@ -60,24 +60,24 @@ or
\ 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_1 \ \
r_ { 21 } & r_ { 22 } & r_ { 23 } & t_2 \ \
r_ { 31 } & r_ { 32 } & r_ { 33 } & t_3
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 \ \
Y \ \
Z \ \
X_w \ \
Y_w \ \
Z_w \ \
1
\ end { bmatrix } \ f ]
where :
- \ f $ ( X , Y , Z ) \ f $ are the coordinates of a 3 D point in the world coordinate space
- \ f $ ( X_w , Y_w , Z_w ) \ f $ are the coordinates of a 3 D point in the world coordinate space
- \ f $ ( u , v ) \ f $ are the coordinates of the projection point in pixels
- \ f $ A \ f $ is a camera matrix , or a matrix of intrinsic parameters
- \ f $ ( cx , cy ) \ f $ is a principal point that is usually at the image center
- \ f $ fx , fy \ f $ are the focal lengths expressed in pixel units .
- \ f $ ( c_ x , c_ y ) \ f $ is a principal point that is usually at the image center
- \ f $ f_ x , f_ y \ f $ are the focal lengths expressed in pixel units .
Thus , if an image from the camera is scaled by a factor , all of these parameters should be scaled
( multiplied / divided , respectively ) by the same factor . The matrix of intrinsic parameters does not
@ -85,15 +85,15 @@ depend on the scene viewed. So, once estimated, it can be re-used as long as the
fixed ( in case of zoom lens ) . The joint rotation - translation matrix \ f $ [ R | t ] \ f $ is called a matrix of
extrinsic parameters . It is used to describe the camera motion around a static scene , or vice versa ,
rigid motion of an object in front of a still camera . That is , \ f $ [ R | t ] \ f $ translates coordinates of a
point \ f $ ( X , Y , Z ) \ f $ to a coordinate system , fixed with respect to the camera . The transformation above
is equivalent to the following ( when \ f $ z \ ne 0 \ f $ ) :
world point \ f $ ( X_w , Y_w , Z_w ) \ f $ to a coordinate system , fixed with respect to the camera .
The transformation above is equivalent to the following ( when \ f $ z \ ne 0 \ f $ ) :
\ f [ \ begin { array } { l }
\ vecthree { x } { y } { z } = R \ vecthree { X } { Y } { Z } + t \ \
x ' = x / z \ \
y ' = y / z \ \
u = f_x * x ' + c_x \ \
v = f_y * y ' + c_y
\ vecthree { X_c } { Y_c } { Z_c } = R \ vecthree { X_w } { Y_w } { Z_w } + t \ \
x ' = X_c / Z_c \ \
y ' = Y_c / Z_c \ \
u = f_x \ times x ' + c_x \ \
v = f_y \ times y ' + c_y
\ end { array } \ f ]
The following figure illustrates the pinhole camera model .
@ -104,14 +104,14 @@ Real lenses usually have some distortion, mostly radial distortion and slight ta
So , the above model is extended as :
\ f [ \ begin { array } { l }
\ vecthree { x } { y } { z } = R \ vecthree { X } { Y } { Z } + t \ \
x ' = x / z \ \
y ' = y / z \ \
\ vecthree { X_c } { Y_c } { Z_c } = R \ vecthree { X_w } { Y_w } { Z_w } + t \ \
x ' = X_c / Z_c \ \
y ' = Y_c / Z_c \ \
x ' ' = 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 ' ' = 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 \ \
\ text { where } \ quad r ^ 2 = x ' ^ 2 + y ' ^ 2 \ \
u = f_x * x ' ' + c_x \ \
v = f_y * y ' ' + c_y
u = f_x \ times x ' ' + c_x \ \
v = f_y \ times y ' ' + c_y
\ end { array } \ f ]
\ 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 $ are radial distortion coefficients . \ f $ p_1 \ f $ and \ f $ p_2 \ f $ are
@ -133,8 +133,8 @@ 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 } \ \
u = f_x * x ' ' ' + c_x \ \
v = f_y * y ' ' ' + c_y
u = f_x \ times x ' ' ' + c_x \ \
v = f_y \ times y ' ' ' + c_y
\ end { array } \ f ]
where the matrix \ f $ R ( \ tau_x , \ tau_y ) \ f $ is defined by two rotations with angular parameter \ f $ \ tau_x \ f $
@ -317,7 +317,7 @@ enum HandEyeCalibrationMethod
@ param jacobian Optional output Jacobian matrix , 3 x9 or 9 x3 , 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 ]
\ 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
@ -325,7 +325,16 @@ Inverse transformation can be also done easily, since
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 3 D geometry
optimization procedures like calibrateCamera , stereoCalibrate , or solvePnP .
optimization procedures like @ ref calibrateCamera , @ ref stereoCalibrate , or @ ref solvePnP .
@ note More information about the computation of the derivative of a 3 D 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 2 D and 3 D 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 ( ) ) ;
@ -634,10 +643,10 @@ Number of input points must be 4. Object points must be defined in the following
- 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
1 xN / Nx1 3 - channel , where N is the number of points . vector \ < Point3f \ > can be also passed here .
1 xN / 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 1 xN / Nx1 2 - channel ,
where N is the number of points . vector \ < Point2f \ > can be also passed here .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { fx } { 0 } { cx } { 0 } { fy } { cy } { 0 } { 0 } { 1 } \ f $ .
where N is the number of points . vector \ < Point2d \ > can be also passed here .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { f_ x } { 0 } { c_ x } { 0 } { f_ y } { c_ y } { 0 } { 0 } { 1 } \ f $ .
@ param distCoeffs Input vector of distortion coefficients
\ 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 $ of
4 , 5 , 8 , 12 or 14 elements . If the vector is NULL / empty , the zero distortion coefficients are
@ -695,7 +704,7 @@ using the perspective projection model \f$ \Pi \f$ and the camera intrinsic para
v \ \
1
\ end { bmatrix } & =
\ bf { A } \ hspace { 0.1 em } \ Pi \ hspace { 0.2 em } ^ { c } \ bf { M } _w
\ bf { A } \ hspace { 0.1 em } \ Pi \ hspace { 0.2 em } ^ { c } \ bf { T } _w
\ begin { bmatrix }
X_ { w } \ \
Y_ { w } \ \
@ -743,7 +752,7 @@ a 3D point expressed in the world frame into the camera frame:
Z_c \ \
1
\ end { bmatrix } & =
\ hspace { 0.2 em } ^ { c } \ bf { M } _w
\ hspace { 0.2 em } ^ { c } \ bf { T } _w
\ begin { bmatrix }
X_ { w } \ \
Y_ { w } \ \
@ -809,9 +818,9 @@ CV_EXPORTS_W bool solvePnP( InputArray objectPoints, InputArray imagePoints,
/** @brief Finds an object pose from 3D-2D point correspondences using the RANSAC scheme.
@ param objectPoints Array of object points in the object coordinate space , Nx3 1 - channel or
1 xN / Nx1 3 - channel , where N is the number of points . vector \ < Point3f \ > can be also passed here .
1 xN / 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 1 xN / Nx1 2 - channel ,
where N is the number of points . vector \ < Point2f \ > can be also passed here .
where N is the number of points . vector \ < Point2d \ > can be also passed here .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { fx } { 0 } { cx } { 0 } { fy } { cy } { 0 } { 0 } { 1 } \ f $ .
@ param distCoeffs Input vector of distortion coefficients
\ 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 $ of
@ -861,7 +870,7 @@ CV_EXPORTS_W bool solvePnPRansac( InputArray objectPoints, InputArray imagePoint
1 x3 / 3 x1 3 - channel . vector \ < Point3f \ > can be also passed here .
@ param imagePoints Array of corresponding image points , 3 x2 1 - channel or 1 x3 / 3 x1 2 - channel .
vector \ < Point2f \ > can be also passed here .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { fx } { 0 } { cx } { 0 } { fy } { cy } { 0 } { 0 } { 1 } \ f $ .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { f_ x } { 0 } { c_ x } { 0 } { f_ y } { c_ y } { 0 } { 0 } { 1 } \ f $ .
@ param distCoeffs Input vector of distortion coefficients
\ 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 $ of
4 , 5 , 8 , 12 or 14 elements . If the vector is NULL / empty , the zero distortion coefficients are
@ -890,10 +899,10 @@ CV_EXPORTS_W int solveP3P( InputArray objectPoints, InputArray imagePoints,
to the camera coordinate frame ) from a 3 D - 2 D point correspondences and starting from an initial solution .
@ param objectPoints Array of object points in the object coordinate space , Nx3 1 - channel or 1 xN / Nx1 3 - channel ,
where N is the number of points . vector \ < Point3f \ > can also be passed here .
where N is the number of points . vector \ < Point3d \ > can also be passed here .
@ param imagePoints Array of corresponding image points , Nx2 1 - channel or 1 xN / Nx1 2 - channel ,
where N is the number of points . vector \ < Point2f \ > can also be passed here .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { fx } { 0 } { cx } { 0 } { fy } { cy } { 0 } { 0 } { 1 } \ f $ .
where N is the number of points . vector \ < Point2d \ > can also be passed here .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { f_ x } { 0 } { c_ x } { 0 } { f_ y } { c_ y } { 0 } { 0 } { 1 } \ f $ .
@ param distCoeffs Input vector of distortion coefficients
\ 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 $ of
4 , 5 , 8 , 12 or 14 elements . If the vector is NULL / empty , the zero distortion coefficients are
@ -918,10 +927,10 @@ CV_EXPORTS_W void solvePnPRefineLM( InputArray objectPoints, InputArray imagePoi
to the camera coordinate frame ) from a 3 D - 2 D point correspondences and starting from an initial solution .
@ param objectPoints Array of object points in the object coordinate space , Nx3 1 - channel or 1 xN / Nx1 3 - channel ,
where N is the number of points . vector \ < Point3f \ > can also be passed here .
where N is the number of points . vector \ < Point3d \ > can also be passed here .
@ param imagePoints Array of corresponding image points , Nx2 1 - channel or 1 xN / Nx1 2 - channel ,
where N is the number of points . vector \ < Point2f \ > can also be passed here .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { fx } { 0 } { cx } { 0 } { fy } { cy } { 0 } { 0 } { 1 } \ f $ .
where N is the number of points . vector \ < Point2d \ > can also be passed here .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { f_ x } { 0 } { c_ x } { 0 } { f_ y } { c_ y } { 0 } { 0 } { 1 } \ f $ .
@ param distCoeffs Input vector of distortion coefficients
\ 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 $ of
4 , 5 , 8 , 12 or 14 elements . If the vector is NULL / empty , the zero distortion coefficients are
@ -960,10 +969,10 @@ Number of input points must be 4 and 2 solutions are returned. Object points mus
Only 1 solution is returned .
@ param objectPoints Array of object points in the object coordinate space , Nx3 1 - channel or
1 xN / Nx1 3 - channel , where N is the number of points . vector \ < Point3f \ > can be also passed here .
1 xN / 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 1 xN / Nx1 2 - channel ,
where N is the number of points . vector \ < Point2f \ > can be also passed here .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { fx } { 0 } { cx } { 0 } { fy } { cy } { 0 } { 0 } { 1 } \ f $ .
where N is the number of points . vector \ < Point2d \ > can be also passed here .
@ param cameraMatrix Input camera matrix \ f $ A = \ vecthreethree { f_ x } { 0 } { c_ x } { 0 } { f_ y } { c_ y } { 0 } { 0 } { 1 } \ f $ .
@ param distCoeffs Input vector of distortion coefficients
\ 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 $ of
4 , 5 , 8 , 12 or 14 elements . If the vector is NULL / empty , the zero distortion coefficients are
@ -1028,7 +1037,7 @@ using the perspective projection model \f$ \Pi \f$ and the camera intrinsic para
v \ \
1
\ end { bmatrix } & =
\ bf { A } \ hspace { 0.1 em } \ Pi \ hspace { 0.2 em } ^ { c } \ bf { M } _w
\ bf { A } \ hspace { 0.1 em } \ Pi \ hspace { 0.2 em } ^ { c } \ bf { T } _w
\ begin { bmatrix }
X_ { w } \ \
Y_ { w } \ \
@ -1076,7 +1085,7 @@ a 3D point expressed in the world frame into the camera frame:
Z_c \ \
1
\ end { bmatrix } & =
\ hspace { 0.2 em } ^ { c } \ bf { M } _w
\ hspace { 0.2 em } ^ { c } \ bf { T } _w
\ begin { bmatrix }
X_ { w } \ \
Y_ { w } \ \
Alexander Alekhin
5 years ago
