Real Time pose estimation of a textured object {#tutorial_real_time_pose}
==============================================

@tableofcontents

@prev_tutorial{tutorial_camera_calibration}
@next_tutorial{tutorial_interactive_calibration}

|    |    |
| -: | :- |
| Original author | Edgar Riba |
| Compatibility | OpenCV >= 3.0 |


Nowadays, augmented reality is one of the top research topic in computer vision and robotics fields.
The most elemental problem in augmented reality is the estimation of the camera pose respect of an
object in the case of computer vision area to perform subsequent 3D rendering or, in robotics, to obtain an object pose for grasping and manipulation. However, this is not a trivial
problem to solve due to the fact that the most common issue in image processing is the computational
cost of applying a lot of algorithms or mathematical operations for solving a problem which is basic
and immediately for humans.

Goal
----

This tutorial explains how to build a real-time application to estimate the camera pose in
order to track a textured object with six degrees of freedom given a 2D image and its 3D textured
model.

The application will have the following parts:

-   Read 3D textured object model and object mesh.
-   Take input from Camera or Video.
-   Extract ORB features and descriptors from the scene.
-   Match scene descriptors with model descriptors using Flann matcher.
-   Pose estimation using PnP + Ransac.
-   Linear Kalman Filter for bad poses rejection.

Theory
------

In computer vision estimate the camera pose from *n* 3D-to-2D point correspondences is a fundamental
and well understood problem. The most general version of the problem requires estimating the six
degrees of freedom of the pose and five calibration parameters: focal length, principal point,
aspect ratio and skew. It could be established with a minimum of 6 correspondences, using the well
known Direct Linear Transform (DLT) algorithm. There are, though, several simplifications to the
problem which turn into an extensive list of different algorithms that improve the accuracy of the
DLT.

The most common simplification is to assume known calibration parameters which is the so-called
Perspective-*n*-Point problem:

![](images/pnp.jpg)

**Problem Formulation:** Given a set of correspondences between 3D points \f$p_i\f$ expressed in a world
reference frame, and their 2D projections \f$u_i\f$ onto the image, we seek to retrieve the pose (\f$R\f$
and \f$t\f$) of the camera w.r.t. the world and the focal length \f$f\f$.

OpenCV provides four different approaches to solve the Perspective-*n*-Point problem which return
\f$R\f$ and \f$t\f$. Then, using the following formula it's possible to project 3D points into the image
plane:

\f[s\ \left [ \begin{matrix}   u \\   v \\  1 \end{matrix} \right ] = \left [ \begin{matrix}   f_x & 0 & c_x \\  0 & f_y & c_y \\   0 & 0 & 1 \end{matrix} \right ] \left [ \begin{matrix}  r_{11} & r_{12} & r_{13} & t_1 \\ r_{21} & r_{22} & r_{23} & t_2 \\  r_{31} & r_{32} & r_{33} & t_3 \end{matrix} \right ] \left [ \begin{matrix}  X \\  Y \\   Z\\ 1 \end{matrix} \right ]\f]

The complete documentation of how to manage with this equations is in @ref calib3d .

Source code
-----------

You can find the source code of this tutorial in the
`samples/cpp/tutorial_code/calib3d/real_time_pose_estimation/` folder of the OpenCV source library.

The tutorial consists of two main programs:

-#  **Model registration**

    This application is intended for users who do not have a 3D textured model of the object to be detected.
    You can use this program to create your own textured 3D model. This program only works for planar
    objects, then if you want to model an object with complex shape you should use a sophisticated
    software to create it.

    The application needs an input image of the object to be registered and its 3D mesh. We have also
    to provide the intrinsic parameters of the camera with which the input image was taken. All the
    files need to be specified using the absolute path or the relative one from your application’s
    working directory. If no files are specified the program will try to open the provided default
    parameters.

    The application starts up extracting the ORB features and descriptors from the input image and
    then uses the mesh along with the [Möller–Trumbore intersection
    algorithm](http://en.wikipedia.org/wiki/M%C3%B6ller%E2%80%93Trumbore_intersection_algorithm/)
    to compute the 3D coordinates of the found features. Finally, the 3D points and the descriptors
    are stored in different lists in a file with YAML format which each row is a different point. The
    technical background on how to store the files can be found in the @ref tutorial_file_input_output_with_xml_yml
    tutorial.

    ![](images/registration.png)

-#  **Model detection**

    The aim of this application is to estimate in real time the object pose given its 3D textured model.

    The application starts up loading the 3D textured model in YAML file format with the same
    structure explained in the model registration program. From the scene, the ORB features and
    descriptors are detected and extracted. Then, is used @ref cv::FlannBasedMatcher with
    @ref cv::flann::GenericIndex to do the matching between the scene descriptors and the model descriptors.
    Using the found matches along with @ref cv::solvePnPRansac function the `R` and `t` of
    the camera are computed. Finally, a KalmanFilter is applied in order to reject bad poses.

    In the case that you compiled OpenCV with the samples, you can find it in `opencv/build/bin/cpp-tutorial-pnp_detection`.
    Then you can run the application and change some parameters:
    @code{.cpp}
    This program shows how to detect an object given its 3D textured model. You can choose to use a recorded video or the webcam.
    Usage:
      ./cpp-tutorial-pnp_detection -help
    Keys:
      'esc' - to quit.
    --------------------------------------------------------------------------

    Usage: cpp-tutorial-pnp_detection [params]

      -c, --confidence (value:0.95)
          RANSAC confidence
      -e, --error (value:2.0)
          RANSAC reprojection error
      -f, --fast (value:true)
          use of robust fast match
      -h, --help (value:true)
          print this message
      --in, --inliers (value:30)
          minimum inliers for Kalman update
      --it, --iterations (value:500)
          RANSAC maximum iterations count
      -k, --keypoints (value:2000)
          number of keypoints to detect
      --mesh
          path to ply mesh
      --method, --pnp (value:0)
          PnP method: (0) ITERATIVE - (1) EPNP - (2) P3P - (3) DLS
      --model
          path to yml model
      -r, --ratio (value:0.7)
          threshold for ratio test
      -v, --video
          path to recorded video
    @endcode
    For example, you can run the application changing the pnp method:
    @code{.cpp}
    ./cpp-tutorial-pnp_detection --method=2
    @endcode

Explanation
-----------

Here is explained in detail the code for the real time application:

-#  **Read 3D textured object model and object mesh.**

    In order to load the textured model I implemented the *class* **Model** which has the function
    *load()* that opens a YAML file and take the stored 3D points with its corresponding descriptors.
    You can find an example of a 3D textured model in
    `samples/cpp/tutorial_code/calib3d/real_time_pose_estimation/Data/cookies_ORB.yml`.
    @code{.cpp}
    /* Load a YAML file using OpenCV */
    void Model::load(const std::string path)
    {
        cv::Mat points3d_mat;

        cv::FileStorage storage(path, cv::FileStorage::READ);
        storage["points_3d"] >> points3d_mat;
        storage["descriptors"] >> descriptors_;

        points3d_mat.copyTo(list_points3d_in_);

        storage.release();

    }
    @endcode
    In the main program the model is loaded as follows:
    @code{.cpp}
    Model model;               // instantiate Model object
    model.load(yml_read_path); // load a 3D textured object model
    @endcode
    In order to read the model mesh I implemented a *class* **Mesh** which has a function *load()*
    that opens a \f$*\f$.ply file and store the 3D points of the object and also the composed triangles.
    You can find an example of a model mesh in
    `samples/cpp/tutorial_code/calib3d/real_time_pose_estimation/Data/box.ply`.
    @code{.cpp}
    /* Load a CSV with *.ply format */
    void Mesh::load(const std::string path)
    {

        // Create the reader
        CsvReader csvReader(path);

        // Clear previous data
        list_vertex_.clear();
        list_triangles_.clear();

        // Read from .ply file
        csvReader.readPLY(list_vertex_, list_triangles_);

        // Update mesh attributes
        num_vertexs_ = list_vertex_.size();
        num_triangles_ = list_triangles_.size();

    }
    @endcode
    In the main program the mesh is loaded as follows:
    @code{.cpp}
    Mesh mesh;                // instantiate Mesh object
    mesh.load(ply_read_path); // load an object mesh
    @endcode
    You can also load different model and mesh:
    @code{.cpp}
    ./cpp-tutorial-pnp_detection --mesh=/absolute_path_to_your_mesh.ply --model=/absolute_path_to_your_model.yml
    @endcode

-#  **Take input from Camera or Video**

    To detect is necessary capture video. It's done loading a recorded video by passing the absolute
    path where it is located in your machine. In order to test the application you can find a recorded
    video in `samples/cpp/tutorial_code/calib3d/real_time_pose_estimation/Data/box.mp4`.
    @code{.cpp}
    cv::VideoCapture cap;                // instantiate VideoCapture
    cap.open(video_read_path);           // open a recorded video

    if(!cap.isOpened())                  // check if we succeeded
    {
       std::cout << "Could not open the camera device" << std::endl;
       return -1;
    }
    @endcode
    Then the algorithm is computed frame per frame:
    @code{.cpp}
    cv::Mat frame, frame_vis;

    while(cap.read(frame) && cv::waitKey(30) != 27)    // capture frame until ESC is pressed
    {

        frame_vis = frame.clone();                     // refresh visualisation frame

        // MAIN ALGORITHM

    }
    @endcode
    You can also load different recorded video:
    @code{.cpp}
    ./cpp-tutorial-pnp_detection --video=/absolute_path_to_your_video.mp4
    @endcode

-#  **Extract ORB features and descriptors from the scene**

    The next step is to detect the scene features and extract it descriptors. For this task I
    implemented a *class* **RobustMatcher** which has a function for keypoints detection and features
    extraction. You can find it in
    `samples/cpp/tutorial_code/calib3d/real_time_pose_estimation/src/RobustMatcher.cpp`. In your
    *RobustMatch* object you can use any of the 2D features detectors of OpenCV. In this case I used
    @ref cv::ORB features because is based on @ref cv::FAST to detect the keypoints and cv::xfeatures2d::BriefDescriptorExtractor
    to extract the descriptors which means that is fast and robust to rotations. You can find more
    detailed information about *ORB* in the documentation.

    The following code is how to instantiate and set the features detector and the descriptors
    extractor:
    @code{.cpp}
    RobustMatcher rmatcher;                                                          // instantiate RobustMatcher

    cv::FeatureDetector * detector = new cv::OrbFeatureDetector(numKeyPoints);       // instantiate ORB feature detector
    cv::DescriptorExtractor * extractor = new cv::OrbDescriptorExtractor();          // instantiate ORB descriptor extractor

    rmatcher.setFeatureDetector(detector);                                           // set feature detector
    rmatcher.setDescriptorExtractor(extractor);                                      // set descriptor extractor
    @endcode
    The features and descriptors will be computed by the *RobustMatcher* inside the matching function.

-#  **Match scene descriptors with model descriptors using Flann matcher**

    It is the first step in our detection algorithm. The main idea is to match the scene descriptors
    with our model descriptors in order to know the 3D coordinates of the found features into the
    current scene.

    Firstly, we have to set which matcher we want to use. In this case is used
    @ref cv::FlannBasedMatcher matcher which in terms of computational cost is faster than the
    @ref cv::BFMatcher matcher as we increase the trained collection of features. Then, for
    FlannBased matcher the index created is *Multi-Probe LSH: Efficient Indexing for High-Dimensional
    Similarity Search* due to *ORB* descriptors are binary.

    You can tune the *LSH* and search parameters to improve the matching efficiency:
    @code{.cpp}
    cv::Ptr<cv::flann::IndexParams> indexParams = cv::makePtr<cv::flann::LshIndexParams>(6, 12, 1); // instantiate LSH index parameters
    cv::Ptr<cv::flann::SearchParams> searchParams = cv::makePtr<cv::flann::SearchParams>(50);       // instantiate flann search parameters

    cv::DescriptorMatcher * matcher = new cv::FlannBasedMatcher(indexParams, searchParams);         // instantiate FlannBased matcher
    rmatcher.setDescriptorMatcher(matcher);                                                         // set matcher
    @endcode
    Secondly, we have to call the matcher by using *robustMatch()* or *fastRobustMatch()* function.
    The difference of using this two functions is its computational cost. The first method is slower
    but more robust at filtering good matches because uses two ratio test and a symmetry test. In
    contrast, the second method is faster but less robust because only applies a single ratio test to
    the matches.

    The following code is to get the model 3D points and its descriptors and then call the matcher in
    the main program:
    @code{.cpp}
    // Get the MODEL INFO

    std::vector<cv::Point3f> list_points3d_model = model.get_points3d();  // list with model 3D coordinates
    cv::Mat descriptors_model = model.get_descriptors();                  // list with descriptors of each 3D coordinate
    @endcode
    @code{.cpp}
    // -- Step 1: Robust matching between model descriptors and scene descriptors

    std::vector<cv::DMatch> good_matches;       // to obtain the model 3D points  in the scene
    std::vector<cv::KeyPoint> keypoints_scene;  // to obtain the 2D points of the scene

    if(fast_match)
    {
        rmatcher.fastRobustMatch(frame, good_matches, keypoints_scene, descriptors_model);
    }
    else
    {
        rmatcher.robustMatch(frame, good_matches, keypoints_scene, descriptors_model);
    }
    @endcode
    The following code corresponds to the *robustMatch()* function which belongs to the
    *RobustMatcher* class. This function uses the given image to detect the keypoints and extract the
    descriptors, match using *two Nearest Neighbour* the extracted descriptors with the given model
    descriptors and vice versa. Then, a ratio test is applied to the two direction matches in order to
    remove these matches which its distance ratio between the first and second best match is larger
    than a given threshold. Finally, a symmetry test is applied in order to remove non symmetrical
    matches.
    @code{.cpp}
    void RobustMatcher::robustMatch( const cv::Mat& frame, std::vector<cv::DMatch>& good_matches,
                                     std::vector<cv::KeyPoint>& keypoints_frame,
                                     const std::vector<cv::KeyPoint>& keypoints_model, const cv::Mat& descriptors_model )
    {

        // 1a. Detection of the ORB features
        this->computeKeyPoints(frame, keypoints_frame);

        // 1b. Extraction of the ORB descriptors
        cv::Mat descriptors_frame;
        this->computeDescriptors(frame, keypoints_frame, descriptors_frame);

        // 2. Match the two image descriptors
        std::vector<std::vector<cv::DMatch> > matches12, matches21;

        // 2a. From image 1 to image 2
        matcher_->knnMatch(descriptors_frame, descriptors_model, matches12, 2); // return 2 nearest neighbours

        // 2b. From image 2 to image 1
        matcher_->knnMatch(descriptors_model, descriptors_frame, matches21, 2); // return 2 nearest neighbours

        // 3. Remove matches for which NN ratio is > than threshold
        // clean image 1 -> image 2 matches
        int removed1 = ratioTest(matches12);
        // clean image 2 -> image 1 matches
        int removed2 = ratioTest(matches21);

        // 4. Remove non-symmetrical matches
        symmetryTest(matches12, matches21, good_matches);

    }
    @endcode
    After the matches filtering we have to subtract the 2D and 3D correspondences from the found scene
    keypoints and our 3D model using the obtained *DMatches* vector. For more information about
    @ref cv::DMatch check the documentation.
    @code{.cpp}
    // -- Step 2: Find out the 2D/3D correspondences

    std::vector<cv::Point3f> list_points3d_model_match;    // container for the model 3D coordinates found in the scene
    std::vector<cv::Point2f> list_points2d_scene_match;    // container for the model 2D coordinates found in the scene

    for(unsigned int match_index = 0; match_index < good_matches.size(); ++match_index)
    {
        cv::Point3f point3d_model = list_points3d_model[ good_matches[match_index].trainIdx ];   // 3D point from model
        cv::Point2f point2d_scene = keypoints_scene[ good_matches[match_index].queryIdx ].pt;    // 2D point from the scene
        list_points3d_model_match.push_back(point3d_model);                                      // add 3D point
        list_points2d_scene_match.push_back(point2d_scene);                                      // add 2D point
    }
    @endcode
    You can also change the ratio test threshold, the number of keypoints to detect as well as use or
    not the robust matcher:
    @code{.cpp}
    ./cpp-tutorial-pnp_detection --ratio=0.8 --keypoints=1000 --fast=false
    @endcode

-#  **Pose estimation using PnP + Ransac**

    Once with the 2D and 3D correspondences we have to apply a PnP algorithm in order to estimate the
    camera pose. The reason why we have to use @ref cv::solvePnPRansac instead of @ref cv::solvePnP is
    due to the fact that after the matching not all the found correspondences are correct and, as like
    as not, there are false correspondences or also called *outliers*. The [Random Sample
    Consensus](http://en.wikipedia.org/wiki/RANSAC) or *Ransac* is a non-deterministic iterative
    method which estimate parameters of a mathematical model from observed data producing an
    approximate result as the number of iterations increase. After applying *Ransac* all the *outliers*
    will be eliminated to then estimate the camera pose with a certain probability to obtain a good
    solution.

    For the camera pose estimation I have implemented a *class* **PnPProblem**. This *class* has 4
    attributes: a given calibration matrix, the rotation matrix, the translation matrix and the
    rotation-translation matrix. The intrinsic calibration parameters of the camera which you are
    using to estimate the pose are necessary. In order to obtain the parameters you can check
    @ref tutorial_camera_calibration_square_chess and @ref tutorial_camera_calibration tutorials.

    The following code is how to declare the *PnPProblem class* in the main program:
    @code{.cpp}
    // Intrinsic camera parameters: UVC WEBCAM

    double f = 55;                           // focal length in mm
    double sx = 22.3, sy = 14.9;             // sensor size
    double width = 640, height = 480;        // image size

    double params_WEBCAM[] = { width*f/sx,   // fx
                               height*f/sy,  // fy
                               width/2,      // cx
                               height/2};    // cy

    PnPProblem pnp_detection(params_WEBCAM); // instantiate PnPProblem class
    @endcode
    The following code is how the *PnPProblem class* initialises its attributes:
    @code{.cpp}
    // Custom constructor given the intrinsic camera parameters

    PnPProblem::PnPProblem(const double params[])
    {
      _A_matrix = cv::Mat::zeros(3, 3, CV_64FC1);   // intrinsic camera parameters
      _A_matrix.at<double>(0, 0) = params[0];       //      [ fx   0  cx ]
      _A_matrix.at<double>(1, 1) = params[1];       //      [  0  fy  cy ]
      _A_matrix.at<double>(0, 2) = params[2];       //      [  0   0   1 ]
      _A_matrix.at<double>(1, 2) = params[3];
      _A_matrix.at<double>(2, 2) = 1;
      _R_matrix = cv::Mat::zeros(3, 3, CV_64FC1);   // rotation matrix
      _t_matrix = cv::Mat::zeros(3, 1, CV_64FC1);   // translation matrix
      _P_matrix = cv::Mat::zeros(3, 4, CV_64FC1);   // rotation-translation matrix

    }
    @endcode
    OpenCV provides four PnP methods: ITERATIVE, EPNP, P3P and DLS. Depending on the application type,
    the estimation method will be different. In the case that we want to make a real time application,
    the more suitable methods are EPNP and P3P since they are faster than ITERATIVE and DLS at
    finding an optimal solution. However, EPNP and P3P are not especially robust in front of planar
    surfaces and sometimes the pose estimation seems to have a mirror effect. Therefore, in this
    tutorial an ITERATIVE method is used due to the object to be detected has planar surfaces.

    The OpenCV RANSAC implementation wants you to provide three parameters: 1) the maximum number of
    iterations until the algorithm stops, 2) the maximum allowed distance between the observed and
    computed point projections to consider it an inlier and 3) the confidence to obtain a good result.
    You can tune these parameters in order to improve your algorithm performance. Increasing the
    number of iterations will have a more accurate solution, but will take more time to find a
    solution. Increasing the reprojection error will reduce the computation time, but your solution
    will be unaccurate. Decreasing the confidence your algorithm will be faster, but the obtained
    solution will be unaccurate.

    The following parameters work for this application:
    @code{.cpp}
    // RANSAC parameters

    int iterationsCount = 500;        // number of Ransac iterations.
    float reprojectionError = 2.0;    // maximum allowed distance to consider it an inlier.
    float confidence = 0.95;          // RANSAC successful confidence.
    @endcode
    The following code corresponds to the *estimatePoseRANSAC()* function which belongs to the
    *PnPProblem class*. This function estimates the rotation and translation matrix given a set of
    2D/3D correspondences, the desired PnP method to use, the output inliers container and the Ransac
    parameters:
    @code{.cpp}
    // Estimate the pose given a list of 2D/3D correspondences with RANSAC and the method to use

    void PnPProblem::estimatePoseRANSAC( const std::vector<cv::Point3f> &list_points3d,        // list with model 3D coordinates
                                         const std::vector<cv::Point2f> &list_points2d,        // list with scene 2D coordinates
                                         int flags, cv::Mat &inliers, int iterationsCount,     // PnP method; inliers container
                                         float reprojectionError, float confidence )           // RANSAC parameters
    {
        cv::Mat distCoeffs = cv::Mat::zeros(4, 1, CV_64FC1);    // vector of distortion coefficients
        cv::Mat rvec = cv::Mat::zeros(3, 1, CV_64FC1);          // output rotation vector
        cv::Mat tvec = cv::Mat::zeros(3, 1, CV_64FC1);          // output translation vector

        bool useExtrinsicGuess = false;   // if true the function uses the provided rvec and tvec values as
                                          // initial approximations of the rotation and translation vectors

        cv::solvePnPRansac( list_points3d, list_points2d, _A_matrix, distCoeffs, rvec, tvec,
                            useExtrinsicGuess, iterationsCount, reprojectionError, confidence,
                            inliers, flags );

        Rodrigues(rvec,_R_matrix);                   // converts Rotation Vector to Matrix
        _t_matrix = tvec;                            // set translation matrix

        this->set_P_matrix(_R_matrix, _t_matrix);    // set rotation-translation matrix

    }
    @endcode
    In the following code are the 3rd and 4th steps of the main algorithm. The first, calling the
    above function and the second taking the output inliers vector from RANSAC to get the 2D scene
    points for drawing purpose. As seen in the code we must be sure to apply RANSAC if we have
    matches, in the other case, the function @ref cv::solvePnPRansac crashes due to any OpenCV *bug*.
    @code{.cpp}
    if(good_matches.size() > 0) // None matches, then RANSAC crashes
    {

        // -- Step 3: Estimate the pose using RANSAC approach
        pnp_detection.estimatePoseRANSAC( list_points3d_model_match, list_points2d_scene_match,
                                          pnpMethod, inliers_idx, iterationsCount, reprojectionError, confidence );


        // -- Step 4: Catch the inliers keypoints to draw
        for(int inliers_index = 0; inliers_index < inliers_idx.rows; ++inliers_index)
        {
        int n = inliers_idx.at<int>(inliers_index);         // i-inlier
        cv::Point2f point2d = list_points2d_scene_match[n]; // i-inlier point 2D
        list_points2d_inliers.push_back(point2d);           // add i-inlier to list
    }
    @endcode
    Finally, once the camera pose has been estimated we can use the \f$R\f$ and \f$t\f$ in order to compute
    the 2D projection onto the image of a given 3D point expressed in a world reference frame using
    the showed formula on *Theory*.

    The following code corresponds to the *backproject3DPoint()* function which belongs to the
    *PnPProblem class*. The function backproject a given 3D point expressed in a world reference frame
    onto a 2D image:
    @code{.cpp}
    // Backproject a 3D point to 2D using the estimated pose parameters

    cv::Point2f PnPProblem::backproject3DPoint(const cv::Point3f &point3d)
    {
        // 3D point vector [x y z 1]'
        cv::Mat point3d_vec = cv::Mat(4, 1, CV_64FC1);
        point3d_vec.at<double>(0) = point3d.x;
        point3d_vec.at<double>(1) = point3d.y;
        point3d_vec.at<double>(2) = point3d.z;
        point3d_vec.at<double>(3) = 1;

        // 2D point vector [u v 1]'
        cv::Mat point2d_vec = cv::Mat(4, 1, CV_64FC1);
        point2d_vec = _A_matrix * _P_matrix * point3d_vec;

        // Normalization of [u v]'
        cv::Point2f point2d;
        point2d.x = point2d_vec.at<double>(0) / point2d_vec.at<double>(2);
        point2d.y = point2d_vec.at<double>(1) / point2d_vec.at<double>(2);

        return point2d;
    }
    @endcode
    The above function is used to compute all the 3D points of the object *Mesh* to show the pose of
    the object.

    You can also change RANSAC parameters and PnP method:
    @code{.cpp}
    ./cpp-tutorial-pnp_detection --error=0.25 --confidence=0.90 --iterations=250 --method=3
    @endcode

-#  **Linear Kalman Filter for bad poses rejection**

    Is it common in computer vision or robotics fields that after applying detection or tracking
    techniques, bad results are obtained due to some sensor errors. In order to avoid these bad
    detections in this tutorial is explained how to implement a Linear Kalman Filter. The Kalman
    Filter will be applied after detected a given number of inliers.

    You can find more information about what [Kalman
    Filter](http://en.wikipedia.org/wiki/Kalman_filter) is. In this tutorial it's used the OpenCV
    implementation of the @ref cv::KalmanFilter based on
    [Linear Kalman Filter for position and orientation tracking](http://campar.in.tum.de/Chair/KalmanFilter)
    to set the dynamics and measurement models.

    Firstly, we have to define our state vector which will have 18 states: the positional data (x,y,z)
    with its first and second derivatives (velocity and acceleration), then rotation is added in form
    of three euler angles (roll, pitch, jaw) together with their first and second derivatives (angular
    velocity and acceleration)

    \f[X = (x,y,z,\dot x,\dot y,\dot z,\ddot x,\ddot y,\ddot z,\psi,\theta,\phi,\dot \psi,\dot \theta,\dot \phi,\ddot \psi,\ddot \theta,\ddot \phi)^T\f]

    Secondly, we have to define the number of measurements which will be 6: from \f$R\f$ and \f$t\f$ we can
    extract \f$(x,y,z)\f$ and \f$(\psi,\theta,\phi)\f$. In addition, we have to define the number of control
    actions to apply to the system which in this case will be *zero*. Finally, we have to define the
    differential time between measurements which in this case is \f$1/T\f$, where *T* is the frame rate of
    the video.
    @code{.cpp}
    cv::KalmanFilter KF;         // instantiate Kalman Filter

    int nStates = 18;            // the number of states
    int nMeasurements = 6;       // the number of measured states
    int nInputs = 0;             // the number of action control

    double dt = 0.125;           // time between measurements (1/FPS)

    initKalmanFilter(KF, nStates, nMeasurements, nInputs, dt);    // init function
    @endcode
    The following code corresponds to the *Kalman Filter* initialisation. Firstly, is set the process
    noise, the measurement noise and the error covariance matrix. Secondly, are set the transition
    matrix which is the dynamic model and finally the measurement matrix, which is the measurement
    model.

    You can tune the process and measurement noise to improve the *Kalman Filter* performance. As the
    measurement noise is reduced the faster will converge doing the algorithm sensitive in front of
    bad measurements.
    @code{.cpp}
    void initKalmanFilter(cv::KalmanFilter &KF, int nStates, int nMeasurements, int nInputs, double dt)
    {

      KF.init(nStates, nMeasurements, nInputs, CV_64F);                 // init Kalman Filter

      cv::setIdentity(KF.processNoiseCov, cv::Scalar::all(1e-5));       // set process noise
      cv::setIdentity(KF.measurementNoiseCov, cv::Scalar::all(1e-4));   // set measurement noise
      cv::setIdentity(KF.errorCovPost, cv::Scalar::all(1));             // error covariance


                     /* DYNAMIC MODEL */

      //  [1 0 0 dt  0  0 dt2   0   0 0 0 0  0  0  0   0   0   0]
      //  [0 1 0  0 dt  0   0 dt2   0 0 0 0  0  0  0   0   0   0]
      //  [0 0 1  0  0 dt   0   0 dt2 0 0 0  0  0  0   0   0   0]
      //  [0 0 0  1  0  0  dt   0   0 0 0 0  0  0  0   0   0   0]
      //  [0 0 0  0  1  0   0  dt   0 0 0 0  0  0  0   0   0   0]
      //  [0 0 0  0  0  1   0   0  dt 0 0 0  0  0  0   0   0   0]
      //  [0 0 0  0  0  0   1   0   0 0 0 0  0  0  0   0   0   0]
      //  [0 0 0  0  0  0   0   1   0 0 0 0  0  0  0   0   0   0]
      //  [0 0 0  0  0  0   0   0   1 0 0 0  0  0  0   0   0   0]
      //  [0 0 0  0  0  0   0   0   0 1 0 0 dt  0  0 dt2   0   0]
      //  [0 0 0  0  0  0   0   0   0 0 1 0  0 dt  0   0 dt2   0]
      //  [0 0 0  0  0  0   0   0   0 0 0 1  0  0 dt   0   0 dt2]
      //  [0 0 0  0  0  0   0   0   0 0 0 0  1  0  0  dt   0   0]
      //  [0 0 0  0  0  0   0   0   0 0 0 0  0  1  0   0  dt   0]
      //  [0 0 0  0  0  0   0   0   0 0 0 0  0  0  1   0   0  dt]
      //  [0 0 0  0  0  0   0   0   0 0 0 0  0  0  0   1   0   0]
      //  [0 0 0  0  0  0   0   0   0 0 0 0  0  0  0   0   1   0]
      //  [0 0 0  0  0  0   0   0   0 0 0 0  0  0  0   0   0   1]

      // position
      KF.transitionMatrix.at<double>(0,3) = dt;
      KF.transitionMatrix.at<double>(1,4) = dt;
      KF.transitionMatrix.at<double>(2,5) = dt;
      KF.transitionMatrix.at<double>(3,6) = dt;
      KF.transitionMatrix.at<double>(4,7) = dt;
      KF.transitionMatrix.at<double>(5,8) = dt;
      KF.transitionMatrix.at<double>(0,6) = 0.5*std::pow(dt,2);
      KF.transitionMatrix.at<double>(1,7) = 0.5*std::pow(dt,2);
      KF.transitionMatrix.at<double>(2,8) = 0.5*std::pow(dt,2);

      // orientation
      KF.transitionMatrix.at<double>(9,12) = dt;
      KF.transitionMatrix.at<double>(10,13) = dt;
      KF.transitionMatrix.at<double>(11,14) = dt;
      KF.transitionMatrix.at<double>(12,15) = dt;
      KF.transitionMatrix.at<double>(13,16) = dt;
      KF.transitionMatrix.at<double>(14,17) = dt;
      KF.transitionMatrix.at<double>(9,15) = 0.5*std::pow(dt,2);
      KF.transitionMatrix.at<double>(10,16) = 0.5*std::pow(dt,2);
      KF.transitionMatrix.at<double>(11,17) = 0.5*std::pow(dt,2);


           /* MEASUREMENT MODEL */

      //  [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
      //  [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
      //  [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
      //  [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0]
      //  [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0]
      //  [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0]

      KF.measurementMatrix.at<double>(0,0) = 1;  // x
      KF.measurementMatrix.at<double>(1,1) = 1;  // y
      KF.measurementMatrix.at<double>(2,2) = 1;  // z
      KF.measurementMatrix.at<double>(3,9) = 1;  // roll
      KF.measurementMatrix.at<double>(4,10) = 1; // pitch
      KF.measurementMatrix.at<double>(5,11) = 1; // yaw

    }
    @endcode
    In the following code is the 5th step of the main algorithm. When the obtained number of inliers
    after *Ransac* is over the threshold, the measurements matrix is filled and then the *Kalman
    Filter* is updated:
    @code{.cpp}
    // -- Step 5: Kalman Filter

    // GOOD MEASUREMENT
    if( inliers_idx.rows >= minInliersKalman )
    {

        // Get the measured translation
        cv::Mat translation_measured(3, 1, CV_64F);
        translation_measured = pnp_detection.get_t_matrix();

        // Get the measured rotation
        cv::Mat rotation_measured(3, 3, CV_64F);
        rotation_measured = pnp_detection.get_R_matrix();

        // fill the measurements vector
        fillMeasurements(measurements, translation_measured, rotation_measured);

    }

    // Instantiate estimated translation and rotation
    cv::Mat translation_estimated(3, 1, CV_64F);
    cv::Mat rotation_estimated(3, 3, CV_64F);

    // update the Kalman filter with good measurements
    updateKalmanFilter( KF, measurements,
                  translation_estimated, rotation_estimated);
    @endcode
    The following code corresponds to the *fillMeasurements()* function which converts the measured
    [Rotation Matrix to Eulers
    angles](http://euclideanspace.com/maths/geometry/rotations/conversions/matrixToEuler/index.htm)
    and fill the measurements matrix along with the measured translation vector:
    @code{.cpp}
    void fillMeasurements( cv::Mat &measurements,
                       const cv::Mat &translation_measured, const cv::Mat &rotation_measured)
    {
        // Convert rotation matrix to euler angles
        cv::Mat measured_eulers(3, 1, CV_64F);
        measured_eulers = rot2euler(rotation_measured);

        // Set measurement to predict
        measurements.at<double>(0) = translation_measured.at<double>(0); // x
        measurements.at<double>(1) = translation_measured.at<double>(1); // y
        measurements.at<double>(2) = translation_measured.at<double>(2); // z
        measurements.at<double>(3) = measured_eulers.at<double>(0);      // roll
        measurements.at<double>(4) = measured_eulers.at<double>(1);      // pitch
        measurements.at<double>(5) = measured_eulers.at<double>(2);      // yaw
    }
    @endcode
    The following code corresponds to the *updateKalmanFilter()* function which update the Kalman
    Filter and set the estimated Rotation Matrix and translation vector. The estimated Rotation Matrix
    comes from the estimated [Euler angles to Rotation
    Matrix](http://euclideanspace.com/maths/geometry/rotations/conversions/eulerToMatrix/index.htm).
    @code{.cpp}
    void updateKalmanFilter( cv::KalmanFilter &KF, cv::Mat &measurement,
                         cv::Mat &translation_estimated, cv::Mat &rotation_estimated )
    {

        // First predict, to update the internal statePre variable
        cv::Mat prediction = KF.predict();

        // The "correct" phase that is going to use the predicted value and our measurement
        cv::Mat estimated = KF.correct(measurement);

        // Estimated translation
        translation_estimated.at<double>(0) = estimated.at<double>(0);
        translation_estimated.at<double>(1) = estimated.at<double>(1);
        translation_estimated.at<double>(2) = estimated.at<double>(2);

        // Estimated euler angles
        cv::Mat eulers_estimated(3, 1, CV_64F);
        eulers_estimated.at<double>(0) = estimated.at<double>(9);
        eulers_estimated.at<double>(1) = estimated.at<double>(10);
        eulers_estimated.at<double>(2) = estimated.at<double>(11);

        // Convert estimated quaternion to rotation matrix
        rotation_estimated = euler2rot(eulers_estimated);

    }
    @endcode
    The 6th step is set the estimated rotation-translation matrix:
    @code{.cpp}
    // -- Step 6: Set estimated projection matrix
    pnp_detection_est.set_P_matrix(rotation_estimated, translation_estimated);
    @endcode
    The last and optional step is draw the found pose. To do it I implemented a function to draw all
    the mesh 3D points and an extra reference axis:
    @code{.cpp}
    // -- Step X: Draw pose

    drawObjectMesh(frame_vis, &mesh, &pnp_detection, green);                // draw current pose
    drawObjectMesh(frame_vis, &mesh, &pnp_detection_est, yellow);           // draw estimated pose

    double l = 5;
    std::vector<cv::Point2f> pose_points2d;
    pose_points2d.push_back(pnp_detection_est.backproject3DPoint(cv::Point3f(0,0,0)));    // axis center
    pose_points2d.push_back(pnp_detection_est.backproject3DPoint(cv::Point3f(l,0,0)));    // axis x
    pose_points2d.push_back(pnp_detection_est.backproject3DPoint(cv::Point3f(0,l,0)));    // axis y
    pose_points2d.push_back(pnp_detection_est.backproject3DPoint(cv::Point3f(0,0,l)));    // axis z
    draw3DCoordinateAxes(frame_vis, pose_points2d);                                       // draw axes
    @endcode
    You can also modify the minimum inliers to update Kalman Filter:
    @code{.cpp}
    ./cpp-tutorial-pnp_detection --inliers=20
    @endcode

Results
-------

The following videos are the results of pose estimation in real time using the explained detection
algorithm using the following parameters:
@code{.cpp}
// Robust Matcher parameters

int numKeyPoints = 2000;      // number of detected keypoints
float ratio = 0.70f;          // ratio test
bool fast_match = true;       // fastRobustMatch() or robustMatch()


// RANSAC parameters

int iterationsCount = 500;    // number of Ransac iterations.
int reprojectionError = 2.0;  // maximum allowed distance to consider it an inlier.
float confidence = 0.95;      // ransac successful confidence.


// Kalman Filter parameters

int minInliersKalman = 30;    // Kalman threshold updating
@endcode
You can watch the real time pose estimation on the [YouTube
here](http://www.youtube.com/user/opencvdev/videos).

@youtube{XNATklaJlSQ}
@youtube{YLS9bWek78k}
