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#ifndef OPENCV_CALIB3D_HPP

#define OPENCV_CALIB3D_HPP

#include "opencv2/core.hpp"

#include "opencv2/core/types.hpp"

#include "opencv2/features2d.hpp"

#include "opencv2/core/affine.hpp"

#include "opencv2/core/utils/logger.hpp"

/**

  @defgroup calib3d Camera Calibration and 3D Reconstruction

The functions in this section use a so-called pinhole camera model. The view of a scene

is obtained by projecting a scene's 3D point \f$P_w\f$ into the image plane using a perspective

transformation which forms the corresponding pixel \f$p\f$. Both \f$P_w\f$ and \f$p\f$ are

represented in homogeneous coordinates, i.e. as 3D and 2D homogeneous vector respectively. You will

find a brief introduction to projective geometry, homogeneous vectors and homogeneous

transformations at the end of this section's introduction. For more succinct notation, we often drop

the 'homogeneous' and say vector instead of homogeneous vector.

The distortion-free projective transformation given by a  pinhole camera model is shown below.

\f[s \; p = A \begin{bmatrix} R|t \end{bmatrix} P_w,\f]

where \f$P_w\f$ is a 3D point expressed with respect to the world coordinate system,

\f$p\f$ is a 2D pixel in the image plane, \f$A\f$ is the camera intrinsic matrix,

\f$R\f$ and \f$t\f$ are the rotation and translation that describe the change of coordinates from

world to camera coordinate systems (or camera frame) and \f$s\f$ is the projective transformation's

arbitrary scaling and not part of the camera model.

The camera intrinsic matrix \f$A\f$ (notation used as in @cite Zhang2000 and also generally notated

as \f$K\f$) projects 3D points given in the camera coordinate system to 2D pixel coordinates, i.e.

\f[p = A P_c.\f]

The camera intrinsic matrix \f$A\f$ is composed of the focal lengths \f$f_x\f$ and \f$f_y\f$, which are

expressed in pixel units, and the principal point \f$(c_x, c_y)\f$, that is usually close to the

image center:

\f[A = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1},\f]

and thus

\f[s \vecthree{u}{v}{1} = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1} \vecthree{X_c}{Y_c}{Z_c}.\f]

The matrix of intrinsic parameters does not depend on the scene viewed. So, once estimated, it can

be re-used as long as the focal length is fixed (in case of a zoom lens). Thus, if an image from the

camera is scaled by a factor, all of these parameters need to be scaled (multiplied/divided,

respectively) by the same factor.

The joint rotation-translation matrix \f$[R|t]\f$ is the matrix product of a projective

transformation and a homogeneous transformation. The 3-by-4 projective transformation maps 3D points

represented in camera coordinates to 2D points in the image plane and represented in normalized

camera coordinates \f$x' = X_c / Z_c\f$ and \f$y' = Y_c / Z_c\f$:

\f[Z_c \begin{bmatrix}

x' \\

y' \\

1

\end{bmatrix} = \begin{bmatrix}

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0

\end{bmatrix}

\begin{bmatrix}

X_c \\

Y_c \\

Z_c \\

1

\end{bmatrix}.\f]

The homogeneous transformation is encoded by the extrinsic parameters \f$R\f$ and \f$t\f$ and

represents the change of basis from world coordinate system \f$w\f$ to the camera coordinate sytem

\f$c\f$. Thus, given the representation of the point \f$P\f$ in world coordinates, \f$P_w\f$, we

obtain \f$P\f$'s representation in the camera coordinate system, \f$P_c\f$, by

\f[P_c = \begin{bmatrix}

R & t \\

0 & 1

\end{bmatrix} P_w,\f]

This homogeneous transformation is composed out of \f$R\f$, a 3-by-3 rotation matrix, and \f$t\f$, a

3-by-1 translation vector:

\f[\begin{bmatrix}

R & t \\

0 & 1

\end{bmatrix} = \begin{bmatrix}

r_{11} & r_{12} & r_{13} & t_x \\

r_{21} & r_{22} & r_{23} & t_y \\

r_{31} & r_{32} & r_{33} & t_z \\

0 & 0 & 0 & 1

\end{bmatrix},

\f]

and therefore

\f[\begin{bmatrix}

X_c \\

Y_c \\

Z_c \\

1

\end{bmatrix} = \begin{bmatrix}

r_{11} & r_{12} & r_{13} & t_x \\

r_{21} & r_{22} & r_{23} & t_y \\

r_{31} & r_{32} & r_{33} & t_z \\

0 & 0 & 0 & 1

\end{bmatrix}

\begin{bmatrix}

X_w \\

Y_w \\

Z_w \\

1

\end{bmatrix}.\f]

Combining the projective transformation and the homogeneous transformation, we obtain the projective

transformation that maps 3D points in world coordinates into 2D points in the image plane and in

normalized camera coordinates:

\f[Z_c \begin{bmatrix}

x' \\

y' \\

1

\end{bmatrix} = \begin{bmatrix} R|t \end{bmatrix} \begin{bmatrix}

X_w \\

Y_w \\

Z_w \\

1

\end{bmatrix} = \begin{bmatrix}

r_{11} & r_{12} & r_{13} & t_x \\

r_{21} & r_{22} & r_{23} & t_y \\

r_{31} & r_{32} & r_{33} & t_z

\end{bmatrix}

\begin{bmatrix}

X_w \\

Y_w \\

Z_w \\

1

\end{bmatrix},\f]

with \f$x' = X_c / Z_c\f$ and \f$y' = Y_c / Z_c\f$. Putting the equations for instrincs and extrinsics together, we can write out

\f$s \; p = A \begin{bmatrix} R|t \end{bmatrix} P_w\f$ as

\f[s \vecthree{u}{v}{1} = \vecthreethree{f_x}{0}{c_x}{0}{f_y}{c_y}{0}{0}{1}

\begin{bmatrix}

r_{11} & r_{12} & r_{13} & t_x \\

r_{21} & r_{22} & r_{23} & t_y \\

r_{31} & r_{32} & r_{33} & t_z

\end{bmatrix}

\begin{bmatrix}

X_w \\

Y_w \\

Z_w \\

1

\end{bmatrix}.\f]

If \f$Z_c \ne 0\f$, the transformation above is equivalent to the following,

\f[\begin{bmatrix}

u \\

v

\end{bmatrix} = \begin{bmatrix}

f_x X_c/Z_c + c_x \\

f_y Y_c/Z_c + c_y

\end{bmatrix}\f]

with

\f[\vecthree{X_c}{Y_c}{Z_c} = \begin{bmatrix}

R|t

\end{bmatrix} \begin{bmatrix}

X_w \\

Y_w \\

Z_w \\

1

\end{bmatrix}.\f]

The following figure illustrates the pinhole camera model.

![Pinhole camera model](pics/pinhole_camera_model.png) { width=70% }

Real lenses usually have some distortion, mostly radial distortion, and slight tangential distortion.

So, the above model is extended as:

\f[\begin{bmatrix}

u \\

v

\end{bmatrix} = \begin{bmatrix}

f_x x'' + c_x \\

f_y y'' + c_y

\end{bmatrix}\f]

where

\f[\begin{bmatrix}

x'' \\

y''

\end{bmatrix} = \begin{bmatrix}

x' \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} + 2 p_1 x' y' + p_2(r^2 + 2 x'^2) + s_1 r^2 + s_2 r^4 \\

y' \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} + p_1 (r^2 + 2 y'^2) + 2 p_2 x' y' + s_3 r^2 + s_4 r^4 \\

\end{bmatrix}\f]

with

\f[r^2 = x'^2 + y'^2\f]

and

\f[\begin{bmatrix}

x'\\

y'

\end{bmatrix} = \begin{bmatrix}

X_c/Z_c \\

Y_c/Z_c

\end{bmatrix},\f]

if \f$Z_c \ne 0\f$.

The distortion parameters are the radial coefficients \f$k_1\f$, \f$k_2\f$, \f$k_3\f$, \f$k_4\f$, \f$k_5\f$, and \f$k_6\f$

,\f$p_1\f$ and \f$p_2\f$ are the tangential distortion coefficients, and \f$s_1\f$, \f$s_2\f$, \f$s_3\f$, and \f$s_4\f$,

are the thin prism distortion coefficients. Higher-order coefficients are not considered in OpenCV.

The next figures show two common types of radial distortion: barrel distortion

(\f$ 1 + k_1 r^2 + k_2 r^4 + k_3 r^6 \f$ monotonically decreasing)

and pincushion distortion (\f$ 1 + k_1 r^2 + k_2 r^4 + k_3 r^6 \f$ monotonically increasing).

Radial distortion is always monotonic for real lenses,

and if the estimator produces a non-monotonic result,

this should be considered a calibration failure.

More generally, radial distortion must be monotonic and the distortion function must be bijective.

A failed estimation result may look deceptively good near the image center

but will work poorly in e.g. AR/SFM applications.

The optimization method used in OpenCV camera calibration does not include these constraints as

the framework does not support the required integer programming and polynomial inequalities.

See [issue #15992](https://github.com/opencv/opencv/issues/15992) for additional information.

![](pics/distortion_examples.png)

![](pics/distortion_examples2.png)

In some cases, the image sensor may be tilted in order to focus an oblique plane in front of the

camera (Scheimpflug principle). This can be useful for particle image velocimetry (PIV) or

triangulation with a laser fan. The tilt causes a perspective distortion of \f$x''\f$ and

\f$y''\f$. This distortion can be modeled in the following way, see e.g. @cite Louhichi07.

\f[\begin{bmatrix}

u \\

v

\end{bmatrix} = \begin{bmatrix}

f_x x''' + c_x \\

f_y y''' + c_y

\end{bmatrix},\f]

where

\f[s\vecthree{x'''}{y'''}{1} =

\vecthreethree{R_{33}(\tau_x, \tau_y)}{0}{-R_{13}(\tau_x, \tau_y)}

{0}{R_{33}(\tau_x, \tau_y)}{-R_{23}(\tau_x, \tau_y)}

{0}{0}{1} R(\tau_x, \tau_y) \vecthree{x''}{y''}{1}\f]

and the matrix \f$R(\tau_x, \tau_y)\f$ is defined by two rotations with angular parameter

\f$\tau_x\f$ and \f$\tau_y\f$, respectively,

\f[

R(\tau_x, \tau_y) =

\vecthreethree{\cos(\tau_y)}{0}{-\sin(\tau_y)}{0}{1}{0}{\sin(\tau_y)}{0}{\cos(\tau_y)}

\vecthreethree{1}{0}{0}{0}{\cos(\tau_x)}{\sin(\tau_x)}{0}{-\sin(\tau_x)}{\cos(\tau_x)} =

\vecthreethree{\cos(\tau_y)}{\sin(\tau_y)\sin(\tau_x)}{-\sin(\tau_y)\cos(\tau_x)}

{0}{\cos(\tau_x)}{\sin(\tau_x)}

{\sin(\tau_y)}{-\cos(\tau_y)\sin(\tau_x)}{\cos(\tau_y)\cos(\tau_x)}.

\f]

In the functions below the coefficients are passed or returned as

\f[(k_1, k_2, p_1, p_2[, k_3[, k_4, k_5, k_6 [, s_1, s_2, s_3, s_4[, \tau_x, \tau_y]]]])\f]

vector. That is, if the vector contains four elements, it means that \f$k_3=0\f$ . The distortion

coefficients do not depend on the scene viewed. Thus, they also belong to the intrinsic camera

parameters. And they remain the same regardless of the captured image resolution. If, for example, a

camera has been calibrated on images of 320 x 240 resolution, absolutely the same distortion

coefficients can be used for 640 x 480 images from the same camera while \f$f_x\f$, \f$f_y\f$,

\f$c_x\f$, and \f$c_y\f$ need to be scaled appropriately.

The functions below use the above model to do the following:

-   Project 3D points to the image plane given intrinsic and extrinsic parameters.

-   Compute extrinsic parameters given intrinsic parameters, a few 3D points, and their

projections.

-   Estimate intrinsic and extrinsic camera parameters from several views of a known calibration

pattern (every view is described by several 3D-2D point correspondences).

-   Estimate the relative position and orientation of the stereo camera "heads" and compute the

*rectification* transformation that makes the camera optical axes parallel.

<B> Homogeneous Coordinates </B><br>

Homogeneous Coordinates are a system of coordinates that are used in projective geometry. Their use

allows to represent points at infinity by finite coordinates and simplifies formulas when compared

to the cartesian counterparts, e.g. they have the advantage that affine transformations can be

expressed as linear homogeneous transformation.

One obtains the homogeneous vector \f$P_h\f$ by appending a 1 along an n-dimensional cartesian

vector \f$P\f$ e.g. for a 3D cartesian vector the mapping \f$P \rightarrow P_h\f$ is:

\f[\begin{bmatrix}

X \\

Y \\

Z

\end{bmatrix} \rightarrow \begin{bmatrix}

X \\

Y \\

Z \\

1

\end{bmatrix}.\f]

For the inverse mapping \f$P_h \rightarrow P\f$, one divides all elements of the homogeneous vector

by its last element, e.g. for a 3D homogeneous vector one gets its 2D cartesian counterpart by:

\f[\begin{bmatrix}

X \\

Y \\

W

\end{bmatrix} \rightarrow \begin{bmatrix}

X / W \\

Y / W

\end{bmatrix},\f]

if \f$W \ne 0\f$.

Due to this mapping, all multiples \f$k P_h\f$, for \f$k \ne 0\f$, of a homogeneous point represent

the same point \f$P_h\f$. An intuitive understanding of this property is that under a projective

transformation, all multiples of \f$P_h\f$ are mapped to the same point. This is the physical

observation one does for pinhole cameras, as all points along a ray through the camera's pinhole are

projected to the same image point, e.g. all points along the red ray in the image of the pinhole

camera model above would be mapped to the same image coordinate. This property is also the source

for the scale ambiguity s in the equation of the pinhole camera model.

As mentioned, by using homogeneous coordinates we can express any change of basis parameterized by

\f$R\f$ and \f$t\f$ as a linear transformation, e.g. for the change of basis from coordinate system

0 to coordinate system 1 becomes:

\f[P_1 = R P_0 + t \rightarrow P_{h_1} = \begin{bmatrix}

R & t \\

0 & 1

\end{bmatrix} P_{h_0}.\f]

<B> Homogeneous Transformations, Object frame / Camera frame </B><br>

Change of basis or computing the 3D coordinates from one frame to another frame can be achieved easily using

the following notation:

\f[

\mathbf{X}_c = \hspace{0.2em}

{}^{c}\mathbf{T}_o \hspace{0.2em} \mathbf{X}_o

\f]

\f[

\begin{bmatrix}

X_c \\

Y_c \\

Z_c \\

1

\end{bmatrix} =

\begin{bmatrix}

{}^{c}\mathbf{R}_o & {}^{c}\mathbf{t}_o \\

0_{1 \times 3} & 1

\end{bmatrix}

\begin{bmatrix}

X_o \\

Y_o \\

Z_o \\

1

\end{bmatrix}

\f]

For a 3D points (\f$ \mathbf{X}_o \f$) expressed in the object frame, the homogeneous transformation matrix

\f$ {}^{c}\mathbf{T}_o \f$ allows computing the corresponding coordinate (\f$ \mathbf{X}_c \f$) in the camera frame.

This transformation matrix is composed of a 3x3 rotation matrix \f$ {}^{c}\mathbf{R}_o \f$ and a 3x1 translation vector

\f$ {}^{c}\mathbf{t}_o \f$.

The 3x1 translation vector \f$ {}^{c}\mathbf{t}_o \f$ is the position of the object frame in the camera frame and the

3x3 rotation matrix \f$ {}^{c}\mathbf{R}_o \f$ the orientation of the object frame in the camera frame.

With this simple notation, it is easy to chain the transformations. For instance, to compute the 3D coordinates of a point

expressed in the object frame in the world frame can be done with:

\f[

\mathbf{X}_w = \hspace{0.2em}

{}^{w}\mathbf{T}_c \hspace{0.2em} {}^{c}\mathbf{T}_o \hspace{0.2em}

\mathbf{X}_o =

{}^{w}\mathbf{T}_o \hspace{0.2em} \mathbf{X}_o

\f]

Similarly, computing the inverse transformation can be done with:

\f[

\mathbf{X}_o = \hspace{0.2em}

{}^{o}\mathbf{T}_c \hspace{0.2em} \mathbf{X}_c =

\left( {}^{c}\mathbf{T}_o \right)^{-1} \hspace{0.2em} \mathbf{X}_c

\f]

The inverse of an homogeneous transformation matrix is then:

\f[

{}^{o}\mathbf{T}_c = \left( {}^{c}\mathbf{T}_o \right)^{-1} =

\begin{bmatrix}

{}^{c}\mathbf{R}^{\top}_o & - \hspace{0.2em} {}^{c}\mathbf{R}^{\top}_o \hspace{0.2em} {}^{c}\mathbf{t}_o \\

0_{1 \times 3} & 1

\end{bmatrix}

\f]

One can note that the inverse of a 3x3 rotation matrix is directly its matrix transpose.

![Perspective projection, from object to camera frame](pics/pinhole_homogeneous_transformation.jpg) { width=70% }

This figure summarizes the whole process. The object pose returned for instance by the @ref solvePnP function

or pose from fiducial marker detection is this \f$ {}^{c}\mathbf{T}_o \f$ transformation.

The camera intrinsic matrix \f$ \mathbf{K} \f$ allows projecting the 3D point expressed in the camera frame onto the image plane

assuming a perspective projection model (pinhole camera model). Image coordinates extracted from classical image processing functions

assume a (u,v) top-left coordinates frame.

\note

- for an online video course on this topic, see for instance:

  - ["3.3.1. Homogeneous Transformation Matrices", Modern Robotics, Kevin M. Lynch and Frank C. Park](https://modernrobotics.northwestern.edu/nu-gm-book-resource/3-3-1-homogeneous-transformation-matrices/)

- the 3x3 rotation matrix is composed of 9 values but describes a 3 dof transformation

- some additional properties of the 3x3 rotation matrix are:

  - \f$ \mathrm{det} \left( \mathbf{R} \right) = 1 \f$

  - \f$ \mathbf{R} \mathbf{R}^{\top} = \mathbf{R}^{\top} \mathbf{R} = \mathrm{I}_{3 \times 3} \f$

  - interpolating rotation can be done using the [Slerp (spherical linear interpolation)](https://en.wikipedia.org/wiki/Slerp) method

- quick conversions between the different rotation formalisms can be done using this [online tool](https://www.andre-gaschler.com/rotationconverter/)

<B> Intrinsic parameters from camera lens specifications </B><br>

When dealing with industrial cameras, the camera intrinsic matrix or more precisely \f$ \left(f_x, f_y \right) \f$

can be deduced, approximated from the camera specifications:

\f[

f_x = \frac{f_{\text{mm}}}{\text{pixel_size_in_mm}} = \frac{f_{\text{mm}}}{\text{sensor_size_in_mm} / \text{nb_pixels}}

\f]

In a same way, the physical focal length can be deduced from the angular field of view:

\f[

f_{\text{mm}} = \frac{\text{sensor_size_in_mm}}{2 \times \tan{\frac{\text{fov}}{2}}}

\f]

This latter conversion can be useful when using a rendering software to mimic a physical camera device.

@note

    -    See also #calibrationMatrixValues

<B> Additional references, notes </B><br>

@note

    -   Many functions in this module take a camera intrinsic matrix as an input parameter. Although all

        functions assume the same structure of this parameter, they may name it differently. The

        parameter's description, however, will be clear in that a camera intrinsic matrix with the structure

        shown above is required.

    -   A calibration sample for 3 cameras in a horizontal position can be found at

        opencv_source_code/samples/cpp/3calibration.cpp

    -   A calibration sample based on a sequence of images can be found at

        opencv_source_code/samples/cpp/calibration.cpp

    -   A calibration sample in order to do 3D reconstruction can be found at

        opencv_source_code/samples/cpp/build3dmodel.cpp

    -   A calibration example on stereo calibration can be found at

        opencv_source_code/samples/cpp/stereo_calib.cpp

    -   A calibration example on stereo matching can be found at

        opencv_source_code/samples/cpp/stereo_match.cpp

    -   (Python) A camera calibration sample can be found at

        opencv_source_code/samples/python/calibrate.py

  @{

    @defgroup calib3d_fisheye Fisheye camera model

    Definitions: Let P be a point in 3D of coordinates X in the world reference frame (stored in the

    matrix X) The coordinate vector of P in the camera reference frame is:

    \f[Xc = R X + T\f]

    where R is the rotation matrix corresponding to the rotation vector om: R = rodrigues(om); call x, y

    and z the 3 coordinates of Xc:

    \f[\begin{array}{l} x = Xc_1 \\ y = Xc_2 \\ z = Xc_3 \end{array} \f]

    The pinhole projection coordinates of P is [a; b] where

    \f[\begin{array}{l} a = x / z \ and \ b = y / z \\ r^2 = a^2 + b^2 \\ \theta = atan(r) \end{array} \f]

    Fisheye distortion:

    \f[\theta_d = \theta (1 + k_1 \theta^2 + k_2 \theta^4 + k_3 \theta^6 + k_4 \theta^8)\f]

    The distorted point coordinates are [x'; y'] where

    \f[\begin{array}{l} x' = (\theta_d / r) a \\ y' = (\theta_d / r) b \end{array} \f]

    Finally, conversion into pixel coordinates: The final pixel coordinates vector [u; v] where:

    \f[\begin{array}{l} u = f_x (x' + \alpha y') + c_x \\

    v = f_y y' + c_y \end{array} \f]

    Summary:

    Generic camera model @cite Kannala2006 with perspective projection and without distortion correction

  @}

 */

namespace cv

{

//! @addtogroup calib3d

//! @{

//! type of the robust estimation algorithm

enum { LMEDS  = 4,  //!< least-median of squares algorithm

       RANSAC = 8,  //!< RANSAC algorithm

       RHO    = 16, //!< RHO algorithm

       USAC_DEFAULT  = 32, //!< USAC algorithm, default settings

       USAC_PARALLEL = 33, //!< USAC, parallel version

       USAC_FM_8PTS = 34,  //!< USAC, fundamental matrix 8 points

       USAC_FAST = 35,     //!< USAC, fast settings

       USAC_ACCURATE = 36, //!< USAC, accurate settings

       USAC_PROSAC = 37,   //!< USAC, sorted points, runs PROSAC

       USAC_MAGSAC = 38    //!< USAC, runs MAGSAC++

     };

enum SolvePnPMethod {

    SOLVEPNP_ITERATIVE   = 0, //!< Pose refinement using non-linear Levenberg-Marquardt minimization scheme @cite Madsen04 @cite Eade13 \n

                              //!< Initial solution for non-planar "objectPoints" needs at least 6 points and uses the DLT algorithm. \n

                              //!< Initial solution for planar "objectPoints" needs at least 4 points and uses pose from homography decomposition.

    SOLVEPNP_EPNP        = 1, //!< EPnP: Efficient Perspective-n-Point Camera Pose Estimation @cite lepetit2009epnp

    SOLVEPNP_P3P         = 2, //!< Revisiting the P3P Problem @cite ding2023revisiting

    SOLVEPNP_DLS         = 3, //!< **Broken implementation. Using this flag will fallback to EPnP.** \n

                              //!< A Direct Least-Squares (DLS) Method for PnP @cite hesch2011direct

    SOLVEPNP_UPNP        = 4, //!< **Broken implementation. Using this flag will fallback to EPnP.** \n

                              //!< Exhaustive Linearization for Robust Camera Pose and Focal Length Estimation @cite penate2013exhaustive

    SOLVEPNP_AP3P        = 5, //!< An Efficient Algebraic Solution to the Perspective-Three-Point Problem @cite Ke17

    SOLVEPNP_IPPE        = 6, //!< Infinitesimal Plane-Based Pose Estimation @cite Collins14 \n

                              //!< Object points must be coplanar.

    SOLVEPNP_IPPE_SQUARE = 7, //!< Infinitesimal Plane-Based Pose Estimation @cite Collins14 \n

                              //!< This is a special case suitable for marker pose estimation.\n

                              //!< 4 coplanar object points must be defined in the following order:

                              //!<   - point 0: [-squareLength / 2,  squareLength / 2, 0]

                              //!<   - point 1: [ squareLength / 2,  squareLength / 2, 0]

                              //!<   - point 2: [ squareLength / 2, -squareLength / 2, 0]

                              //!<   - point 3: [-squareLength / 2, -squareLength / 2, 0]

    SOLVEPNP_SQPNP       = 8, //!< SQPnP: A Consistently Fast and Globally OptimalSolution to the Perspective-n-Point Problem @cite Terzakis2020SQPnP

#ifndef CV_DOXYGEN

    SOLVEPNP_MAX_COUNT        //!< Used for count

#endif

};

enum { CALIB_CB_ADAPTIVE_THRESH = 1,

       CALIB_CB_NORMALIZE_IMAGE = 2,

       CALIB_CB_FILTER_QUADS    = 4,

       CALIB_CB_FAST_CHECK      = 8,

       CALIB_CB_EXHAUSTIVE      = 16,

       CALIB_CB_ACCURACY        = 32,

       CALIB_CB_LARGER          = 64,

       CALIB_CB_MARKER          = 128,

       CALIB_CB_PLAIN           = 256

     };

enum { CALIB_CB_SYMMETRIC_GRID  = 1,

       CALIB_CB_ASYMMETRIC_GRID = 2,

       CALIB_CB_CLUSTERING      = 4

     };

enum { CALIB_NINTRINSIC          = 18,

       CALIB_USE_INTRINSIC_GUESS = 0x00001,

       CALIB_FIX_ASPECT_RATIO    = 0x00002,

       CALIB_FIX_PRINCIPAL_POINT = 0x00004,

       CALIB_ZERO_TANGENT_DIST   = 0x00008,

       CALIB_FIX_FOCAL_LENGTH    = 0x00010,

       CALIB_FIX_K1              = 0x00020,

       CALIB_FIX_K2              = 0x00040,

       CALIB_FIX_K3              = 0x00080,

       CALIB_FIX_K4              = 0x00800,

       CALIB_FIX_K5              = 0x01000,

       CALIB_FIX_K6              = 0x02000,

       CALIB_RATIONAL_MODEL      = 0x04000,

       CALIB_THIN_PRISM_MODEL    = 0x08000,

       CALIB_FIX_S1_S2_S3_S4     = 0x10000,

       CALIB_TILTED_MODEL        = 0x40000,

       CALIB_FIX_TAUX_TAUY       = 0x80000,

       CALIB_USE_QR              = 0x100000, //!< use QR instead of SVD decomposition for solving. Faster but potentially less precise

       CALIB_FIX_TANGENT_DIST    = 0x200000,

       // only for stereo

       CALIB_FIX_INTRINSIC       = 0x00100,

       CALIB_SAME_FOCAL_LENGTH   = 0x00200,

       // for stereo rectification

       CALIB_ZERO_DISPARITY      = 0x00400,

       CALIB_USE_LU              = (1 << 17), //!< use LU instead of SVD decomposition for solving. much faster but potentially less precise

       CALIB_USE_EXTRINSIC_GUESS = (1 << 22), //!< for stereoCalibrate

       CALIB_DISABLE_SCHUR_COMPLEMENT = (1 << 23)  //!< disable Schur complement (use Bouguet calibration engine)

     };

//! the algorithm for finding fundamental matrix

enum { FM_7POINT = 1, //!< 7-point algorithm

       FM_8POINT = 2, //!< 8-point algorithm

       FM_LMEDS  = 4, //!< least-median algorithm. 7-point algorithm is used.

       FM_RANSAC = 8  //!< RANSAC algorithm. It needs at least 15 points. 7-point algorithm is used.

     };

enum HandEyeCalibrationMethod

{

    CALIB_HAND_EYE_TSAI         = 0, //!< A New Technique for Fully Autonomous and Efficient 3D Robotics Hand/Eye Calibration @cite Tsai89

    CALIB_HAND_EYE_PARK         = 1, //!< Robot Sensor Calibration: Solving AX = XB on the Euclidean Group @cite Park94

    CALIB_HAND_EYE_HORAUD       = 2, //!< Hand-eye Calibration @cite Horaud95

    CALIB_HAND_EYE_ANDREFF      = 3, //!< On-line Hand-Eye Calibration @cite Andreff99

    CALIB_HAND_EYE_DANIILIDIS   = 4  //!< Hand-Eye Calibration Using Dual Quaternions @cite Daniilidis98

};

enum RobotWorldHandEyeCalibrationMethod

{

    CALIB_ROBOT_WORLD_HAND_EYE_SHAH = 0, //!< Solving the robot-world/hand-eye calibration problem using the kronecker product @cite Shah2013SolvingTR

    CALIB_ROBOT_WORLD_HAND_EYE_LI   = 1  //!< Simultaneous robot-world and hand-eye calibration using dual-quaternions and kronecker product @cite Li2010SimultaneousRA

};

enum SamplingMethod { SAMPLING_UNIFORM=0, SAMPLING_PROGRESSIVE_NAPSAC=1, SAMPLING_NAPSAC=2,

        SAMPLING_PROSAC=3 };

enum LocalOptimMethod {LOCAL_OPTIM_NULL=0, LOCAL_OPTIM_INNER_LO=1, LOCAL_OPTIM_INNER_AND_ITER_LO=2,

        LOCAL_OPTIM_GC=3, LOCAL_OPTIM_SIGMA=4};

enum ScoreMethod {SCORE_METHOD_RANSAC=0, SCORE_METHOD_MSAC=1, SCORE_METHOD_MAGSAC=2, SCORE_METHOD_LMEDS=3};

enum NeighborSearchMethod { NEIGH_FLANN_KNN=0, NEIGH_GRID=1, NEIGH_FLANN_RADIUS=2 };

enum PolishingMethod { NONE_POLISHER=0, LSQ_POLISHER=1, MAGSAC=2, COV_POLISHER=3 };

struct CV_EXPORTS_W_SIMPLE UsacParams

{ // in alphabetical order

    CV_WRAP UsacParams();

    CV_PROP_RW double confidence;

    CV_PROP_RW bool isParallel;

    CV_PROP_RW int loIterations;

    CV_PROP_RW LocalOptimMethod loMethod;

    CV_PROP_RW int loSampleSize;

    CV_PROP_RW int maxIterations;

    CV_PROP_RW NeighborSearchMethod neighborsSearch;

    CV_PROP_RW int randomGeneratorState;

    CV_PROP_RW SamplingMethod sampler;

    CV_PROP_RW ScoreMethod score;

    CV_PROP_RW double threshold;

    CV_PROP_RW PolishingMethod final_polisher;

    CV_PROP_RW int final_polisher_iterations;

};

/** @brief Converts a rotation matrix to a rotation vector or vice versa.

@param src Input rotation vector (3x1 or 1x3) or rotation matrix (3x3).

@param dst Output rotation matrix (3x3) or rotation vector (3x1 or 1x3), respectively.

@param jacobian Optional output Jacobian matrix, 3x9 or 9x3, which is a matrix of partial

derivatives of the output array components with respect to the input array components.

\f[\begin{array}{l} \theta \leftarrow norm(r) \\ r  \leftarrow r/ \theta \\ R =  \cos(\theta) I + (1- \cos{\theta} ) r r^T +  \sin(\theta) \vecthreethree{0}{-r_z}{r_y}{r_z}{0}{-r_x}{-r_y}{r_x}{0} \end{array}\f]

Inverse transformation can be also done easily, since

\f[\sin ( \theta ) \vecthreethree{0}{-r_z}{r_y}{r_z}{0}{-r_x}{-r_y}{r_x}{0} = \frac{R - R^T}{2}\f]

A rotation vector is a convenient and most compact representation of a rotation matrix (since any

rotation matrix has just 3 degrees of freedom). The representation is used in the global 3D geometry

optimization procedures like @ref calibrateCamera, @ref stereoCalibrate, or @ref solvePnP .

@note More information about the computation of the derivative of a 3D rotation matrix with respect to its exponential coordinate

can be found in:

    - A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates, Guillermo Gallego, Anthony J. Yezzi @cite Gallego2014ACF

@note Useful information on SE(3) and Lie Groups can be found in:

    - A tutorial on SE(3) transformation parameterizations and on-manifold optimization, Jose-Luis Blanco @cite blanco2010tutorial

    - Lie Groups for 2D and 3D Transformation, Ethan Eade @cite Eade17

    - A micro Lie theory for state estimation in robotics, Joan Solà, Jérémie Deray, Dinesh Atchuthan @cite Sol2018AML

 */

CV_EXPORTS_W void Rodrigues( InputArray src, OutputArray dst, OutputArray jacobian = noArray() );

/** Levenberg-Marquardt solver. Starting with the specified vector of parameters it

    optimizes the target vector criteria "err"

    (finds local minima of each target vector component absolute value).

    When needed, it calls user-provided callback.

*/

class CV_EXPORTS LMSolver : public Algorithm

{

public:

    class CV_EXPORTS Callback

    {

    public:

        virtual ~Callback() {}

        /**

         computes error and Jacobian for the specified vector of parameters

         @param param the current vector of parameters

         @param err output vector of errors: err_i = actual_f_i - ideal_f_i

         @param J output Jacobian: J_ij = d(ideal_f_i)/d(param_j)

         when J=noArray(), it means that it does not need to be computed.

         Dimensionality of error vector and param vector can be different.

         The callback should explicitly allocate (with "create" method) each output array

         (unless it's noArray()).

        */

        virtual bool compute(InputArray param, OutputArray err, OutputArray J) const = 0;

    };

    /**

       Runs Levenberg-Marquardt algorithm using the passed vector of parameters as the start point.

       The final vector of parameters (whether the algorithm converged or not) is stored at the same

       vector. The method returns the number of iterations used. If it's equal to the previously specified

       maxIters, there is a big chance the algorithm did not converge.

       @param param initial/final vector of parameters.

       Note that the dimensionality of parameter space is defined by the size of param vector,

       and the dimensionality of optimized criteria is defined by the size of err vector

       computed by the callback.

    */

    virtual int run(InputOutputArray param) const = 0;

    /**

       Sets the maximum number of iterations

       @param maxIters the number of iterations

    */

    virtual void setMaxIters(int maxIters) = 0;

    /**

       Retrieves the current maximum number of iterations

    */

    virtual int getMaxIters() const = 0;

    /**

       Creates Levenberg-Marquard solver

       @param cb callback

       @param maxIters maximum number of iterations that can be further

         modified using setMaxIters() method.

    */

    static Ptr<LMSolver> create(const Ptr<LMSolver::Callback>& cb, int maxIters);

    static Ptr<LMSolver> create(const Ptr<LMSolver::Callback>& cb, int maxIters, double eps);

};

/** @example samples/cpp/tutorial_code/features2D/Homography/pose_from_homography.cpp

An example program about pose estimation from coplanar points

Check @ref tutorial_homography "the corresponding tutorial" for more details

*/

/** @brief Finds a perspective transformation between two planes.

@param srcPoints Coordinates of the points in the original plane, a matrix of the type CV_32FC2

or vector\<Point2f\> .

@param dstPoints Coordinates of the points in the target plane, a matrix of the type CV_32FC2 or

a vector\<Point2f\> .

@param method Method used to compute a homography matrix. The following methods are possible:

-   **0** - a regular method using all the points, i.e., the least squares method

-   @ref RANSAC - RANSAC-based robust method

-   @ref LMEDS - Least-Median robust method

-   @ref RHO - PROSAC-based robust method

@param ransacReprojThreshold Maximum allowed reprojection error to treat a point pair as an inlier

(used in the RANSAC and RHO methods only). That is, if

\f[\| \texttt{dstPoints} _i -  \texttt{convertPointsHomogeneous} ( \texttt{H} \cdot \texttt{srcPoints} _i) \|_2  >  \texttt{ransacReprojThreshold}\f]

then the point \f$i\f$ is considered as an outlier. If srcPoints and dstPoints are measured in pixels,

it usually makes sense to set this parameter somewhere in the range of 1 to 10.

@param mask Optional output mask set by a robust method ( RANSAC or LMeDS ). Note that the input

mask values are ignored.

@param maxIters The maximum number of RANSAC iterations.

@param confidence Confidence level, between 0 and 1.

The function finds and returns the perspective transformation \f$H\f$ between the source and the

destination planes:

\f[s_i  \vecthree{x'_i}{y'_i}{1} \sim H  \vecthree{x_i}{y_i}{1}\f]

so that the back-projection error

\f[\sum _i \left ( x'_i- \frac{h_{11} x_i + h_{12} y_i + h_{13}}{h_{31} x_i + h_{32} y_i + h_{33}} \right )^2+ \left ( y'_i- \frac{h_{21} x_i + h_{22} y_i + h_{23}}{h_{31} x_i + h_{32} y_i + h_{33}} \right )^2\f]

is minimized. If the parameter method is set to the default value 0, the function uses all the point

pairs to compute an initial homography estimate with a simple least-squares scheme.

However, if not all of the point pairs ( \f$srcPoints_i\f$, \f$dstPoints_i\f$ ) fit the rigid perspective

transformation (that is, there are some outliers), this initial estimate will be poor. In this case,

you can use one of the three robust methods. The methods RANSAC, LMeDS and RHO try many different

random subsets of the corresponding point pairs (of four pairs each, collinear pairs are discarded), estimate the homography matrix

using this subset and a simple least-squares algorithm, and then compute the quality/goodness of the

computed homography (which is the number of inliers for RANSAC or the least median re-projection error for

LMeDS). The best subset is then used to produce the initial estimate of the homography matrix and

the mask of inliers/outliers.

Regardless of the method, robust or not, the computed homography matrix is refined further (using

inliers only in case of a robust method) with the Levenberg-Marquardt method to reduce the

re-projection error even more.

The methods RANSAC and RHO can handle practically any ratio of outliers but need a threshold to

distinguish inliers from outliers. The method LMeDS does not need any threshold but it works

correctly only when there are more than 50% of inliers. Finally, if there are no outliers and the

noise is rather small, use the default method (method=0).

The function is used to find initial intrinsic and extrinsic matrices. Homography matrix is

determined up to a scale. If \f$h_{33}\f$ is non-zero, the matrix is normalized so that \f$h_{33}=1\f$.

@note Whenever an \f$H\f$ matrix cannot be estimated, an empty one will be returned.

@sa

getAffineTransform, estimateAffine2D, estimateAffinePartial2D, getPerspectiveTransform, warpPerspective,

perspectiveTransform

 */

CV_EXPORTS_W Mat findHomography( InputArray srcPoints, InputArray dstPoints,

                                 int method = 0, double ransacReprojThreshold = 3,

                                 OutputArray mask=noArray(), const int maxIters = 2000,

                                 const double confidence = 0.995);

/** @overload */

CV_EXPORTS Mat findHomography( InputArray srcPoints, InputArray dstPoints,

                               OutputArray mask, int method = 0, double ransacReprojThreshold = 3 );

CV_EXPORTS_W Mat findHomography(InputArray srcPoints, InputArray dstPoints, OutputArray mask,

                   const UsacParams &params);

/** @brief Computes an RQ decomposition of 3x3 matrices.

@param src 3x3 input matrix.

@param mtxR Output 3x3 upper-triangular matrix.

@param mtxQ Output 3x3 orthogonal matrix.

@param Qx Optional output 3x3 rotation matrix around x-axis.

@param Qy Optional output 3x3 rotation matrix around y-axis.

@param Qz Optional output 3x3 rotation matrix around z-axis.

The function computes a RQ decomposition using the given rotations. This function is used in

#decomposeProjectionMatrix to decompose the left 3x3 submatrix of a projection matrix into a camera

and a rotation matrix.

It optionally returns three rotation matrices, one for each axis, and the three Euler angles in

degrees (as the return value) that could be used in OpenGL. Note, there is always more than one

sequence of rotations about the three principal axes that results in the same orientation of an

object, e.g. see @cite Slabaugh . Returned three rotation matrices and corresponding three Euler angles

are only one of the possible solutions.

 */

CV_EXPORTS_W Vec3d RQDecomp3x3( InputArray src, OutputArray mtxR, OutputArray mtxQ,

                                OutputArray Qx = noArray(),

                                OutputArray Qy = noArray(),

                                OutputArray Qz = noArray());

/** @brief Decomposes a projection matrix into a rotation matrix and a camera intrinsic matrix.

@param projMatrix 3x4 input projection matrix P.

@param cameraMatrix Output 3x3 camera intrinsic matrix \f$\cameramatrix{A}\f$.

@param rotMatrix Output 3x3 external rotation matrix R.

@param transVect Output 4x1 vector representing the camera position in homogeneous coordinates.

To obtain the translation vector, use t = -rotMatrix * transVect[:3].

@param rotMatrixX Optional 3x3 rotation matrix around x-axis.

@param rotMatrixY Optional 3x3 rotation matrix around y-axis.

@param rotMatrixZ Optional 3x3 rotation matrix around z-axis.

@param eulerAngles Optional three-element vector containing three Euler angles of rotation in

degrees.

The function computes a decomposition of a projection matrix into a calibration and a rotation

matrix and the position of a camera.

It optionally returns three rotation matrices, one for each axis, and three Euler angles that could

be used in OpenGL. Note, there is always more than one sequence of rotations about the three

principal axes that results in the same orientation of an object, e.g. see @cite Slabaugh . Returned

three rotation matrices and corresponding three Euler angles are only one of the possible solutions.

The function is based on #RQDecomp3x3 .

 */

CV_EXPORTS_W void decomposeProjectionMatrix( InputArray projMatrix, OutputArray cameraMatrix,

                                             OutputArray rotMatrix, OutputArray transVect,

                                             OutputArray rotMatrixX = noArray(),

                                             OutputArray rotMatrixY = noArray(),

                                             OutputArray rotMatrixZ = noArray(),

                                             OutputArray eulerAngles =noArray() );

/** @brief Computes partial derivatives of the matrix product for each multiplied matrix.

@param A First multiplied matrix.

@param B Second multiplied matrix.

@param dABdA First output derivative matrix d(A\*B)/dA of size

\f$\texttt{A.rows*B.cols} \times {A.rows*A.cols}\f$ .

@param dABdB Second output derivative matrix d(A\*B)/dB of size

\f$\texttt{A.rows*B.cols} \times {B.rows*B.cols}\f$ .

The function computes partial derivatives of the elements of the matrix product \f$A*B\f$ with regard to

the elements of each of the two input matrices. The function is used to compute the Jacobian

matrices in #stereoCalibrate but can also be used in any other similar optimization function.

 */

CV_EXPORTS_W void matMulDeriv( InputArray A, InputArray B, OutputArray dABdA, OutputArray dABdB );

/** @brief Combines two rotation-and-shift transformations.

@param rvec1 First rotation vector.

@param tvec1 First translation vector.

@param rvec2 Second rotation vector.

@param tvec2 Second translation vector.

@param rvec3 Output rotation vector of the superposition.

@param tvec3 Output translation vector of the superposition.

@param dr3dr1 Optional output derivative of rvec3 with regard to rvec1

@param dr3dt1 Optional output derivative of rvec3 with regard to tvec1

@param dr3dr2 Optional output derivative of rvec3 with regard to rvec2

@param dr3dt2 Optional output derivative of rvec3 with regard to tvec2

@param dt3dr1 Optional output derivative of tvec3 with regard to rvec1

@param dt3dt1 Optional output derivative of tvec3 with regard to tvec1

@param dt3dr2 Optional output derivative of tvec3 with regard to rvec2

@param dt3dt2 Optional output derivative of tvec3 with regard to tvec2

The functions compute:

\f[\begin{array}{l} \texttt{rvec3} =  \mathrm{rodrigues} ^{-1} \left ( \mathrm{rodrigues} ( \texttt{rvec2} )  \cdot \mathrm{rodrigues} ( \texttt{rvec1} ) \right )  \\ \texttt{tvec3} =  \mathrm{rodrigues} ( \texttt{rvec2} )  \cdot \texttt{tvec1} +  \texttt{tvec2} \end{array} ,\f]

where \f$\mathrm{rodrigues}\f$ denotes a rotation vector to a rotation matrix transformation, and

\f$\mathrm{rodrigues}^{-1}\f$ denotes the inverse transformation. See #Rodrigues for details.

Also, the functions can compute the derivatives of the output vectors with regards to the input

vectors (see #matMulDeriv ). The functions are used inside #stereoCalibrate but can also be used in

your own code where Levenberg-Marquardt or another gradient-based solver is used to optimize a

function that contains a matrix multiplication.

 */

CV_EXPORTS_W void composeRT( InputArray rvec1, InputArray tvec1,

                             InputArray rvec2, InputArray tvec2,

                             OutputArray rvec3, OutputArray tvec3,

                             OutputArray dr3dr1 = noArray(), OutputArray dr3dt1 = noArray(),

                             OutputArray dr3dr2 = noArray(), OutputArray dr3dt2 = noArray(),

                             OutputArray dt3dr1 = noArray(), OutputArray dt3dt1 = noArray(),

                             OutputArray dt3dr2 = noArray(), OutputArray dt3dt2 = noArray() );

/** @brief Projects 3D points to an image plane.

The function computes the 2D projections of 3D points to the image plane, given intrinsic and

extrinsic camera parameters. Optionally, the function computes Jacobians -matrices of partial

derivatives of image points coordinates (as functions of all the input parameters) with respect to

the particular parameters, intrinsic and/or extrinsic. The Jacobians are used during the global

optimization in @ref calibrateCamera, @ref solvePnP, and @ref stereoCalibrate. The function itself

can also be used to compute a re-projection error, given the current intrinsic and extrinsic

parameters.

@note **Coordinate Systems:**

- **Input (`objectPoints`)**: 3D points in the **world coordinate frame**.

- **Output (`imagePoints`)**: 2D projections in **pixel coordinates** of the image plane, with distortion applied.

  The coordinates \f$(u, v)\f$ are measured in pixels from the top-left corner of the image.

The transformation chain is: World coordinates → Camera coordinates (via rvec/tvec) → Normalized camera coordinates

→ Distortion applied → Pixel coordinates (via cameraMatrix).

@param objectPoints Array of object points expressed wrt. the world coordinate frame. A 3xN/Nx3

1-channel or 1xN/Nx1 3-channel (or vector\<Point3f\> ), where N is the number of points in the view.

@param rvec The rotation vector (@ref Rodrigues) that, together with tvec, performs a change of

basis from world to camera coordinate system, see @ref calibrateCamera for details.

@param tvec The translation vector, see parameter description above.

@param cameraMatrix Camera intrinsic matrix \f$\cameramatrix{A}\f$ .

@param distCoeffs Input vector of distortion coefficients

\f$\distcoeffs\f$ . If the vector is empty, the zero distortion coefficients are assumed.

@param imagePoints Output array of image points in **pixel coordinates**, 1xN/Nx1 2-channel, or

vector\<Point2f\> .

@param jacobian Optional output 2Nx(10+\<numDistCoeffs\>) jacobian matrix of derivatives of image

points with respect to components of the rotation vector, translation vector, focal lengths,

coordinates of the principal point and the distortion coefficients. In the old interface different

components of the jacobian are returned via different output parameters.

@param aspectRatio Optional "fixed aspect ratio" parameter. If the parameter is not 0, the

function assumes that the aspect ratio (\f$f_x / f_y\f$) is fixed and correspondingly adjusts the

jacobian matrix.

@note By setting rvec = tvec = \f$[0, 0, 0]\f$, or by setting cameraMatrix to a 3x3 identity matrix,

or by passing zero distortion coefficients, one can get various useful partial cases of the

function. This means, one can compute the distorted coordinates for a sparse set of points or apply

a perspective transformation (and also compute the derivatives) in the ideal zero-distortion setup.

 */

CV_EXPORTS_W void projectPoints( InputArray objectPoints,

                                 InputArray rvec, InputArray tvec,

                                 InputArray cameraMatrix, InputArray distCoeffs,

                                 OutputArray imagePoints,

                                 OutputArray jacobian = noArray(),

                                 double aspectRatio = 0 );

/** @example samples/cpp/tutorial_code/features2D/Homography/homography_from_camera_displacement.cpp

An example program about homography from the camera displacement

Check @ref tutorial_homography "the corresponding tutorial" for more details

*/

/** @brief Finds an object pose \f$ {}^{c}\mathbf{T}_o \f$ from 3D-2D point correspondences:

![Perspective projection, from object to camera frame](pics/pinhole_homogeneous_transformation.jpg){ width=50% }

@see @ref calib3d_solvePnP

This function returns the rotation and the translation vectors that transform a 3D point expressed in the object

coordinate frame to the camera coordinate frame, using different methods:

- P3P methods (@ref SOLVEPNP_P3P, @ref SOLVEPNP_AP3P): need 4 input points to return a unique solution.

- @ref SOLVEPNP_IPPE Input points must be >= 4 and object points must be coplanar.

- @ref SOLVEPNP_IPPE_SQUARE Special case suitable for marker pose estimation.

Number of input points must be 4. Object points must be defined in the following order:

  - point 0: [-squareLength / 2,  squareLength / 2, 0]

  - point 1: [ squareLength / 2,  squareLength / 2, 0]

  - point 2: [ squareLength / 2, -squareLength / 2, 0]

  - point 3: [-squareLength / 2, -squareLength / 2, 0]

- for all the other flags, number of input points must be >= 4 and object points can be in any configuration.

@param objectPoints Array of object points in the object coordinate space, Nx3 1-channel or

1xN/Nx1 3-channel, where N is the number of points. vector\<Point3d\> can be also passed here.

@param imagePoints Array of corresponding image points, Nx2 1-channel or 1xN/Nx1 2-channel,

where N is the number of points. vector\<Point2d\> can be also passed here.

@param cameraMatrix Input camera intrinsic matrix \f$\cameramatrix{A}\f$ .

@param distCoeffs Input vector of distortion coefficients

\f$\distcoeffs\f$. If the vector is NULL/empty, the zero distortion coefficients are

assumed.

@param rvec Output rotation vector (see @ref Rodrigues ) that, together with tvec, brings points from

the model coordinate system to the camera coordinate system.

@param tvec Output translation vector.

@param useExtrinsicGuess Parameter used for #SOLVEPNP_ITERATIVE. If true (1), the function uses

the provided rvec and tvec values as initial approximations of the rotation and translation

vectors, respectively, and further optimizes them.

@param flags Method for solving a PnP problem: see @ref calib3d_solvePnP_flags

More information about Perspective-n-Points is described in @ref calib3d_solvePnP

@note

   -   An example of how to use solvePnP for planar augmented reality can be found at

        opencv_source_code/samples/python/plane_ar.py

   -   If you are using Python:

        - Numpy array slices won't work as input because solvePnP requires contiguous

        arrays (enforced by the assertion using cv::Mat::checkVector() around line 55 of

        modules/calib3d/src/solvepnp.cpp version 2.4.9)

        - The P3P algorithm requires image points to be in an array of shape (N,1,2) due

        to its calling of #undistortPoints (around line 75 of modules/calib3d/src/solvepnp.cpp version 2.4.9)

        which requires 2-channel information.

        - Thus, given some data D = np.array(...) where D.shape = (N,M), in order to use a subset of

        it as, e.g., imagePoints, one must effectively copy it into a new array: imagePoints =

        np.ascontiguousarray(D[:,:2]).reshape((N,1,2))

   -   The methods @ref SOLVEPNP_DLS and @ref SOLVEPNP_UPNP cannot be used as the current implementations are

       unstable and sometimes give completely wrong results. If you pass one of these two

       flags, @ref SOLVEPNP_EPNP method will be used instead.

   -   The minimum number of points is 4 in the general case. In the case of @ref SOLVEPNP_P3P and @ref SOLVEPNP_AP3P

       methods, it is required to use exactly 4 points (the first 3 points are used to estimate all the solutions

       of the P3P problem, the last one is used to retain the best solution that minimizes the reprojection error).

   -   With @ref SOLVEPNP_ITERATIVE method and `useExtrinsicGuess=true`, the minimum number of points is 3 (3 points

       are sufficient to compute a pose but there are up to 4 solutions). The initial solution should be close to the

       global solution to converge. The function returns true if some solution is found. User code is responsible for

       solution quality assessment.

   -   With @ref SOLVEPNP_IPPE input points must be >= 4 and object points must be coplanar.

   -   With @ref SOLVEPNP_IPPE_SQUARE this is a special case suitable for marker pose estimation.

       Number of input points must be 4. Object points must be defined in the following order:

         - point 0: [-squareLength / 2,  squareLength / 2, 0]

         - point 1: [ squareLength / 2,  squareLength / 2, 0]

         - point 2: [ squareLength / 2, -squareLength / 2, 0]

         - point 3: [-squareLength / 2, -squareLength / 2, 0]

   -   With @ref SOLVEPNP_SQPNP input points must be >= 3

 */

CV_EXPORTS_W bool solvePnP( InputArray objectPoints, InputArray imagePoints,

                            InputArray cameraMatrix, InputArray distCoeffs,

                            OutputArray rvec, OutputArray tvec,

                            bool useExtrinsicGuess = false, int flags = SOLVEPNP_ITERATIVE );

/** @brief Finds an object pose \f$ {}^{c}\mathbf{T}_o \f$ from 3D-2D point correspondences using the RANSAC scheme to deal with bad matches.

![Perspective projection, from object to camera frame](pics/pinhole_homogeneous_transformation.jpg){ width=50% }

@see @ref calib3d_solvePnP

@param objectPoints Array of object points in the object coordinate space, Nx3 1-channel or

1xN/Nx1 3-channel, where N is the number of points. vector\<Point3d\> can be also passed here.

@param imagePoints Array of corresponding image points, Nx2 1-channel or 1xN/Nx1 2-channel,

where N is the number of points. vector\<Point2d\> can be also passed here.

@param cameraMatrix Input camera intrinsic matrix \f$\cameramatrix{A}\f$ .

@param distCoeffs Input vector of distortion coefficients

\f$\distcoeffs\f$. If the vector is NULL/empty, the zero distortion coefficients are

assumed.

@param rvec Output rotation vector (see @ref Rodrigues ) that, together with tvec, brings points from

the model coordinate system to the camera coordinate system.

@param tvec Output translation vector.

@param useExtrinsicGuess Parameter used for @ref SOLVEPNP_ITERATIVE. If true (1), the function uses

the provided rvec and tvec values as initial approximations of the rotation and translation

vectors, respectively, and further optimizes them.

@param iterationsCount Number of iterations.

@param reprojectionError Inlier threshold value used by the RANSAC procedure. The parameter value

is the maximum allowed distance between the observed and computed point projections to consider it

an inlier.

@param confidence The probability that the algorithm produces a useful result.

@param inliers Output vector that contains indices of inliers in objectPoints and imagePoints .

@param flags Method for solving a PnP problem (see @ref solvePnP ).

The function estimates an object pose given a set of object points, their corresponding image

projections, as well as the camera intrinsic matrix and the distortion coefficients. This function finds such

a pose that minimizes reprojection error, that is, the sum of squared distances between the observed

projections imagePoints and the projected (using @ref projectPoints ) objectPoints. The use of RANSAC

makes the function resistant to outliers.

@note

   -   An example of how to use solvePnPRansac for object detection can be found at

        @ref tutorial_real_time_pose

   -   The default method used to estimate the camera pose for the Minimal Sample Sets step

       is #SOLVEPNP_EPNP. Exceptions are:

         - if you choose #SOLVEPNP_P3P or #SOLVEPNP_AP3P, these methods will be used.

         - if the number of input points is equal to 4, #SOLVEPNP_P3P is used.

   -   The method used to estimate the camera pose using all the inliers is defined by the

       flags parameters unless it is equal to #SOLVEPNP_P3P or #SOLVEPNP_AP3P. In this case,

       the method #SOLVEPNP_EPNP will be used instead.

 */

CV_EXPORTS_W bool solvePnPRansac( InputArray objectPoints, InputArray imagePoints,

                                  InputArray cameraMatrix, InputArray distCoeffs,

                                  OutputArray rvec, OutputArray tvec,

                                  bool useExtrinsicGuess = false, int iterationsCount = 100,

                                  float reprojectionError = 8.0, double confidence = 0.99,

                                  OutputArray inliers = noArray(), int flags = SOLVEPNP_ITERATIVE );