将 3D 圆柱体放入点云中

Question

我有点困惑。我有来自点云的 x,y,z 点，我想使用 scipy 和 python 拟合一个圆柱体。

我通常在互联网上找到的圆柱方程是

(x - a)**2 + (y - b)**2 = r**2

但我也希望 z 作为该方程中的参数存在，所以我也使用

 (x - a)**2 + (y - b)**2 + (z - c)**2 = r**2

但这是球体的方程。我都用过这两种方法来进行实验。 我的问题是，当我使用球体公式绘制预测点（x_test、y_test、z_predicted）时，这些点与 x_test、y_test、z_test 不接近。当我使用典型的圆柱公式时，也会发生同样的情况。 所以我问，在可能不与任何轴平行的圆柱体上拟合 x、y、z 的正确公式是什么？

我给出了代码和一些实验要点。

from scipy.optimize import minimize
from sklearn.preprocessing import StandardScaler
import pandas as pd
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from sklearn.model_selection import train_test_split

from cfg_fit_cylinder import *

# Define the cylinder equation
def cylinder_equation(params, xyz):
    a, b, c, r = params
    x, y, z = xyz
    return (x - a)**2 + (y - b)**2 + (z - c)**2 - r**2

def cost_function(params, xyz_data):
    a, b, c, r = params
    total_error = 0
    for x, y, z in xyz_data:
        error = ((x - a)**2 + (y - b)**2 + (z - c)**2 - r**2)**2
        total_error += error
    return total_error

# Plot 3D points
def plot_3d_points(vertices, xyz_test, a, b, c, r):
    fig = plt.figure(figsize=(12, 6))
    ax1 = fig.add_subplot(121, projection='3d')  # For the 3D points
    ax2 = fig.add_subplot(122, projection='3d')  # For the cylinder surface

    x, y, z = vertices['x'], vertices['y'], vertices['z'] # Unscaled inital points
    ax1.scatter(x, y, z, c='b', marker='o')
    xyz_test = scaler.inverse_transform(xyz_test)
    ax1.scatter(xyz_test[:, 0], xyz_test[:, 1], xyz_test[:, 2], c='r', marker='x')
    ax1.set_xlabel('X Label')
    ax1.set_ylabel('Y Label')
    ax1.set_zlabel('Z Label')

    # Also unscaled
    resolution = 100
    theta = np.linspace(0, 2 * np.pi, resolution)
    z_surface = np.linspace(min(z), max(z), resolution) #here
    Z, Theta = np.meshgrid(z_surface, theta)
    X = a + r * np.cos(Theta)
    Y = b + r * np.sin(Theta)
    ax2.plot_surface(X, Y, Z, cmap='Greens', alpha=1)

    predicted_z = np.array([cylinder_equation([a, b, c, r], xyz) for xyz in xyz_test]) # Unscaled test points/parameters
    shift_z = predicted_z - xyz_test[:, 2]

    # Plot the points with the z-shift
    ax1.scatter(xyz_test[:, 0], xyz_test[:, 1], xyz_test[:, 2] + shift_z, c='black', marker='x')

    plt.show()


if __name__ == "__main__":
    # Load data
    points = pd.read_csv(PATH_TO_POINT_CLOUD, header=None)
    points.columns = ['x', 'y', 'z']

    # Scale data
    scaler = StandardScaler()
    points_scaled = scaler.fit_transform(points)

    params_list = []
    rmse_list = []
    result_list = []
    num_samples = [i for i in range(7, 3300, 100)]

    for num in num_samples:
        points_scaled_sampled = points_scaled[np.random.choice(len(points_scaled), num, replace=False)]

        train, test = train_test_split(points_scaled_sampled, test_size=0.3) # here

        # Initial guess based on data statistics
        initial_guess = [np.mean(train[:, 0]), np.mean(train[:, 1]), np.mean(train[:, 2]), 
                        np.std(np.linalg.norm(train - np.mean(train, axis=0), axis=1))]

        # Use the 'trust-constr' method for optimization
        result = minimize(cost_function, initial_guess, args=(train,), method='trust-constr')
        result_list.append(result)
        print('Optimization result:')
        print(result)
        print("========")
        params = result.x
        params_list.append(params)

        cylinder_values = np.array([cylinder_equation(params, xyz) for xyz in train])

        ssr = np.mean(cylinder_values**2)
        rmse = np.sqrt(ssr)
        rmse_list.append(rmse)

        print(f'RMSE, Samples: {round(rmse, 3)}, {num}')

    print('========')
    print(f'Min RMSE, samples: {round(min(rmse_list), 2)}, {num_samples[np.argmin(rmse_list)]}, Max RMSE, samples: {round(max(rmse_list), 2)}, {num_samples[np.argmax(rmse_list)]}')

    min_params = params_list[np.argmin(rmse_list)]
    a_min, b_min, c_min, r_min = min_params[0], min_params[1], min_params[2], min_params[3]

    # Denormalize parameters
    denormalized_params = scaler.inverse_transform([[a_min, b_min, c_min]])[0]
    a_min, b_min, c_min = denormalized_params

    # Denormalize r
    r_min = r_min * scaler.scale_[2] + scaler.mean_[2]

    plot_3d_points(points, test, a_min, b_min, c_min, r_min)
    plt.plot(num_samples, rmse_list)

以及一些要点

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Answer 1

(x - a)^2 + (y - b)^2 = r^2 .

该公式的含义是针对 xy 平面上半径为 r、圆心位于 (a, b) 的圆。自由不受约束的 z 坐标可以具有任何值。因此，该公式适用于以 r 为半径、以 z 轴为方向穿过点 (a, b, 0) 的圆柱体。

为了在三维空间中表示任意圆柱体，必须将位置和轴方向加上圆柱体的半径。

您的坐标值已保存为 xyz 文件并使用 FindSurface Web 演示 (https://developers.curvsurf.com/WebDemo/) 进行处理。