Robotics-Robot Vision (estimation)

Previously, the core of robot vision was estimation. The features and registration are also prepared for estimation. Once the registration is complete, we can estimate the position and posture of the robot from the image. With the position, posture, we can make three-dimensional reconstruction of things to splice. The problem of estimating robot pose from visual information can be divided into three categories: 1, field attractions on the same plane. 2, the field attractions in three-dimensional space. 3, two point cloud registration. One major prerequisite for all problems is knowing the camera's internal matrix K.

1. Pose estimation by single-response matrix

The single-should matrix refers to the mapping relationship from R2--R2. But in estimating the problem, if we can get this mapping relationship, we can restore the transformation matrix from the world coordinate system x_w to the camera coordinate system x_c. This transformation matrix expresses the posture of the camera relative to the x_w.

H = s*k*[r1 R2 t]--assumes a Z coordinate of 0 on the plane

S*[R1 R2 T] = k-1*h--using a single-response matrix to find the rotation and translation vectors

R3 = r1xr2--Recovery R3

S is not important, only needs to be normalized to K-1*H1 can be obtained.

So, the most important thing is how to find out two scenarios in the single-should. In the front I mention from vanishing point to get the single-relationship, but if not from the rectangular---quadrilateral mapping, we do not have vanishing point can be found.

Here is an elegant to bursting approach. Based on matrix transformation and singular value decomposition. JB Shi is really worthy of Daniel. The 32 sentences are so simple as to tell the question.

　　

Since there are 8 degrees of freedom in the H matrix, each pair of single points can provide two equations, so 4 single-point can uniquely determine the single-matrix H. Ax = 0, which we have already learned in the proposed unity chapter. X is the minimum singular value for the column of the V Matrix. Here is the first occurrence of singular value decomposition.

So far, we have restored the H matrix. The [R1 R2 T] can be lifted according to normal thinking. However, our H-matrix is optimized by singular value decomposition, and the inverse R1 R2 does not necessarily satisfy the orthogonal condition, nor does it meet the equal-length condition. So, we're going to fit the RT matrix once more.

The goal of this fitting is min (Ros3-r '). where r ' = [k-1h (:, 1:2) x]. The method is still singular value decomposition, R = UV '. This is the second occurrence of singular value decomposition.

2. Estimation of position and pose by projective transformation

The premise of the estimation of position and posture by the single-matrix is that all the points are on a plane. And the estimation of the position and pose by the projective transformation discards this premise, so the previous section is a special case of this section. This issue is a scientific name for the PNP issue: Perspective-n-point.

Imitating the above ideas, we can still write the following forms:

　　

There are 12 unknowns in the projective matrix, 9 from the rotation matrix, and 3 from the translation vector. Each of the points can provide 2 equations. So as long as 6 scene points, we can use singular value decomposition to obtain the value of the P-matrix. Similarly, after obtaining the P-matrix, t = k-1*p, and finally using singular value decomposition to fix T.

In common sense, however, this problem has only 6 degrees of freedom (3 translation, 3 rotation). We use 6 points to actually be a dirty method.

3. Pose estimation by two point clouds

This can be a lot more of a RGBD camera that is now very hot. In this paper, three-dimensional images of the same object are obtained from different angles, and the transformation relationship between the two positions is derived. The premise of this problem is that the point can be one by one corresponding to the solution. If the point cannot correspond, that is the ICP algorithm problem.

This problem is a scientific name: Procrustes problem. From Greek mythology. In Chinese, it can be called the shoe problem. How to rotate and pan the foot, and then put it in the shoe. Its mathematical description is as follows: Reduce the difference between AB by selecting the appropriate r,t.

　　

T actually very good guess, if two point regiment can overlap, then its center of gravity is certainly coincident. So t represents a vector between two points of gravity. This problem has the following variants:

　　

The matrix analysis shows that the 2 norm of a vector has the following variants:

By matrix analysis, the last two items are actually equal (the cyclic invariance of traces and transpose invariance)

Then the goal of optimization can be converted to:

　　

The trace is the amount associated with the singular value (the same as the change trace)

　　

Obviously, if the trace Z is as large as possible, then there is only one case where z is the unit array, and the trace of the unit array is the largest in the rotation matrix. So the analytic solution of R is as follows:

　　

At this point, we have obtained the analytic solution of 3d--3d pose estimation!

　　

　　

　　

Robotics-Robot Vision (estimation)