[Image] Camera calibration (2) Zhang Zhengyu Calibration Deduction _ Image Calibration

Original articles, welcome to reprint. Reprint Please specify: Reproduced from Cheung's blog

Original link: http://blog.csdn.net/humanking7/article/details/44756235

The previous article spoke about the camera's calibration of the basic knowledge of visual measurement in four coordinate systems . The following focuses on the famous Zhang Zhengyu calibration Method . => 1. Preliminary knowledge => 1.1. From pixel coordinate system (U,V) to World coordinate system (XW,YW,YW)

Here directly to take the results of the post, the middle of the other coordinate system to eliminate the direct relationship, directly given, as follows:

The formula is as follows:

=> 1.2. Symbolic provisions (notation)

In order to be consistent with Professor Zhang Zhengyu 's thesis, now unify the formula symbol.

=> 1.3. Mathematical basis in derivation

1th : the rotation vector R is the orthogonal matrix, so the following properties:

2nd : It's s . It is a scale factor, it appears only for convenient operation, and for homogeneous coordinates, scaling factor does not change the coordinate value . => 2. Drama => 2.1. STEP.1 of the calibration plane to the image plane (homography)

At the beginning did not know the paper homography do not know where the holy, search to the last explanation:

Because the tensor calibration is based on the plane checkerboard calibration, so want to understand the tensor calibration, the first should be from the two plane of the single homography (map) mapping start.

homography : A projection map that is defined as a plane to another plane in computer vision. First of all, the image plane and the calibration of the chessboard Gopping surface of the single nature.

Because the calibration is planar , we can construct the world coordinate system on the plane of Z = 0 . Then the single response is calculated. Let Z = 0 convert the upper form to the following (direct interception of the derivation in the paper):

Analysis:

H is a 3x3 matrix and has an element as the homogeneous coordinate. Therefore, H has 8 unknown quantities to be solved (it can be analyzed that A has 5 unknowns, followed by [R1,r2,t] has three unknowns, a total of 8).

(X,y) as the coordinates of the calibration, can be controlled by the designer, is known quantity . (u,v) is pixel coordinates, we can get directly through the camera. A set of corresponding (x,y) => (u,v) We can get two sets of equations.

Now there are 8 unknown quantities to be asked, so at least 8 equations . So you need at least 4 sets of corresponding points . So there are 4 groups (X,y) => (u,v) can be calculated, the image plane to the world plane of the single response matrix H , which is the Zhang Zhengyu calibration using four corner points of the board as a calibration object (. Don't know if it's right). => 2.2. STEP.2 using constraint conditions to solve internal reference matrix a

From the STEP1, we can get the single response matrix H using 4 points. But h is the internal reference of the Matrix and the outer parameters . We want to finally get the internal reference and the outer parameters separately. So we need to find a way to get internal reference first. Then the foreign ginseng is then solved.

The h1,h2 in the upper-form is solved by solving the single response matrix H , so the unknown quantity is only internal reference matrix a . A contains 5 parameters , and if you need to fully solve the 5 unknowns, then you need 3 different single response matrices H (because 3 different single response matrices H can produce 2 equations under 6 constraints ), then how to get 3 different single-response matrices H . That's a picture of 3 different calibration planes, and most of us get different calibration photos by changing the relative position between the camera and the calibration board. (if 2 photos are calibrated, a internal reference r=0 will be shed)

Of course, this is only the first reason that the Zhang Zhengyu calibration method constantly transforms the calibration plate azimuth. The second reason is the maximum likelihood estimate (Maximum-likelihood estimation) mentioned by Zhang Zhengyu (which I have to learn).

Now it's time to start the math class:

First make

We can see that Matrix B is a symmetric matrix with only 6 valid elements, so a 6-D vector b , and then a simplified formula (that is, the two constraints)

By bringing the result of the operation into two constraints, we can get the equation group:

The English part also proposed the function of 3 pictures (do not understand can again look at the previous analysis)

After B is solved by using the formula above, B is obtained, and the camera Internal reference matrix A can be obtained by Cholesky decomposition . => 2.3. STEP.3 using Internal reference matrix A to solve the external parameter matrix

Already have internal reference matrix A, through the following formula, we can solve the outer parameter matrix.

=> 3. Summary

The above is the mathematical principle and derivation of the Zhang Zhengyu calibration method, but Zhang Zhengyu himself also says that there is no actual physical meaning, but provides an initial value for the subsequent maximum likelihood parameter estimation. The maximum likelihood algorithm used to improve the calibration accuracy in Zhang Zhengyu calibration is also studied, and I hope to post a blog later.

The younger brother talents, if has the mistake, hoped everybody corrects the instruction.