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Perspective-n-Point (PnP) pose computation

Pose computation overview

The pose computation problem [183] consists in solving for the rotation and translation that minimizes the reprojection error from 3D-2D point correspondences.

The solvePnP and related functions estimate the object pose given a set of object points, their corresponding image projections, as well as the camera intrinsic matrix and the distortion coefficients, see the figure below (more precisely, the X-axis of the camera frame is pointing to the right, the Y-axis downward and the Z-axis forward).

Points expressed in the world frame Xw are projected into the image plane [u,v] using the perspective projection model Π and the camera intrinsic parameters matrix A (also denoted K in the literature):

[uv1]=AΠcTw[XwYwZw1][uv1]=[fx0cx0fycy001][100001000010][r11r12r13txr21r22r23tyr31r32r33tz0001][XwYwZw1]

The estimated pose is thus the rotation (rvec) and the translation (tvec) vectors that allow transforming a 3D point expressed in the world frame into the camera frame:

[XcYcZc1]=cTw[XwYwZw1][XcYcZc1]=[r11r12r13txr21r22r23tyr31r32r33tz0001][XwYwZw1]

Pose computation methods

Refer to the cv::SolvePnPMethod enum documentation for the list of possible values. Some details about each method are described below:

  • cv::SOLVEPNP_ITERATIVE Iterative method is based on a Levenberg-Marquardt optimization. In this case the 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 cv::projectPoints ) "objectPoints". Initial solution for non-planar "objectPoints" needs at least 6 points and uses the DLT algorithm. Initial solution for planar "objectPoints" needs at least 4 points and uses pose from homography decomposition.
  • cv::SOLVEPNP_P3P Method is based on the paper of X.S. Gao, X.-R. Hou, J. Tang, H.-F. Chang "Complete Solution Classification for the Perspective-Three-Point Problem" ([97]). In this case the function requires exactly four object and image points.
  • cv::SOLVEPNP_AP3P Method is based on the paper of T. Ke, S. Roumeliotis "An Efficient Algebraic Solution to the Perspective-Three-Point Problem" ([145]). In this case the function requires exactly four object and image points.
  • cv::SOLVEPNP_EPNP Method has been introduced by F. Moreno-Noguer, V. Lepetit and P. Fua in the paper "EPnP: Efficient Perspective-n-Point Camera Pose Estimation" ([158]).
  • cv::SOLVEPNP_DLS Broken implementation. Using this flag will fallback to EPnP.
    Method is based on the paper of J. Hesch and S. Roumeliotis. "A Direct Least-Squares (DLS) Method for PnP" ([125]).
  • cv::SOLVEPNP_UPNP Broken implementation. Using this flag will fallback to EPnP.
    Method is based on the paper of A. Penate-Sanchez, J. Andrade-Cetto, F. Moreno-Noguer. "Exhaustive Linearization for Robust Camera Pose and Focal Length Estimation" ([215]). In this case the function also estimates the parameters fx and fy assuming that both have the same value. Then the cameraMatrix is updated with the estimated focal length.
  • cv::SOLVEPNP_IPPE Method is based on the paper of T. Collins and A. Bartoli. "Infinitesimal Plane-Based Pose Estimation" ([61]). This method requires coplanar object points.
  • cv::SOLVEPNP_IPPE_SQUARE Method is based on the paper of Toby Collins and Adrien Bartoli. "Infinitesimal Plane-Based Pose Estimation" ([61]). This method is suitable for marker pose estimation. It requires 4 coplanar object points 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]
  • cv::SOLVEPNP_SQPNP Method is based on the paper "A Consistently Fast and Globally Optimal Solution to the Perspective-n-Point Problem" by G. Terzakis and M.Lourakis ([269]). It requires 3 or more points.

P3P

The cv::solveP3P() computes an object pose from exactly 3 3D-2D point correspondences. A P3P problem has up to 4 solutions.

Note
The solutions are sorted by reprojection errors (lowest to highest).

PnP

The cv::solvePnP() 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 (cv::SOLVEPNP_P3P, cv::SOLVEPNP_AP3P): need 4 input points to return a unique solution.
  • cv::SOLVEPNP_IPPE Input points must be >= 4 and object points must be coplanar.
  • cv::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.

Generic PnP

The cv::solvePnPGeneric() allows retrieving all the possible solutions.

Currently, only cv::SOLVEPNP_P3P, cv::SOLVEPNP_AP3P, cv::SOLVEPNP_IPPE, cv::SOLVEPNP_IPPE_SQUARE, cv::SOLVEPNP_SQPNP can return multiple solutions.

RANSAC PnP

The cv::solvePnPRansac() computes the object pose wrt. the camera frame using a RANSAC scheme to deal with outliers.

More information can be found in [326]

Pose refinement

Pose refinement consists in estimating the rotation and translation that minimizes the reprojection error using a non-linear minimization method and starting from an initial estimate of the solution. OpenCV proposes cv::solvePnPRefineLM() and cv::solvePnPRefineVVS() for this problem.

cv::solvePnPRefineLM() uses a non-linear Levenberg-Marquardt minimization scheme [179] [77] and the current implementation computes the rotation update as a perturbation and not on SO(3).

cv::solvePnPRefineVVS() uses a Gauss-Newton non-linear minimization scheme [183] and with an update of the rotation part computed using the exponential map.

Note
at least three 3D-2D point correspondences are necessary.