PCL源码剖析之MarchingCubes算法

MarchingCubes算法简介

MarchingCubes(移动立方体)算法是目前三围数据场等值面生成中最常用的方法。它实际上是一个分而治之的方法，把等值面的抽取分布于每个体素中进行。对于每个被处理的体素，以三角面片逼近其内部的等值面片。每个体素是一个小立方体，构造三角面片的处理过程对每个体素都“扫描”一遍，就好像一个处理器在这些体素上移动一样，由此得名移动立方体算法。

MC算法主要有三步：1.将点云数据转换为体素网格数据；2.使用线性插值对每个体素抽取等值面；3.对等值面进行网格三角化

PCL源码剖析之MarchingCubesHoppe

PCL中使用MarchingCubesHoppe类进行三维重建执行的函数体为performReconstruction()，其代码如下：

template <typename PointNT> void
pcl::MarchingCubes<PointNT>::performReconstruction (pcl::PolygonMesh &output)
{
if (!(iso_level_ >= 0 && iso_level_ < 1))
{
PCL_ERROR ("[pcl::%s::performReconstruction] Invalid iso level %f! Please use a number between 0 and 1.\n", getClassName ().c_str (), iso_level_);
output.cloud.width = output.cloud.height = 0;
output.cloud.data.clear ();
output.polygons.clear ();
return;
}
// Create grid
grid_ = std::vector<float> (res_x_*res_y_*res_z_, 0.0f);
// Populate tree
tree_->setInputCloud (input_);
getBoundingBox ();
// Transform the point cloud into a voxel grid
// This needs to be implemented in a child class
voxelizeData ();
// Run the actual marching cubes algorithm, store it into a point cloud,
// and copy the point cloud + connectivity into output
pcl::PointCloud<PointNT> cloud;
for (int x = 1; x < res_x_-1; ++x)
for (int y = 1; y < res_y_-1; ++y)
for (int z = 1; z < res_z_-1; ++z)
{
Eigen::Vector3i index_3d (x, y, z);
std::vector<float> leaf_node;
getNeighborList1D (leaf_node, index_3d);
createSurface (leaf_node, index_3d, cloud);
}
pcl::toPCLPointCloud2 (cloud, output.cloud);
output.polygons.resize (cloud.size () / 3);
for (size_t i = 0; i < output.polygons.size (); ++i)
{
pcl::Vertices v;
v.vertices.resize (3);
for (int j = 0; j < 3; ++j)
v.vertices[j] = static_cast<int> (i) * 3 + j;
output.polygons[i] = v;
}
}

可以看出PCL将会产生res_x_ * res_y_ * res_z_个网格，即为Resolution分辨率。voxelizeData ();即将点云数据转换为体素网格数据，其实现如下：

template <typename PointNT> void
pcl::MarchingCubesHoppe<PointNT>::voxelizeData ()
{
for (int x = 0; x < res_x_; ++x)
for (int y = 0; y < res_y_; ++y)
for (int z = 0; z < res_z_; ++z)
{
std::vector<int> nn_indices;
std::vector<float> nn_sqr_dists;
Eigen::Vector3f point;
point[0] = min_p_[0] + (max_p_[0] - min_p_[0]) * float (x) / float (res_x_);
point[1] = min_p_[1] + (max_p_[1] - min_p_[1]) * float (y) / float (res_y_);
point[2] = min_p_[2] + (max_p_[2] - min_p_[2]) * float (z) / float (res_z_);
PointNT p;
p.getVector3fMap () = point;
tree_->nearestKSearch (p, 1, nn_indices, nn_sqr_dists);
grid_[x * res_y_*res_z_ + y * res_z_ + z] = input_->points[nn_indices[0]].getNormalVector3fMap ().dot (
point - input_->points[nn_indices[0]].getVector3fMap ());
}
}

该函数对每个体素网格数据进行赋值，其值为一符号距离函数值，其定义为：f(Pi) = (Pi - Oi) * N(Oi), 这里Pi为给定的点，Oi为Pi周围K近邻点集(输入点云的子集)的中心， N(Oi)为点Oi的法向量，中间的*为数量积；求出的值其实是点Oi到过点Pi的有向切平面的距离，图示如下：

点q处的法向量是单位法向量，所以点q到切平面的距离是dot(N(p), vec(p, q))

从代码中可以看出，这里的K = 1，即求出最近邻点。

下面的代码描述了对每个体素网格的处理过程，其主要过程是计算出每个体素网格与等值面的交点，然后按一定顺序将交点连接，从而形成三角面片。

for (int x = 1; x < res_x_-1; ++x)
for (int y = 1; y < res_y_-1; ++y)
for (int z = 1; z < res_z_-1; ++z)
{
Eigen::Vector3i index_3d (x, y, z);
std::vector<float> leaf_node;
getNeighborList1D (leaf_node, index_3d);
createSurface (leaf_node, index_3d, cloud);
}

getNeighorList1D(leaf_node, index_3d);即是求出当前体素网格的8个顶点对应符号距离函数值，即数组grid_中对应的值。实现代码如下：

template <typename PointNT> void
pcl::MarchingCubes<PointNT>::getNeighborList1D (std::vector<float> &leaf,
Eigen::Vector3i &index3d)
{
leaf = std::vector<float> (8, 0.0f);
leaf[0] = getGridValue (index3d);
leaf[1] = getGridValue (index3d + Eigen::Vector3i (1, 0, 0));
leaf[2] = getGridValue (index3d + Eigen::Vector3i (1, 0, 1));
leaf[3] = getGridValue (index3d + Eigen::Vector3i (0, 0, 1));
leaf[4] = getGridValue (index3d + Eigen::Vector3i (0, 1, 0));
leaf[5] = getGridValue (index3d + Eigen::Vector3i (1, 1, 0));
leaf[6] = getGridValue (index3d + Eigen::Vector3i (1, 1, 1));
leaf[7] = getGridValue (index3d + Eigen::Vector3i (0, 1, 1));
}

createSurface (leaf_node, index_3d, cloud);即是求出每个体素网格与等值面的交点，然后按一定顺序将交点连接，从而形成三角面片。下面分片段剖析createSurface ()函数代码。

int cubeindex = 0;
Eigen::Vector3f vertex_list[12];
if (leaf_node[0] < iso_level_) cubeindex |= 1;
if (leaf_node[1] < iso_level_) cubeindex |= 2;
if (leaf_node[2] < iso_level_) cubeindex |= 4;
if (leaf_node[3] < iso_level_) cubeindex |= 8;
if (leaf_node[4] < iso_level_) cubeindex |= 16;
if (leaf_node[5] < iso_level_) cubeindex |= 32;
if (leaf_node[6] < iso_level_) cubeindex |= 64;
if (leaf_node[7] < iso_level_) cubeindex |= 128;

此段代码是将8个顶点的标量值与等值面相比较，如果标量值小于等值面值(即顶点在等值面下面)，则将cubeindex相应的位置为1。这样就可以知道8个顶点中哪些在等值面下，哪些在等值面之上了。
立方体中顶点与棱边的编号如下所示：

例如，如果顶点3的值在等值面值之下并且所有其他顶点的值都在等值面之上，那么我们可以通过切割边2、3、11来创建一个三角面片。

算法使用一个边表将cubeindex映射为一个12bit的数值，每一位与一条边相关，如果边没有被等值面切割则设为0，切割则设为1。如果没有边被切割那么表返回0，这种情况发生在当cubeindex = 0(所有顶点在等值面之下)或0xff(所有顶点在等值面之上)。举个之前的例子，如果只有顶点3在等值面下面，cubeindex将会等于0000 1000 或8。边表edgeTable定义在marching_cubes.h文件中：

const unsigned int edgeTable[256] = {
0x0 , 0x109, 0x203, 0x30a, 0x406, 0x50f, 0x605, 0x70c,
0x80c, 0x905, 0xa0f, 0xb06, 0xc0a, 0xd03, 0xe09, 0xf00,
0x190, 0x99 , 0x393, 0x29a, 0x596, 0x49f, 0x795, 0x69c,
0x99c, 0x895, 0xb9f, 0xa96, 0xd9a, 0xc93, 0xf99, 0xe90,
0x230, 0x339, 0x33 , 0x13a, 0x636, 0x73f, 0x435, 0x53c,
0xa3c, 0xb35, 0x83f, 0x936, 0xe3a, 0xf33, 0xc39, 0xd30,
0x3a0, 0x2a9, 0x1a3, 0xaa , 0x7a6, 0x6af, 0x5a5, 0x4ac,
0xbac, 0xaa5, 0x9af, 0x8a6, 0xfaa, 0xea3, 0xda9, 0xca0,
0x460, 0x569, 0x663, 0x76a, 0x66 , 0x16f, 0x265, 0x36c,
0xc6c, 0xd65, 0xe6f, 0xf66, 0x86a, 0x963, 0xa69, 0xb60,
0x5f0, 0x4f9, 0x7f3, 0x6fa, 0x1f6, 0xff , 0x3f5, 0x2fc,
0xdfc, 0xcf5, 0xfff, 0xef6, 0x9fa, 0x8f3, 0xbf9, 0xaf0,
0x650, 0x759, 0x453, 0x55a, 0x256, 0x35f, 0x55 , 0x15c,
0xe5c, 0xf55, 0xc5f, 0xd56, 0xa5a, 0xb53, 0x859, 0x950,
0x7c0, 0x6c9, 0x5c3, 0x4ca, 0x3c6, 0x2cf, 0x1c5, 0xcc ,
0xfcc, 0xec5, 0xdcf, 0xcc6, 0xbca, 0xac3, 0x9c9, 0x8c0,
0x8c0, 0x9c9, 0xac3, 0xbca, 0xcc6, 0xdcf, 0xec5, 0xfcc,
0xcc , 0x1c5, 0x2cf, 0x3c6, 0x4ca, 0x5c3, 0x6c9, 0x7c0,
0x950, 0x859, 0xb53, 0xa5a, 0xd56, 0xc5f, 0xf55, 0xe5c,
0x15c, 0x55 , 0x35f, 0x256, 0x55a, 0x453, 0x759, 0x650,
0xaf0, 0xbf9, 0x8f3, 0x9fa, 0xef6, 0xfff, 0xcf5, 0xdfc,
0x2fc, 0x3f5, 0xff , 0x1f6, 0x6fa, 0x7f3, 0x4f9, 0x5f0,
0xb60, 0xa69, 0x963, 0x86a, 0xf66, 0xe6f, 0xd65, 0xc6c,
0x36c, 0x265, 0x16f, 0x66 , 0x76a, 0x663, 0x569, 0x460,
0xca0, 0xda9, 0xea3, 0xfaa, 0x8a6, 0x9af, 0xaa5, 0xbac,
0x4ac, 0x5a5, 0x6af, 0x7a6, 0xaa , 0x1a3, 0x2a9, 0x3a0,
0xd30, 0xc39, 0xf33, 0xe3a, 0x936, 0x83f, 0xb35, 0xa3c,
0x53c, 0x435, 0x73f, 0x636, 0x13a, 0x33 , 0x339, 0x230,
0xe90, 0xf99, 0xc93, 0xd9a, 0xa96, 0xb9f, 0x895, 0x99c,
0x69c, 0x795, 0x49f, 0x596, 0x29a, 0x393, 0x99 , 0x190,
0xf00, 0xe09, 0xd03, 0xc0a, 0xb06, 0xa0f, 0x905, 0x80c,
0x70c, 0x605, 0x50f, 0x406, 0x30a, 0x203, 0x109, 0x0
};

edgeTable[8] = 0x80c = 1000 0000 1100。这就表示边2、3、11与等值面相交。
判断出边与等值面相交之后，就要确定具体的交点。这里PCL使用线性插值，线性插值具体见线性插值。代码如下：

// Find the vertices where the surface intersects the cube
if (edgeTable[cubeindex] & 1)
interpolateEdge (p[0], p[1], leaf_node[0], leaf_node[1], vertex_list[0]);
if (edgeTable[cubeindex] & 2)
interpolateEdge (p[1], p[2], leaf_node[1], leaf_node[2], vertex_list[1]);
if (edgeTable[cubeindex] & 4)
interpolateEdge (p[2], p[3], leaf_node[2], leaf_node[3], vertex_list[2]);
if (edgeTable[cubeindex] & 8)
interpolateEdge (p[3], p[0], leaf_node[3], leaf_node[0], vertex_list[3]);
if (edgeTable[cubeindex] & 16)
interpolateEdge (p[4], p[5], leaf_node[4], leaf_node[5], vertex_list[4]);
if (edgeTable[cubeindex] & 32)
interpolateEdge (p[5], p[6], leaf_node[5], leaf_node[6], vertex_list[5]);
if (edgeTable[cubeindex] & 64)
interpolateEdge (p[6], p[7], leaf_node[6], leaf_node[7], vertex_list[6]);
if (edgeTable[cubeindex] & 128)
interpolateEdge (p[7], p[4], leaf_node[7], leaf_node[4], vertex_list[7]);
if (edgeTable[cubeindex] & 256)
interpolateEdge (p[0], p[4], leaf_node[0], leaf_node[4], vertex_list[8]);
if (edgeTable[cubeindex] & 512)
interpolateEdge (p[1], p[5], leaf_node[1], leaf_node[5], vertex_list[9]);
if (edgeTable[cubeindex] & 1024)
interpolateEdge (p[2], p[6], leaf_node[2], leaf_node[6], vertex_list[10]);
if (edgeTable[cubeindex] & 2048)
interpolateEdge (p[3], p[7], leaf_node[3], leaf_node[7], vertex_list[11]);

文章参考于：http://www.doc88.com/p-5475997688638.html

http://paulbourke.net/geometry/polygonise/

http://books.google.com.hk/books?id=4k4kvDwP-lgC&printsec=frontcover&hl=zh-CN#v=onepage&q&f=false