vector_to_rel_pose (Operator)

Name

`vector_to_rel_pose` — Compute the relative orientation between two cameras given image point correspondences and known camera parameters and reconstruct 3D space points.

Signature

`vector_to_rel_pose( : : Rows1, Cols1, Rows2, Cols2, CovRR1, CovRC1, CovCC1, CovRR2, CovRC2, CovCC2, CamPar1, CamPar2, Method : RelPose, CovRelPose, Error, X, Y, Z, CovXYZ)`

Description

For a stereo configuration with known camera parameters the geometric relation between the two images is defined by the relative pose. The operator `vector_to_rel_pose` computes the relative pose from in general at least six point correspondences in the image pair. `RelPose` indicates the relative pose of camera 1 with respect to camera 2 (see `create_pose` for more information about poses and their representations.). This is in accordance with the explicit calibration of a stereo setup using the operator `calibrate_cameras`. Now, let R,t be the rotation and translation of the relative pose. Then, the essential matrix E is defined as , where denotes the 3x3 skew-symmetric matrix realising the cross product with the vector t. The pose can be determined from the epipolar constraint:

Note, that the essential matrix is a projective entity and thus is defined up to a scaling factor. From this follows that the translation vector of the relative pose can only be determined up to scale too. In fact, the computed translation vector will always be normalized to unit length. As a consequence, a threedimensional reconstruction of the scene, here in terms of points given by their coordinates (`X`,`Y`,`Z`), can be carried out only up to a single global scaling factor. If the absolute 3D coordinates of the reconstruction are to be achieved the unknown scaling factor can be computed from a gauge, which has to be visible in both images. For example, a simple gauge can be given by any known distance between points in the scene.

The operator `vector_to_rel_pose` is designed to deal with a camera model that includes lens distortions. This is in constrast to the operator `vector_to_essential_matrix`, which encompasses only straight line preserving cameras. The camera parameters are passed by the arguments `CamPar1`, `CamPar2`. The 3D direction vectors and are calculated from the point coordinates (`Rows1`,`Cols1`) and (`Rows2`,`Cols2`) by inverting the process of projection (see Calibration / Multi-View). The point correspondences are typically determined by applying the operator `match_rel_pose_ransac`.

The parameter `Method` decides whether the relative orientation between the cameras is of a special type and which algorithm is to be applied for its computation. If `Method` is either 'normalized_dlt' or 'gold_standard' the relative orientation is arbitrary. Choosing 'trans_normalized_dlt' or 'trans_gold_standard' means that the relative motion between the cameras is a pure translation. The typical application for this special motion case is the scenario of a single fixed camera looking onto a moving conveyor belt. In this case the minimum required number of corresponding points is just two instead of six in the general case.

The relative pose is computed by a linear algorithm if 'normalized_dlt' or 'trans_normalized_dlt' is chosen. With 'gold_standard' or 'trans_gold_standard' the algorithm gives a statistically optimal result. Here, 'normalized_dlt' and 'gold_standard' stand for direct-linear-transformation and gold-standard-algorithm respectively. All methods return the coordinates (`X`,`Y`,`Z`) of the reconstructed 3D points. The optimal methods also return the covariances of the 3D points in `CovXYZ`. Let n be the number of points then the 3x3 covariance matrices are concatenated and stored in a tuple of length 9n. Additionally, the optimal methods return the 6x6 covariance matrix of the pose `CovRelPose`.

If an optimal gold-standard-algorithm is chosen the covariances of the image points (`CovRR1`, `CovRC1`, `CovCC1`, `CovRR2`, `CovRC2`, `CovCC2`) can be incorporated in the computation. They can be provided for example by the operator `points_foerstner`. If the point covariances are unknown, which is the default, empty tuples are input. In this case the optimization algorithm internally assumes uniform and equal covariances for all points.

The value `Error` indicates the overall quality of the optimization process and is the root-mean-square euclidian distance in pixels between the points and their corresponding epipolar lines.

For the operator `vector_to_rel_pose` a special configuration of scene points and cameras exists: if all 3D points lie in a single plane and additionally are all closer to one of the two cameras then the solution in the relative pose is not unique but twofold. As a consequence both solutions are computed and returned by the operator. This means that all output parameters are of double length and the values of the second solution are simply concatenated behind the values of the first one.

Execution Information

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Processed without parallelization.

Parameters

`Rows1` (input_control)  number-array `→` (real / integer)

Input points in image 1 (row coordinate).

Restriction: `length(Rows1) >= 6 || length(Rows1) >= 2`

`Cols1` (input_control)  number-array `→` (real / integer)

Input points in image 1 (column coordinate).

Restriction: `length(Cols1) == length(Rows1)`

`Rows2` (input_control)  number-array `→` (real / integer)

Input points in image 2 (row coordinate).

Restriction: `length(Rows2) == length(Rows1)`

`Cols2` (input_control)  number-array `→` (real / integer)

Input points in image 2 (column coordinate).

Restriction: `length(Cols2) == length(Rows1)`

`CovRR1` (input_control)  number-array `→` (real / integer)

Row coordinate variance of the points in image 1.

Default value: []

`CovRC1` (input_control)  number-array `→` (real / integer)

Covariance of the points in image 1.

Default value: []

`CovCC1` (input_control)  number-array `→` (real / integer)

Column coordinate variance of the points in image 1.

Default value: []

`CovRR2` (input_control)  number-array `→` (real / integer)

Row coordinate variance of the points in image 2.

Default value: []

`CovRC2` (input_control)  number-array `→` (real / integer)

Covariance of the points in image 2.

Default value: []

`CovCC2` (input_control)  number-array `→` (real / integer)

Column coordinate variance of the points in image 2.

Default value: []

`CamPar1` (input_control)  campar `→` (real / integer / string)

Camera parameters of the 1st camera.

`CamPar2` (input_control)  campar `→` (real / integer / string)

Camera parameters of the 2nd camera.

`Method` (input_control)  string `→` (string)

Algorithm for the computation of the relative pose and for special pose types.

Default value: 'normalized_dlt'

List of values: 'gold_standard', 'normalized_dlt', 'trans_gold_standard', 'trans_normalized_dlt'

`RelPose` (output_control)  pose `→` (real / integer)

Computed relative orientation of the cameras (3D pose).

`CovRelPose` (output_control)  real-array `→` (real)

6x6 covariance matrix of the relative camera orientation.

`Error` (output_control)  real(-array) `→` (real)

Root-Mean-Square of the epipolar distance error.

`X` (output_control)  real-array `→` (real)

X coordinates of the reconstructed 3D points.

`Y` (output_control)  real-array `→` (real)

Y coordinates of the reconstructed 3D points.

`Z` (output_control)  real-array `→` (real)

Z coordinates of the reconstructed 3D points.

`CovXYZ` (output_control)  real-array `→` (real)

Covariance matrices of the reconstructed 3D points.

Possible Predecessors

`match_rel_pose_ransac`

Possible Successors

`gen_binocular_rectification_map`, `rel_pose_to_fundamental_matrix`

Alternatives

`vector_to_essential_matrix`, `vector_to_fundamental_matrix`, `binocular_calibration`

`camera_calibration`