《PCL点云库学习VS2010(X64)》Part 34 旋转平移矩阵用法

《PCL点云库学习&VS2010(X64)》Part 34 旋转平移矩阵用法

1、变换与投影矩阵讲解：

https://en.wikipedia.org/wiki/Transformation_matrix

http://web.iitd.ac.in/~hegde/cad/lecture/L6_3dtrans.pdf

点云变换主要涉及平移、旋转、缩放、反射、剪切、视角转换、投影等，这里主要介绍平移与旋转。

2、使用Eigen::Matrix4f 进行变换

#include <pcl/io/pcd_io.h>
#include <pcl/common/transforms.h>
#include <pcl/visualization/pcl_visualizer.h>int
main(int argc, char** argv)
{// Objects for storing the point clouds.pcl::PointCloud<pcl::PointXYZ>::Ptr cloud(new pcl::PointCloud<pcl::PointXYZ>);pcl::PointCloud<pcl::PointXYZ>::Ptr transformed(new pcl::PointCloud<pcl::PointXYZ>);// Read a PCD file from disk.if (pcl::io::loadPCDFile<pcl::PointXYZ>(argv[1], *cloud) != 0){return -1;}// Transformation matrix object, initialized to the identity matrix// (a null transformation).Eigen::Matrix4f transformation = Eigen::Matrix4f::Identity();// Set a rotation around the Z axis (right hand rule).float theta = 90.0f * (M_PI / 180.0f); // 90 degrees.transformation(0, 0) = cos(theta);transformation(0, 1) = -sin(theta);transformation(1, 0) = sin(theta);transformation(1, 1) = cos(theta);// Set a translation on the X axis.transformation(0, 3) = 1.0f; // 1 meter (positive direction).pcl::transformPointCloud(*cloud, *transformed, transformation);// Visualize both the original and the result.pcl::visualization::PCLVisualizer viewer(argv[1]);viewer.addPointCloud(cloud, "original");// The transformed one's points will be red in color.pcl::visualization::PointCloudColorHandlerCustom<pcl::PointXYZ> colorHandler(transformed, 255, 0, 0);viewer.addPointCloud(transformed, colorHandler, "transformed");// Add 3D colored axes to help see the transformation.viewer.addCoordinateSystem(1.0, 0);while (!viewer.wasStopped()){viewer.spinOnce();}
}

2、使用Eigen::Affine3进行变换

#include <iostream>#include <pcl/io/pcd_io.h>
#include <pcl/io/ply_io.h>
#include <pcl/point_cloud.h>
#include <pcl/console/parse.h>
#include <pcl/common/transforms.h>
#include <pcl/visualization/pcl_visualizer.h>// This function displays the help
void
showHelp(char * program_name)
{std::cout << std::endl;std::cout << "Usage: " << program_name << " cloud_filename.[pcd|ply]" << std::endl;std::cout << "-h:  Show this help." << std::endl;
}// This is the main function
int
main (int argc, char** argv)
{// Show helpif (pcl::console::find_switch (argc, argv, "-h") || pcl::console::find_switch (argc, argv, "--help")) {showHelp (argv[0]);return 0;}// Fetch point cloud filename in arguments | Works with PCD and PLY filesstd::vector<int> filenames;bool file_is_pcd = false;filenames = pcl::console::parse_file_extension_argument (argc, argv, ".ply");if (filenames.size () != 1)  {filenames = pcl::console::parse_file_extension_argument (argc, argv, ".pcd");if (filenames.size () != 1) {showHelp (argv[0]);return -1;} else {file_is_pcd = true;}}// Load file | Works with PCD and PLY filespcl::PointCloud<pcl::PointXYZ>::Ptr source_cloud (new pcl::PointCloud<pcl::PointXYZ> ());if (file_is_pcd) {if (pcl::io::loadPCDFile (argv[filenames[0]], *source_cloud) < 0)  {std::cout << "Error loading point cloud " << argv[filenames[0]] << std::endl << std::endl;showHelp (argv[0]);return -1;}} else {if (pcl::io::loadPLYFile (argv[filenames[0]], *source_cloud) < 0)  {std::cout << "Error loading point cloud " << argv[filenames[0]] << std::endl << std::endl;showHelp (argv[0]);return -1;}}/* Reminder: how transformation matrices work :|-------> This column is the translation| 1 0 0 x |  \| 0 1 0 y |   }-> The identity 3x3 matrix (no rotation) on the left| 0 0 1 z |  /| 0 0 0 1 |    -> We do not use this line (and it has to stay 0,0,0,1)METHOD #1: Using a Matrix4fThis is the "manual" method, perfect to understand but error prone !*/Eigen::Matrix4f transform_1 = Eigen::Matrix4f::Identity();// Define a rotation matrix (see https://en.wikipedia.org/wiki/Rotation_matrix)float theta = M_PI/4; // The angle of rotation in radianstransform_1 (0,0) = cos (theta);transform_1 (0,1) = -sin(theta);transform_1 (1,0) = sin (theta);transform_1 (1,1) = cos (theta);//    (row, column)// Define a translation of 2.5 meters on the x axis.transform_1 (0,3) = 2.5;// Print the transformationprintf ("Method #1: using a Matrix4f\n");std::cout << transform_1 << std::endl;/*  METHOD #2: Using a Affine3fThis method is easier and less error prone*/Eigen::Affine3f transform_2 = Eigen::Affine3f::Identity();// Define a translation of 2.5 meters on the x axis.transform_2.translation() << 2.5, 0.0, 0.0;// The same rotation matrix as before; theta radians arround Z axistransform_2.rotate (Eigen::AngleAxisf (theta, Eigen::Vector3f::UnitZ()));// Print the transformationprintf ("\nMethod #2: using an Affine3f\n");std::cout << transform_2.matrix() << std::endl;// Executing the transformationpcl::PointCloud<pcl::PointXYZ>::Ptr transformed_cloud (new pcl::PointCloud<pcl::PointXYZ> ());// You can either apply transform_1 or transform_2; they are the samepcl::transformPointCloud (*source_cloud, *transformed_cloud, transform_2);// Visualizationprintf(  "\nPoint cloud colors :  white  = original point cloud\n""                        red  = transformed point cloud\n");pcl::visualization::PCLVisualizer viewer ("Matrix transformation example");// Define R,G,B colors for the point cloudpcl::visualization::PointCloudColorHandlerCustom<pcl::PointXYZ> source_cloud_color_handler (source_cloud, 255, 255, 255);// We add the point cloud to the viewer and pass the color handlerviewer.addPointCloud (source_cloud, source_cloud_color_handler, "original_cloud");pcl::visualization::PointCloudColorHandlerCustom<pcl::PointXYZ> transformed_cloud_color_handler (transformed_cloud, 230, 20, 20); // Redviewer.addPointCloud (transformed_cloud, transformed_cloud_color_handler, "transformed_cloud");viewer.addCoordinateSystem (1.0, "cloud", 0);viewer.setBackgroundColor(0.05, 0.05, 0.05, 0); // Setting background to a dark greyviewer.setPointCloudRenderingProperties (pcl::visualization::PCL_VISUALIZER_POINT_SIZE, 2, "original_cloud");viewer.setPointCloudRenderingProperties (pcl::visualization::PCL_VISUALIZER_POINT_SIZE, 2, "transformed_cloud");//viewer.setPosition(800, 400); // Setting visualiser window positionwhile (!viewer.wasStopped ()) { // Display the visualiser until 'q' key is pressedviewer.spinOnce ();}return 0;
}

3、使用Eigen::Quaternion

4、总结：

1）使最小二乘方程值最小，从而求得c,R,T三个参数，分别表示局部放大系数、旋转系数和平移系数。

齐次变换后得到一个4x4矩阵：

返回一个矩阵，使上述的最小二乘方程最小。

见文章“Least-squares estimation of transformation parameters between two point patterns”Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573

2）matrix4f的具体结构及意义。

  /* Reminder: how transformation matrices work :|-------> This column is the translation| 1 0 0 x |  \| 0 1 0 y |   }-> The identity 3x3 matrix (no rotation) on the left| 0 0 1 z |  /| 0 0 0 1 |    -> We do not use this line (and it has to stay 0,0,0,1)METHOD #1: Using a Matrix4fThis is the "manual" method, perfect to understand but error prone !*/

由上述的提示段代码可知，左上角的三行三列主要是旋转矩阵参数，最后一列的上三行是平移矩阵。

实际上该4x4矩阵肯定不止这些，具体的如下图所示：

其中p,q,r对应的透视变换参数，左上角三行三列包括局部缩放、剪切、旋转和反射等参数。具体的设置见文章中的第二个pdf链接。最后一行的l,m,n表示沿着x,y,z轴进行平移。s表示全局缩放参数。

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