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answered 2020-10-08 10:44:33 -0600

RashidLadj gravatar image 
Rotation_matrix_T = cv.Rodrigues(rvec)[0]
Rotation_matrix = Rotation_matrix_T.T
Translate_vector = - np.dot(Rotation_matrix , tvec)

Transformation_matrix = np.eye(4)
Transformation_matrix[:3, :3] = Rotation_matrix
Transformation_matrix[:3,  3] = Translate_vector

## method 1 ##
proj_matrix = np.hstack((Rotation_matrix_T, - np.dot(Rotation_matrix_T, Translate_vector)))

## method 2 ##
proj_matrix = np.zeros(3, 4)
Transformation_matrix_inv = np.linalg.inv(Transformation_matrix)
proj_matrix = Transformation_matrix_inv[:3, :4]

# # # # # # # # # # # # # # # # # #
# #     R | s*t                 # # 
# # T = -------                 # # 
# #     0 0 0 1                 # #     
# #                             # #
# #           R.T | -R.T*s*t    # # 
# # inv(T) =  --------------    # # 
# #            0  0  0  1       # # 
# #                             # #
# # P = R.T| -R.T*s*t           # # 
# # P = inv(T)[:3, :4]          # # 
# # # # # # # # # # # # # # # # # #

This projection Matrix don't contain Camera Matrix

proj_matrix_Final_Camera1 = np.dot(cameraMatrix, np.eye(4)[:3, :4])
proj_matrix_Final_Camera2 = np.dot(cameraMatrix, proj_matrix)

Rotation_matrix_T = cv.Rodrigues(rvec)[0]
 Rotation_matrix = Rotation_matrix_T.T
 Translate_vector = - np.dot(Rotation_matrix , tvec)
tvec)
 
Transformation_matrix = np.eye(4)
Transformation_matrix[:3, :3] = Rotation_matrix
Transformation_matrix[:3, 3] = Translate_vector
## method 1 ##
proj_matrix = np.hstack((Rotation_matrix_T, - np.dot(Rotation_matrix_T, Translate_vector)))
## method 2 ##
proj_matrix = np.zeros(3, 4)
Transformation_matrix_inv = np.linalg.inv(Transformation_matrix)
proj_matrix = Transformation_matrix_inv[:3, :4]
# # # # # # # # # # # # # # # # # #
# # R | s*t # #
# # T = ------- # #
# # 0 0 0 1 # #
# # # #
# # R.T | -R.T*s*t # #
# # inv(T) = -------------- # #
# # 0 0 0 1 # #
# # # #
# # P = R.T| -R.T*s*t # #
# # P = inv(T)[:3, :4] # #
# # # # # # # # # # # # # # # # # #
# s = Scale You don't need it now #
# # # # # # # # # # # # # # # # # #

 
This projection Matrix don't contain Camera Matrix
 
proj_matrix_Final_Camera1 = np.dot(cameraMatrix, np.eye(4)[:3, :4])
proj_matrix_Final_Camera2 = np.dot(cameraMatrix, proj_matrix)
click to hide/show revision 3
No.3 Revision

updated 2020-10-08 20:57:57 -0600

supra56 gravatar image

Code:

Rotation_matrix_T = cv.Rodrigues(rvec)[0]
 Rotation_matrix = Rotation_matrix_T.T
 Translate_vector = - np.dot(Rotation_matrix , tvec)
 
tvec)

Transformation_matrix = np.eye(4)
Transformation_matrix[:3, :3] = Rotation_matrix
Transformation_matrix[:3,  3] = Translate_vector

## method 1 ##
proj_matrix = np.hstack((Rotation_matrix_T, - np.dot(Rotation_matrix_T, Translate_vector)))

## method 2 ##
proj_matrix = np.zeros(3, 4)
Transformation_matrix_inv = np.linalg.inv(Transformation_matrix)
proj_matrix = Transformation_matrix_inv[:3, :4]

# # # # # # # # # # # # # # # # # #
# #     R | s*t                 # # 
# # T = -------                 # # 
# #     0 0 0 1                 # #     
# #                             # #
# #           R.T | -R.T*s*t    # # 
# # inv(T) =  --------------    # # 
# #            0  0  0  1       # # 
# #                             # #
# # P = R.T| -R.T*s*t           # # 
# # P = inv(T)[:3, :4]          # # 
# # # # # # # # # # # # # # # # # #
# s = Scale You don't need it now #
# # # # # # # # # # # # # # # # # #

 
This projection Matrix don't contain Camera Matrix
 
proj_matrix_Final_Camera1 = np.dot(cameraMatrix, np.eye(4)[:3, :4])
proj_matrix_Final_Camera2 = np.dot(cameraMatrix, proj_matrix)

Code:

Rotation_matrix_T = cv.Rodrigues(rvec)[0]
Rotation_matrix = Rotation_matrix_T.T
Translate_vector = - np.dot(Rotation_matrix , tvec)

Transformation_matrix = np.eye(4)
Transformation_matrix[:3, :3] = Rotation_matrix
Transformation_matrix[:3,  3] = Translate_vector

## method 1 ##
proj_matrix = np.hstack((Rotation_matrix_T, - np.dot(Rotation_matrix_T, Translate_vector)))

## method 2 ##
proj_matrix = np.zeros(3, 4)
Transformation_matrix_inv = np.linalg.inv(Transformation_matrix)
proj_matrix = Transformation_matrix_inv[:3, :4]

# # # # # # # # # # # # # # # # # #
# #     R | s*t                 # # 
# # T = -------                 # # 
# #     0 0 0 1                 # #     
# #                             # #
# #           R.T | -R.T*s*t    # # 
# # inv(T) =  --------------    # # 
# #            0  0  0  1       # # 
# #                             # #
# # P = R.T| -R.T*s*t           # # 
# # P = inv(T)[:3, :4]          # # 
# # # # # # # # # # # # # # # # # #
# s = Scale You don't need it now #
# # # # # # # # # # # # # # # # # #

This projection Matrix don't contain Camera Matrix

proj_matrix_Final_Camera1 = np.dot(cameraMatrix, np.eye(4)[:3, :4])
proj_matrix_Final_Camera2 = np.dot(cameraMatrix, proj_matrix)

To answer your question: The rvec and tvec returned by solvepnp don't include the values of the camera matrix. The rotation and translation are recovered in this way

Rotation = cv.rodriges (rvec) [0] .T
Traslation = - np.dot (Rotation.T, tvec)