3d 点云中的平面拟合 [英] Plane fitting in a 3d point cloud
问题描述
我正在尝试使用回归公式在 3d 点云中找到平面
你只需要像这样添加一个标量字段:
is_floor = cloud.add_scalar_field(plane_fit")Wich 将为拟合的平面的点添加一个值为 1 的新列.
您可以将标量场可视化:
旧答案
我认为您可以轻松地使用 PCA 将平面拟合到 3D 点而不是回归.
这是一个简单的 PCA 实现:
def PCA(数据，相关性=假，排序=真):"对数据应用主成分分析参数----------数据:数组包含数据的数组.数组必须有 NxM 维，其中每个N 行中的代表不同的单个记录，M 列中的每一列表示为该单个记录记录的不同变量.大批([[V11, ... , V1m],...,[Vn1, ... , Vnm]])相关性(可选):bool设置要计算的矩阵类型(见注释):如果为真，则计算相关矩阵.如果 False(默认)计算协方差矩阵.排序(可选):布尔设置特征值/向量的顺序如果为 True(默认)，它们将被排序(从高值到低值).如果 False 他们不会.退货-------特征值:(1,M) 数组对应矩阵的特征值.特征向量:(M,M) 数组对应矩阵的特征向量.笔记-----当存在不同的幅度时，相关矩阵是更好的选择表示 M 个变量.在其他情况下使用协方差矩阵."均值 = np.mean(数据，轴 = 0)data_adjust = 数据 - 平均值#: 由于 np.cov/corrcoef 语法，数据被转置如果相关性:矩阵 = np.corrcoef(data_adjust.T)别的:矩阵 = np.cov(data_adjust.T)特征值，特征向量 = np.linalg.eig(matrix)如果排序:#: 对特征值和特征向量进行排序sort = eigenvalues.argsort()[::-1]特征值 = 特征值[排序]特征向量 = 特征向量[:,sort]返回特征值、特征向量这是将点拟合到平面的方法:
def best_fitting_plane(points, equation=False):"计算给定点的最佳拟合平面参数----------点:数组对应于我们想要的点的 x,y,z 坐标定义最佳拟合平面.预期格式:大批([[x1,y1,z1],...,[xn,yn,zn]])方程(可选):布尔设置输出平面格式:如果为 True，则返回平面的 a、b、c、d 系数.如果 False(默认)返回 1 个点和 1 个法线向量.退货-------a, b, c, d : 浮动求解平面方程的系数.或者点，正常:数组由 1 个点和 1 个法线向量定义的平面.格式:数组([Px,Py,Pz]), 数组([Nx,Ny,Nz])"w, v = PCA(点)#: 平面的法线是最后一个特征向量正常 = v[:,2]#: 从飞机上得到一个点点= np.mean(点，轴= 0)如果方程:a, b, c = 正常d = -(np.dot(normal, point))返回 a, b, c, d别的:返回点，正常然而，由于此方法对异常值敏感，您可以使用 RANSAC 使拟合稳健异常值.
在 这里有一个 Python 实现.
并且您应该只需要定义一个平面模型类，以便使用它来将平面拟合到 3D 点.
在任何情况下，如果您可以从异常值中清除 3D 点(也许您可以使用 KD-Tree S.O.R 过滤器)，您应该使用 PCA 获得不错的结果.
这是一个S.O.R的实现:
def statistics_outilier_removal(kdtree, k=8, z_max=2 ):"在给定的 KDTree 上计算统计异常值去除过滤器.参数----------kdtree:scipy 的 KDTree 实例KDTree 的结构将用于计算过滤器.k(可选):int最近邻居的数量将用于估计从每个点到他最近的邻居的平均距离.默认值:8z_max(可选):int最大 Z 分数决定该点是异常值还是异常值不是.退货-------sor_filter : 布尔数组指示点应保留或不保留的布尔掩码.布尔掩码的大小将与点数相同在 KD 树中.笔记-----应使用 2 个可选参数(k 和 z_max)来调整过滤到想要的结果.更高的k"值(通常)会导致修剪的点数更高.较低的z_max"值(通常)会导致修剪的点数较高."距离，我 = kdtree.query(kdtree.data, k=k, n_jobs=-1)z_distances = stats.zscore(np.mean(distances,axis=1))sor_filter = abs(z_distances) 您可以使用可能使用 这个实现
I am trying to find planes in a 3d point cloud, using the regression formula Z= aX + bY +C
I implemented least squares and ransac solutions, but the 3 parameters equation limits the plane fitting to 2.5D- the formula can not be applied on planes parallel to the Z-axis.
My question is how can I generalize the plane fitting to full 3d? I want to add the fourth parameter in order to get the full equation aX +bY +c*Z + d how can I avoid the trivial (0,0,0,0) solution?
Thanks!
The Code I'm using:
from sklearn import linear_model
def local_regression_plane_ransac(neighborhood):
"""
Computes parameters for a local regression plane using RANSAC
"""
XY = neighborhood[:,:2]
Z = neighborhood[:,2]
ransac = linear_model.RANSACRegressor(
linear_model.LinearRegression(),
residual_threshold=0.1
)
ransac.fit(XY, Z)
inlier_mask = ransac.inlier_mask_
coeff = model_ransac.estimator_.coef_
intercept = model_ransac.estimator_.intercept_
Update
This functionality is now integrated in https://github.com/daavoo/pyntcloud and makes the plane fitting process much simplier:
Given a point cloud:
You just need to add a scalar field like this:
is_floor = cloud.add_scalar_field("plane_fit")
Wich will add a new column with value 1 for the points of the plane fitted.
You can visualize the scalar field:
Old answer
I think that you could easily use PCA to fit the plane to the 3D points instead of regression.
Here is a simple PCA implementation:
def PCA(data, correlation = False, sort = True):
""" Applies Principal Component Analysis to the data
Parameters
----------
data: array
The array containing the data. The array must have NxM dimensions, where each
of the N rows represents a different individual record and each of the M columns
represents a different variable recorded for that individual record.
array([
[V11, ... , V1m],
...,
[Vn1, ... , Vnm]])
correlation(Optional) : bool
Set the type of matrix to be computed (see Notes):
If True compute the correlation matrix.
If False(Default) compute the covariance matrix.
sort(Optional) : bool
Set the order that the eigenvalues/vectors will have
If True(Default) they will be sorted (from higher value to less).
If False they won't.
Returns
-------
eigenvalues: (1,M) array
The eigenvalues of the corresponding matrix.
eigenvector: (M,M) array
The eigenvectors of the corresponding matrix.
Notes
-----
The correlation matrix is a better choice when there are different magnitudes
representing the M variables. Use covariance matrix in other cases.
"""
mean = np.mean(data, axis=0)
data_adjust = data - mean
#: the data is transposed due to np.cov/corrcoef syntax
if correlation:
matrix = np.corrcoef(data_adjust.T)
else:
matrix = np.cov(data_adjust.T)
eigenvalues, eigenvectors = np.linalg.eig(matrix)
if sort:
#: sort eigenvalues and eigenvectors
sort = eigenvalues.argsort()[::-1]
eigenvalues = eigenvalues[sort]
eigenvectors = eigenvectors[:,sort]
return eigenvalues, eigenvectors
And here is how you could fit the points to a plane:
def best_fitting_plane(points, equation=False):
""" Computes the best fitting plane of the given points
Parameters
----------
points: array
The x,y,z coordinates corresponding to the points from which we want
to define the best fitting plane. Expected format:
array([
[x1,y1,z1],
...,
[xn,yn,zn]])
equation(Optional) : bool
Set the oputput plane format:
If True return the a,b,c,d coefficients of the plane.
If False(Default) return 1 Point and 1 Normal vector.
Returns
-------
a, b, c, d : float
The coefficients solving the plane equation.
or
point, normal: array
The plane defined by 1 Point and 1 Normal vector. With format:
array([Px,Py,Pz]), array([Nx,Ny,Nz])
"""
w, v = PCA(points)
#: the normal of the plane is the last eigenvector
normal = v[:,2]
#: get a point from the plane
point = np.mean(points, axis=0)
if equation:
a, b, c = normal
d = -(np.dot(normal, point))
return a, b, c, d
else:
return point, normal
However as this method is sensitive to outliers you could use RANSAC to make the fit robust to outliers.
There is a Python implementation of ransac here.
And you should only need to define a Plane Model class in order to use it for fitting planes to 3D points.
In any case if you can clean the 3D points from outliers (maybe you could use a KD-Tree S.O.R filter to that) you should get pretty good results with PCA.
Here is an implementation of an S.O.R:
def statistical_outilier_removal(kdtree, k=8, z_max=2 ):
""" Compute a Statistical Outlier Removal filter on the given KDTree.
Parameters
----------
kdtree: scipy's KDTree instance
The KDTree's structure which will be used to
compute the filter.
k(Optional): int
The number of nearest neighbors wich will be used to estimate the
mean distance from each point to his nearest neighbors.
Default : 8
z_max(Optional): int
The maximum Z score wich determines if the point is an outlier or
not.
Returns
-------
sor_filter : boolean array
The boolean mask indicating wherever a point should be keeped or not.
The size of the boolean mask will be the same as the number of points
in the KDTree.
Notes
-----
The 2 optional parameters (k and z_max) should be used in order to adjust
the filter to the desired result.
A HIGHER 'k' value will result(normally) in a HIGHER number of points trimmed.
A LOWER 'z_max' value will result(normally) in a HIGHER number of points trimmed.
"""
distances, i = kdtree.query(kdtree.data, k=k, n_jobs=-1)
z_distances = stats.zscore(np.mean(distances, axis=1))
sor_filter = abs(z_distances) < z_max
return sor_filter
You could feed the function with a KDtree of your 3D points computed maybe using this implementation
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