Research on Binocular Vision Algorithm (ii) camera model and direct linear method (DLT)

first, the mathematical model of the camera

Camera model for all future calibration algorithm key, only this side has a fairly thorough understanding of the future calibration algorithm to have a better understanding. I have studied for a long time, almost every day repeated to read a few times, and finally understand the derivation process.

I think first of all we need to understand the relationship between the four planar coordinate systems in the camera model: the pixel plane coordinate system (U,V), the plane coordinate system (image physical coordinates (x, y), the camera coordinate system (XC,YC,ZC), and the World coordinate system (XW,YW,ZW), which are described in each article that describes the camera model.

When I first began to understand, looking at the pile of formulas was very dizzy, I believe a lot of beginners and I, but think about, just, we assume some parameters, so that the coordinates between the four coordinate systems, so that we can from the picture of a point coordinate all the way back to the world of the point coordinates, This has achieved our goal of three-dimensional reconstruction. And the parameters that we assume are the internal and external parameters that we want to calibrate. 1. The relationship between pixel coordinates and image plane coordinate system

Before determining their relationship, we can assume that each pixel has a physical size of dx and dy in the U-axis and V-axis direction. Take a closer look at their model to introduce the following formula (which is better understood):

Equation 1

Explanation: 1, dx,dy,u0,v0 actually are we assume the parameter, Dxdy represents the sensor chip on the actual size of pixels, is connected to the pixel coordinate system and the real size coordinate system, U0,V0 is the image of the center, the final is to ask for the internal and external parameters.

With this formula we can use the knowledge of linear algebra to represent the equation in matrix form:

Equation 2

Of course, we can also express it in a different matrix form:

Equation 3

2. Relationship between camera coordinate system and world coordinate system

The relationship between these two coordinate systems we can rotate the matrix R and T to get the following relationship:

Equation 4

Explanation: 1, in this formula, R is the 3*3 Matrix, T is 3*1,0 (0,0,0), simplified with LW as the 4*4 matrix.

3. Imaging projection Relationship (camera coordinate system and image plane coordinate system)

In the camera model we can get the following formula:

Equation 5

Explanation: 1,

We also use matrices as a form of representation:

Equation 6

4. Get the formula

And we can combine the above formula to get:

By means of the equation:

Replace ZC to get the following formula:

Explanation: 1, in the formula: L1,l2,..., L11 is 11 factors related to U0,v0,f,kx,ky,u,x,j and xs,ys,zs; The imaging characteristics of the camera are determined. X0,y0,f is called the camera interior azimuth element, U,x,j and Xs,ys,zs is called the camera foreign bits. If more than 6 space points (X i,y i,z i) and their image coordinates (U i,v i) are known, the L1,l2,..., L11 can be solved by the formula, and the camera parameters are obtained. If the parameters of the two or more cameras are known to be L1,l2,..., L11, the spatial coordinates (x, y, z) can be calculated from the upper formula according to the image coordinates of the space points in each camera (J,y j). The process of determining L1,L2,..., L11 is called calibration, and the process of solving (x, Y, z) by (XJ,YJ) is called refactoring.

Since the formula is copied, the following is replaced by P:

The move item can be:

Explanation: 1, from the above visible, by the space of more than 6 known points and their image point coordinates can be obtained p matrix. In the general calibration work, there are dozens of known feature points on the target, so that the number of the equations greatly exceeds the number of unknowns, and the least square method is used to reduce the effect of the error.

2, when the standard fixed point has n, it means that the above equation has 2N, at this time we can use a matrix to represent the 2N equation, the formula is:

where L is:

A is:

U is:

This formula can be written out, so that it is more good-looking, listed in the future can be seen P34 to other values are not affected, so make p34=1.

It says that at 2N points, we can make the results more precise with the least squares method, and the result is:

Equation Explanation: 1, because in the actual coordinates with the image coordinates is known, so A and u are known, so can find L.

2, P34 is 1, so the L is P11 to P33.

So all the p we have to find out, and this goes back to the equation, we use the P parameter to replace the internal and external parameters of the formula to facilitate the solution, we can use the P parameter to reverse the introduction of internal and external parameters, because this part of the better understanding, do not write, paste formula is too troublesome.