# Using a matrix to transform a point cloud

In this tutorial we will learn how to transform a point cloud using a 4x4 matrix. We will apply a rotation and a translation to a loaded point cloud and display then result.

This program is able to load one PCD or PLY file; apply a matrix transformation on it and display the original and transformed point cloud.

# The code

First, create a file, let’s say, `matrix_transform.cpp` in your favorite editor, and place the following code inside it:

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136``` ```#include #include #include #include #include #include #include // 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 help if (pcl::console::find_switch (argc, argv, "-h") || pcl::console::find_switch (argc, argv, "--help")) { showHelp (argv); return 0; } // Fetch point cloud filename in arguments | Works with PCD and PLY files std::vector 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); return -1; } else { file_is_pcd = true; } } // Load file | Works with PCD and PLY files pcl::PointCloud::Ptr source_cloud (new pcl::PointCloud ()); if (file_is_pcd) { if (pcl::io::loadPCDFile (argv[filenames], *source_cloud) < 0) { std::cout << "Error loading point cloud " << argv[filenames] << std::endl << std::endl; showHelp (argv); return -1; } } else { if (pcl::io::loadPLYFile (argv[filenames], *source_cloud) < 0) { std::cout << "Error loading point cloud " << argv[filenames] << std::endl << std::endl; showHelp (argv); 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 Matrix4f This 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 radians transform_1 (0,0) = std::cos (theta); transform_1 (0,1) = -sin(theta); transform_1 (1,0) = sin (theta); transform_1 (1,1) = std::cos (theta); // (row, column) // Define a translation of 2.5 meters on the x axis. transform_1 (0,3) = 2.5; // Print the transformation printf ("Method #1: using a Matrix4f\n"); std::cout << transform_1 << std::endl; /* METHOD #2: Using a Affine3f This 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 around Z axis transform_2.rotate (Eigen::AngleAxisf (theta, Eigen::Vector3f::UnitZ())); // Print the transformation printf ("\nMethod #2: using an Affine3f\n"); std::cout << transform_2.matrix() << std::endl; // Executing the transformation pcl::PointCloud::Ptr transformed_cloud (new pcl::PointCloud ()); // You can either apply transform_1 or transform_2; they are the same pcl::transformPointCloud (*source_cloud, *transformed_cloud, transform_2); // Visualization printf( "\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 cloud pcl::visualization::PointCloudColorHandlerCustom source_cloud_color_handler (source_cloud, 255, 255, 255); // We add the point cloud to the viewer and pass the color handler viewer.addPointCloud (source_cloud, source_cloud_color_handler, "original_cloud"); pcl::visualization::PointCloudColorHandlerCustom transformed_cloud_color_handler (transformed_cloud, 230, 20, 20); // Red viewer.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 grey viewer.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 position while (!viewer.wasStopped ()) { // Display the visualiser until 'q' key is pressed viewer.spinOnce (); } return 0; } ```

# The explanation

Now, let’s break down the code piece by piece.

```#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>
```

We include all the headers we will make use of. #include <pcl/common/transforms.h> allows us to use pcl::transformPointCloud function.

```// 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 function display the help in case the user didn’t provide expected arguments.

```  // Show help
if (pcl::console::find_switch (argc, argv, "-h") || pcl::console::find_switch (argc, argv, "--help")) {
showHelp (argv);
return 0;
}
```

We parse the arguments on the command line, either using -h or –help will display the help. This terminates the program

```  // Fetch point cloud filename in arguments | Works with PCD and PLY files
std::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);
return -1;
} else {
file_is_pcd = true;
}
}
```

We look for .ply or .pcd filenames in the arguments. If not found; terminate the program. The bool file_is_pcd will help us choose between loading PCD or PLY file.

```  // Load file | Works with PCD and PLY files
pcl::PointCloud<pcl::PointXYZ>::Ptr source_cloud (new pcl::PointCloud<pcl::PointXYZ> ());

if (file_is_pcd) {
if (pcl::io::loadPCDFile (argv[filenames], *source_cloud) < 0)  {
std::cout << "Error loading point cloud " << argv[filenames] << std::endl << std::endl;
showHelp (argv);
return -1;
}
} else {
if (pcl::io::loadPLYFile (argv[filenames], *source_cloud) < 0)  {
std::cout << "Error loading point cloud " << argv[filenames] << std::endl << std::endl;
showHelp (argv);
return -1;
}
}
```

We now load the PCD/PLY file and check if the file was loaded successfully. Otherwise terminate the program.

```  /* 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 Matrix4f
This is the "manual" method, perfect to understand but error prone !
*/
Eigen::Matrix4f transform_1 = Eigen::Matrix4f::Identity();
```

This is a first approach to create a transformation. This will help you understand how transformation matrices work. We initialize a 4x4 matrix to identity;

```    |  1  0  0  0  |
i = |  0  1  0  0  |
|  0  0  1  0  |
|  0  0  0  1  |
```

Note

The identity matrix is the equivalent of “1” when multiplying numbers; it changes nothing. It is a square matrix with ones on the main diagonal and zeros elsewhere.

This means no transformation (no rotation and no translation). We do not use the last row of the matrix.

The first 3 rows and columns (top left) components are the rotation matrix. The first 3 rows of the last column is the translation.

```  // Define a rotation matrix (see https://en.wikipedia.org/wiki/Rotation_matrix)
float theta = M_PI/4; // The angle of rotation in radians
transform_1 (0,0) = std::cos (theta);
transform_1 (0,1) = -sin(theta);
transform_1 (1,0) = sin (theta);
transform_1 (1,1) = std::cos (theta);
//    (row, column)

// Define a translation of 2.5 meters on the x axis.
transform_1 (0,3) = 2.5;

// Print the transformation
printf ("Method #1: using a Matrix4f\n");
std::cout << transform_1 << std::endl;
```

Here we defined a 45° (PI/4) rotation around the Z axis and a translation on the X axis. This is the transformation we just defined

```    |  cos(θ) -sin(θ)  0.0 |
R = |  sin(θ)  cos(θ)  0.0 |
|  0.0     0.0     1.0 |

t = < 2.5, 0.0, 0.0 >
```
```  /*  METHOD #2: Using a Affine3f
This 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 around Z axis
transform_2.rotate (Eigen::AngleAxisf (theta, Eigen::Vector3f::UnitZ()));

// Print the transformation
printf ("\nMethod #2: using an Affine3f\n");
std::cout << transform_2.matrix() << std::endl;
```

This second approach is easier to understand and is less error prone. Be careful if you want to apply several rotations; rotations are not commutative ! This means than in most cases: rotA * rotB != rotB * rotA.

```  // Executing the transformation
pcl::PointCloud<pcl::PointXYZ>::Ptr transformed_cloud (new pcl::PointCloud<pcl::PointXYZ> ());
// You can either apply transform_1 or transform_2; they are the same
pcl::transformPointCloud (*source_cloud, *transformed_cloud, transform_2);
```

Now we apply this matrix on the point cloud source_cloud and we save the result in the newly created transformed_cloud.

```  // Visualization
printf(  "\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 cloud
pcl::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 handler

pcl::visualization::PointCloudColorHandlerCustom<pcl::PointXYZ> transformed_cloud_color_handler (transformed_cloud, 230, 20, 20); // Red

viewer.setBackgroundColor(0.05, 0.05, 0.05, 0); // Setting background to a dark grey
viewer.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 position

while (!viewer.wasStopped ()) { // Display the visualiser until 'q' key is pressed
viewer.spinOnce ();
}

return 0;
```

We then visualize the result using the PCLVisualizer. The original point cloud will be displayed white and the transformed one in red. The coordoniates axis will be displayed. We also set the background color of the visualizer and the point display size.

# Compiling and running the program

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```cmake_minimum_required(VERSION 2.6 FATAL_ERROR) project(pcl-matrix_transform) find_package(PCL 1.7 REQUIRED) include_directories(\${PCL_INCLUDE_DIRS}) link_directories(\${PCL_LIBRARY_DIRS}) add_definitions(\${PCL_DEFINITIONS}) add_executable (matrix_transform matrix_transform.cpp) target_link_libraries (matrix_transform \${PCL_LIBRARIES}) ```

After you have made the executable, run it passing a path to a PCD or PLY file. To reproduce the results shown below, you can download the cube.ply file:

```\$ ./matrix_transform cube.ply
```

You will see something similar to this:

```./matrix_transform cube.ply
[pcl::PLYReader] /home/victor/cube.ply:12: property 'list uint8 uint32 vertex_indices' of element 'face' is not handled
Method #1: using a Matrix4f
0.707107 -0.707107         0       2.5
0.707107  0.707107         0         0
0         0         1         0
0         0         0         1

Method #2: using an Affine3f
0.707107 -0.707107         0       2.5
0.707107  0.707107         0         0
0         0         1         0
0         0         0         1

Point cloud colors :  white   = original point cloud
red    = transformed point cloud
``` So now you successfully transformed a point cloud using a transformation matrix.
What if you want to transform a single point ? A vector ?
A point is defined in 3D space with its three coordinates; x,y,z (in a cartesian coordinate system).
How can you multiply a vector (with 3 coordinates) with a 4x4 matrix ? You simply can’t ! If you don’t know why please refer to matrix multiplications on wikipedia.

We need a vector with 4 components. What do you put in the last component ? It depends on what you want to do:

• If you want to transform a point: put 1 at the end of the vector so that the translation is taken in account.

• If you want to transform the direction of a vector: put 0 at the end of the vector to ignore the translation.

Here’s a quick example, we want to transform the following vector:

```[10, 5, 0, 3, 0, -1]
```
Where the first 3 components defines the origin coordinates and the last 3 components the direction.
This vector starts at point 10, 5, 0 and ends at 13, 5, -1.

This is what you need to do to transform the vector:

```[10, 5, 0,  1] * 4x4_transformation_matrix
[3,  0, -1, 0] * 4x4_transformation_matrix
```