Compute the essential matrix for a pair of stereo images by automatically finding correspondences between image points.

Given a set of coordinates of characteristic points (Rows1,Cols1) and (Rows2,Cols2) in the stereo images Image1 and Image2 along with known internal camera parameters, specified by the camera matrices CamMat1 and CamMat2, match_essential_matrix_ransac automatically determines the geometry of the stereo setup and finds the correspondences between the characteristic points. The geometry of the stereo setup is represented by the essential matrix EMatrix and all corresponding points have to fulfill the epipolar constraint.

The operator match_essential_matrix_ransac is designed to deal with a linear camera model. The internal camera parameters are passed by the arguments CamMat1 and CamMat2, which are 3x3 upper triangular matrices desribing an affine transformation. The relation between a vector (X,Y,1), representing the direction from the camera to the viewed 3D space point and its (projective) 2D image coordinates (col,row,1) is:

      / col \              / X \                    /  f/Sx     s  Cx \
      | row |  =  CamMat * | Y |   where   CamMat = |     0  f/Sy  Cy |  .
      \  1  /              \ 1 /                    \     0     0   1 /

Note the column/row ordering in the point coordinates which has to be compliant with the x/y notation of the camera coordinate system. The focal length is denoted by f, Sx,Sy are scaling factors, s describes a skew factor and (Cx,Cy) indicates the principal point. Mainly, these are the elements known from the camera parameters as used for example in camera_calibration. Alternatively, the elements of the camera matrix can be described in a different way, see e.g. stationary_camera_self_calibration. Multiplied by the inverse of the camera matrices the direction vectors in 3D space are obtained from the (projective) image coordinates. For known camera matrices the epipolar constraint is given by:

            T
      / X2 \             / X1 \
      | Y2 | * EMatrix * | Y1 |  =  0  .
      \  1 /             \  1 /

The matching process is based on characteristic points, which can be extracted with point operators like points_foerstner or points_harris. The matching itself is carried out in two steps: first, gray value correlations of mask windows around the input points in the first and the second image are determined and an initial matching between them is generated using the similarity of the windows in both images. Then, the RANSAC algorithm is applied to find the essential matrix that maximizes the number of correspondences under the epipolar constraint.

The size of the mask windows is MaskSize x MaskSize. Three metrics for the correlation can be selected. If GrayMatchMethod has the value 'ssd', the sum of the squared gray value differences is used, 'sad' means the sum of absolute differences, and 'ncc' is the normalized cross correlation. This metric is minimized ('ssd', 'sad') or maximized ('ncc') over all possible point pairs. A thus found matching is only accepted if the value of the metric is below the value of MatchThreshold ('ssd', 'sad') or above that value ('ncc').

To increase the speed of the algorithm, the search area for the matchings can be limited. Only points within a window of 2*RowTolerance x 2*ColTolerance points are considered. The offset of the center of the search window in the second image with respect to the position of the current point in the first image is given by RowMove and ColMove.

If the second camera is rotated around the optical axis with respect to the first camera the parameter Rotation may contain an estimate for the rotation angle or an angle interval in radians. A good guess will increase the quality of the gray value matching. If the actual rotation differs too much from the specified estimate the matching will typically fail. In this case, an angle interval should be specified, and Rotation is a tuple with two elements. The larger the given interval the slower the operator is since the RANSAC algorithm is run over all angle increments within the interval.

After the initial matching is completed a randomized search algorithm (RANSAC) is used to determine the essential matrix EMatrix. It tries to find the essential matrix that is consistent with a maximum number of correspondences. For a point to be accepted, the distance to its corresponding epipolar line must not exceed the threshold DistanceThreshold.

The parameter EstimationMethod decides whether the relative orientation between the cameras is of a special type and which algorithm is to be applied for its computation. If EstimationMethod 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 order to get a unique solution in the correspondence problem the minimum required number of corresponding points is six in the general case and three in the special, translational case.

The essential matrix 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, and returns the covariance of the essential matrix CovEMat as well. Here, 'normalized_dlt' and 'gold_standard' stand for direct-linear-transformation and gold-standard-algorithm respectively. Note, that in general the found correspondences differ depending on the deployed estimation method.

The value Error indicates the overall quality of the estimation procedure and is the mean euclidian distance in pixels between the points and their corresponding epipolar lines.

Point pairs consistent with the mentioned constraints are considered to be in correspondences. Points1 contains the indices of the matched input points from the first image and Points2 contains the indices of the corresponding points in the second image.

For the operator match_essential_matrix_ransac 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 essential matrix is not unique but twofold. As a consequence both solutions are computed and returned by the operator. This means that the output parameters EMatrix and CovEMat are of double length and the values of the second solution are simply concatenated behind the values of the first one.

The parameter RandSeed can be used to control the randomized nature of the RANSAC algorithm, and hence to obtain reproducible results. If RandSeed is set to a positive number the operator yields the same result on every call with the same parameters because the internally used random number generator is initialized with the RandSeed. If RandSeed = 0 the random number generator is initialized with the current time. In this case the results may not be reproducible.

match_essential_matrix_ransac is reentrant and processed without parallelization.

points_foerstner, points_harris

vector_to_essential_matrix

match_fundamental_matrix_ransac, match_rel_pose_ransac, stationary_camera_self_calibration

Richard Hartley, Andrew Zisserman: ``Multiple View Geometry in Computer Vision''; Cambridge University Press, Cambridge; 2003.

Olivier Faugeras, Quang-Tuan Luong: ``The Geometry of Multiple Images: The Laws That Govern the Formation of Multiple Images of a Scene and Some of Their Applications''; MIT Press, Cambridge, MA; 2001.

3D Metrology