*0.1))

    Z_outliers = np.zeros_like(Z_noisy).ravel()
    Z_outliers[outlier_indices] += np.random.uniform(low=-10., high=10., size=outlier_indices.size)
    
    Z_final = Z_noisy.ravel() + Z_outliers.reshape(X.shape)

    point_cloud = np.vstack((X.ravel(), Y.ravel(), Z_final)).T

    fitted_params, inliers = fit_plane_ransac(point_cloud)

    A, B, C, D = fitted_params

    xx, yy = np.meshgrid([-10, 10], [-10, 10])
    zz = (-A * xx - B * yy - D) / C

    ax.scatter(*point_cloud.T, c="blue", marker="o", label="Original Points")
    ax.plot_surface(xx, yy, zz, alpha=.5, color="red")

    plt.show()

这段代码首先生成了一些带有噪声和平面外离群点的人工数据作为测试样本。接着调用了`fit_plane_ransac()`函数执行RANSAC过程寻找最优平面，并最终绘制出了原始散点图以及由RANSAC得到的最佳拟合平面图像[^2]。