CloudCompare——拟合空间球

本文由CSDN点云侠原创，CloudCompare——拟合空间球，爬虫自重。如果你不是在点云侠的博客中看到该文章，那么此处便是不要脸的爬虫与GPT生成的文章。

1.拟合球

  源码里用到了四点定球，具体计算原理如下：

  已知空间内不共面的四个点，设其坐标为$A(x_1,y_1,z_1)$、 $B(x_2,y_2,z_2)$、$C(x_3,y_3,z_3)、D(x_4,y_4,z_4)$，设半径为$r$，球心$O$坐标为$(x,y,z)$。利用四点到球心距离相等的性质得到如下四个方程。
$$\begin{gathered} (x-x_1{)}^2+(y-y_1{)}^2+(z-z_1{)}^2=r^2; \\ (x-x_2{)}^2+(y-y_2{)}^2+(z-z_2{)}^2=r^2; \\ (x-x_3{)}^2+(y-y_3{)}^2+(z-z_3{)}^2=r^2; \\ (x-x_4{)}^2+(y-y_4{)}^2+(z-z_4{)}^2=r^2; \end{gathered}$$

展开得:
$$\begin{gathered} x^2+y^2+z^2-2(x_{1}x+y_{1}y+z_{1}z)+x_1^2+y_1^2+z_1^2=r^2① \\ {x}^2+y^2+z^2-2(x_{2}x+y_{2}y+z_{2}z)+x_2^2+y_2^2+z_2^2=r^2② \\ {x}^2+y^2+z^2-2(x_{3}x+y_{3}y+z_{3}z)+x_3^2+y_3^2+z_3^2=r^2③ \\ {x}^2+y^2+z^2-2(x_{4}x+y_{4}y+z_{4}z)+x_4^2+y_4^2+z_4^2=r^2④ \end{gathered}$$

分别作①-②、③ - ④、② - ③得：
$$\begin{gathered} (x_1-x_2)x+(y_1-y_2)y+(z_1-z_2)z=1/2(x_1^2-x_2^2+y_1^2-y_2^2+z_1^2-z_2^2) \\ (x_3-x_4)x+(y_3-y_4)y+(z_3-z_4)z=1/2(x_3^2-x_4^2+y_3^2-y_4^2+z_3^2-z_4^2) \\ (x_2-x_3)x+(y_2-y_3)y+(z_2-z_3)z=1/2(x_2^2-x_3^2+y_2^2-y_3^2+z_2^2-z_3^2) \end{gathered}$$

其对应的系数行列式可设为：

$$D=\left|\begin{array}{ccc}a& b& c\\ a_1& b_1& c_1\\ a_2& b_2& c_2\end{array}\right|$$

则：$$\begin{gathered} a=(x_1-x_2),b=(y_1-y_2),c=(z_1-z_2), \\ {a}_1=(x_3-x_4),b_1=(y_3-y_4)，c_1=(z_3-z_4), \\ {a}_2=(x_2-x_3),b_2=(y_2-y_3)，c_2=(z_2-z_3) \end{gathered}$$

常数项行列式为:

$$L=\left|\begin{array}{c}P\\ Q\\ R\end{array}\right|$$

则：
$$P=\frac{1}{2}(x_1^2-x_2^2+y_1^2-y_2^2+z_1^2-z_2^2)$$
$$Q=\frac{1}{2}(x_3^2-x_4^2+y_3^2-y_4^2+z_3^2-z_4^2)$$
$$R=\frac{1}{2}(x_2^2-x_3^2+y_2^2-y_3^2+z_2^2-z_3^2)$$

现设：
$$Dx=\left|\begin{array}{ccc}P& b& c\\ Q& b_1& c_1\\ R& b_2& c_2\end{array}\right|$$

$$Dy=\left|\begin{array}{ccc}a& P& c\\ a_1& Q& c_1\\ a_2& R& c_2\end{array}\right|$$

$$Dz=\left|\begin{array}{ccc}a& b& P\\ a_1& b_1& Q\\ a_2& b_2& R\end{array}\right|$$

由线性代数中的克拉默法则可知:
$$x=\frac{Dx}{D}$$

$$y=\frac{Dy}{D}$$

$$z=\frac{Dz}{D}$$

2.软件操作

  通过菜单栏的'Tools > Fit > Sphere'找到该功能。

  选择一个或多个点云，然后启动此工具。CloudCompare将在每个点云上拟合球体基元。在控制台中，将输出以下信息：

• center（也可以在球体实体属性中找到球体边界框的中心）
• radius（也可以在sphere实体属性中找到）
• 球体拟合RMS（在默认球体实体名称中调用）注意：理论上球体拟合算法可以处理高达50％的异常值。

球形点云

拟合结果

控制台输出

3.算法源码

GeometricalAnalysisTools::ErrorCode GeometricalAnalysisTools::DetectSphereRobust(GenericIndexedCloudPersist* cloud,double outliersRatio,CCVector3& center,PointCoordinateType& radius,double& rms,GenericProgressCallback* progressCb/*=nullptr*/,double confidence/*=0.99*/,unsigned seed/*=0*/)
{if (!cloud){assert(false);return InvalidInput;}unsigned n = cloud->size();if (n < 4)return NotEnoughPoints;assert(confidence < 1.0);confidence = std::min(confidence, 1.0 - FLT_EPSILON);//we'll need an array (sorted) to compute the mediansstd::vector<PointCoordinateType> values;try{values.resize(n);}catch (const std::bad_alloc&){//not enough memoryreturn NotEnoughMemory;}//number of samplesunsigned m = 1;const unsigned p = 4;if (n > p){m = static_cast<unsigned>(log(1.0 - confidence) / log(1.0 - pow(1.0 - outliersRatio, static_cast<double>(p))));}//for progress notificationNormalizedProgress nProgress(progressCb, m);if (progressCb){if (progressCb->textCanBeEdited()){char buffer[64];sprintf(buffer, "Least Median of Squares samples: %u", m);progressCb->setInfo(buffer);progressCb->setMethodTitle("Detect sphere");}progressCb->update(0);progressCb->start();}//now we are going to randomly extract a subset of 4 points and test the resulting sphere each timeif (seed == 0){std::random_device randomGenerator;   // non-deterministic generatorseed = randomGenerator();}std::mt19937 gen(seed);  // to seed mersenne twister.std::uniform_int_distribution<unsigned> dist(0, n - 1);unsigned sampleCount = 0;unsigned attempts = 0;double minError = -1.0;std::vector<unsigned> indexes;indexes.resize(p);while (sampleCount < m && attempts < 2*m){//get 4 random (different) indexesfor (unsigned j = 0; j < p; ++j){bool isOK = false;while (!isOK){indexes[j] = dist(gen);isOK = true;for (unsigned k = 0; k < j && isOK; ++k)if (indexes[j] == indexes[k])isOK = false;}}assert(p == 4);const CCVector3* A = cloud->getPoint(indexes[0]);const CCVector3* B = cloud->getPoint(indexes[1]);const CCVector3* C = cloud->getPoint(indexes[2]);const CCVector3* D = cloud->getPoint(indexes[3]);++attempts;CCVector3 thisCenter;PointCoordinateType thisRadius;if (ComputeSphereFrom4(*A, *B, *C, *D, thisCenter, thisRadius) != NoError)continue;//compute residualsfor (unsigned i = 0; i < n; ++i){PointCoordinateType error = (*cloud->getPoint(i) - thisCenter).norm() - thisRadius;values[i] = error*error;}const unsigned int	medianIndex = n / 2;std::nth_element(values.begin(), values.begin() + medianIndex, values.end());//the error is the median of the squared residualsdouble error = static_cast<double>(values[medianIndex]);//we keep track of the solution with the least errorif (error < minError || minError < 0.0){minError = error;center = thisCenter;radius = thisRadius;}++sampleCount;if (progressCb && !nProgress.oneStep()){//progress canceled by the userreturn ProcessCancelledByUser;}}//too many failures?!if (sampleCount < m){return ProcessFailed;}//last step: robust estimationReferenceCloud candidates(cloud);if (n > p){//e robust standard deviation estimate (see Zhang's report)double sigma = 1.4826 * (1.0 + 5.0 /(n-p)) * sqrt(minError);//compute the least-squares best-fitting sphere with the points//having residuals below 2.5 sigmadouble maxResidual = 2.5 * sigma;if (candidates.reserve(n)){//compute residuals and select the pointsfor (unsigned i = 0; i < n; ++i){PointCoordinateType error = (*cloud->getPoint(i) - center).norm() - radius;if (error < maxResidual)candidates.addPointIndex(i);}candidates.resize(candidates.size());//eventually estimate the robust sphere parameters with least squares (iterative)if (RefineSphereLS(&candidates, center, radius)){//replace input cloud by this subset!cloud = &candidates;n = cloud->size();}}else{//not enough memory!//we'll keep the rough estimate...}}//update residuals{double residuals = 0;for (unsigned i = 0; i < n; ++i){const CCVector3* P = cloud->getPoint(i);double e = (*P - center).norm() - radius;residuals += e*e;}rms = sqrt(residuals/n);}return NoError;
}

4.相关代码

[1]C++实现：PCL RANSAC拟合空间3D球体
[2]python实现：Open3D——RANSAC三维点云球面拟合
[3] Open3D 最小二乘拟合球
[4] Open3D 非线性最小二乘拟合球