How can I find the angle between two chessboard planes?

Question

I have two chessboard poses obtained with solvePnp:

Mat rotationVector1, translationVector1;
solvePnP(chess1WorldPoints, chess1ImagePoints, intrinsicMatrix, distortCoefficients, rotationVector1, translationVector1);

Mat rotationVector2, translationVector2;
solvePnP(chess2WorldPoints, chess2ImagePoints, intrinsicMatrix, distortCoefficients, rotationVector2, translationVector2);

How can I check if the planes of the poses are parallel, or find the angle between these planes?

More info

I tried obtaining Euler angles and computing the difference between each alpha, beta and gamma but that only tells me relative rotation for each axis I think:

Vec3d eulerAnglesPose1; 
Mat rotationMatrix1;
Rodrigues(rotationVector1, rotationMatrix1);
getEulerAngles(rotationMatrix1, eulerAngles1);

Vec3d eulerAnglesPose2;
Mat rotationMatrix2;
Rodrigues(rotationVector2, rotationMatrix2);
getEulerAngles(rotationMatrix2, eulerAngles2);

I used the getEulerAngles implementation from Camera Rotation SolvePnp :

void getEulerAngles(Mat &rotCamerMatrix, Vec3d &eulerAngles) 
{
    Mat cameraMatrix, rotMatrix, transVect, rotMatrixX, rotMatrixY, rotMatrixZ;
    double* _r = rotCamerMatrix.ptr<double>();
    double projMatrix[12] = 
    { 
     _r[0],_r[1],_r[2],0,
     _r[3],_r[4],_r[5],0,
     _r[6],_r[7],_r[8],0 
    };

    decomposeProjectionMatrix(Mat(3, 4, CV_64FC1, projMatrix), cameraMatrix, rotMatrix, transVect, rotMatrixX, rotMatrixY, rotMatrixZ, eulerAngles);
}

Edit

In my case a rotation-translation pair (R, T) gives the correspondence between a coordinate system where the camera is at (0,0,0) (the camera coordinate system) to a coordinate system where (0,0,0) is something I defined in the first two parameters of solvePnp (the world coordinate system). So I have two world coordinate systems relative to the same camera coordinate system. If I could switch from coord. system 2 to coord. system 1 I could use the Z=0 planes for each one to find the normals and solve my problem.

I think that for example switching from coord. system 2 to camera system should be done like in this post:

Rinv = R' (just the transpose as it's a rotation matrix)
Tinv = -Rinv * T (T is 3x1 column vector)

Then if Pw = [X Y Z] is a point in world coord. system 2 I can get its camera system coords.with:

Pc = [ Rinv Tinv] * [X Y Z 1] transposed.
Pc looks like [a b c d]

Following the same logic again I can get the coordinates of Pc relative to coord. system 1:

Pw1 = [ R1 T1] * Pc

Should I normalize Pc or just normalize Pw1 at the end?

So, you have two rotation matrices. _The angle_ is about a rotation from one to the other? If **R1** = **R** * **R2**, then **R** = **R1** * **R2** ^-1. (To invert a rotation matrix it must be transposed only.) A rotation matrix can be decomposed into Euler angles (or Brian-Tait-angles). Alternatively, you can express it as one rotation (with axis/angle) or convert it to a quaternion (which expresses the same). If you just want to determine the angle of check board _planes_, then what you actually need is the angle of _normals_. (This is determining the angle between two vectors.) — Scheff's Cat, Sep 21 '18 at 10:32
adding to @Scheff comment, you can check if 2 planes are parallel by checking the normals of each plane (it should be the same, or maybe pointing towards the other direction) — api55, Sep 21 '18 at 10:37
As I mentioned _normals_ : I don't know much about OpenCV to tell this precisely but I would expect that the _normal_ in quest is just one of the unit axes provided in the rotation matrix (e.g. the z axis which is found in the 3rd matrix row or column usually). — Scheff's Cat, Sep 21 '18 at 10:40

score 0 · Accepted Answer · answered Sep 27 '18 at 14:32

I found how to translate points between coordinate systems in this OpenCV Demo.

The explanation from "Demo 3: Homography from the camera displacement" (the section spanning from title right until the first lines of code) shows how to translate between coordinate systems using matrix multiplication. I just had to apply it to my situation (I had ^CM_O1 and ^CM_O2 and needed to find ^O1M_O2).

This way I can get two planes in the same coord. system, get their normals and find the angle between them.

Also it helped to realize that the extrinsic matrix [R T] translates a 3D point from the world coord. system to the camera coord. system (where camera is at (0,0,0)), not the other way around.

How can I find the angle between two chessboard planes?

1 Answers1