A Survey of Calibration Methods for Traditional Cameras Based on Line Structure Light

In this paper, the pinhole model is used as a camera model for research. The 2D calibration target is represented by m = (u, v,1)^T, the 3D calibration target is represented by M = [X, Y, X, 1] ^T , and the corresponding homogeneous vectors are, m ˜ = [ u , v , 1 ] T , M ˜ = [ x , y , z , 1 ] T , respectively. Then the relationship between the 3D point M and its projection point m is as follows[7]: (1) s m ˜ = A [ R , t ] M ˜

Where R is the rotation matrix, t = [t_x t_y t_z] ^T for the translation vector, and describes the external parameters of the camera that the camera is calibrated. f_u = a_x/d_x, f_v = a_y/d_y, d_x, d_y indicates the physical size of each pixel in the Y-axis Y-axis direction. f_u, f_v, u₀, v₀ is only related to the internal parameters of the camera, which is the internal parameters that the camera needs to calibrate.

This paper introduces the calibration of 3D calibration targets, using the classic Tsai[8].two-step calibration algorithm. The Tsai two-step method is based on the calibration method of the radial correction constraint (RAC. Radial Alignment Constraint).

The camera distortion model shown in Figure 7 below, The following model includes five coordinate systems, which are camera coordinate system O_c, image pixel coordinate system O_i, world coordinate system O_W actual image physical coordinate system O_d, and ideal image physical coordinate system O_u. The solution process assumes that u0, v0 is known to only consider second-order radial distortion. The main point is both the center of the image and the center of the radial distortion.

Figure 7.

The first step : uses a radial alignment constraint (RAC) linear solution.

Step 2: Find the remaining parameters for nonlinear optimization.

Advantages: Applicable to any camera model, high calibration accuracy.

Insufficient: calibration needs to calibrate targets, which is difficult to achieve in some applications.

In this paper, the research method of two-dimensional calibration target is explained by Zhang Zhengyou calibration method[9][10]. To facilitate the operation, the template is defined on a plane parallel to the X-Y plane (Z = 0) in the world coordinate system..

From the above formula (1): (2) s [ u v 1 ] = [ α γ u 0 0 0 β v 0 0 0 0 1 0 ] [ R t 0 T 1 ] [ x w y w z w 1 ] (3) R = [ r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 ]

Where is the u₀, v₀ principal point coordinate, α, β is the vector of the (u, v), the coordinate axis in the image, and γ is the perpendicularity of the two coordinate axes. Let the template plane Z_W = 0 be in the world coordinate system, you can get: (4) s m ˜ = H M ˜

Where H = [h₁ h₂ h₃] γA[r₁ r₂ r₃], γ is the scaling factor scalar, r₁, r₂ is the two column vectors of the rotation matrix, and t is the translation matrix.

By the nature of the rotation matrix, a constraint matrix is available for each image: (5) { h 1 T A − T A − 1 h 2 = 0 h 1 T A − T A − 1 h 1 = h 2 T A − T A − 1 h 2 }

make: (6) B = A − T A − 1 = [ B 11 B 12 B 13 B 21 B 22 B 23 B 31 B 32 B 33 ] = [ 1 α 2 − γ α 2 β v 0 γ − u 0 β α 2 β γ α 2 β γ 2 α 2 β 2 + 1 β 2 − γ ( v 0 γ − u 0 β ) α 2 β 2 − v 0 β 2 v 0 γ − u 0 β α 2 β − γ ( v 0 γ − u 0 β ) α 2 β 2 − v 0 β 2 γ ( v 0 γ − u 0 β ) 2 α 2 β 2 + v 0 2 β 2 + 1 ]

From (6): B is a symmetric matrix, which can be represented by the following 6D vector: (7) b = [ B 11 B 12 B 13 B 22 B 23 B 33 ] T

Let the ith column vector h_i = [h _i1 h _i2 h _i3] in H be obtained: (8) h i T B h j = v i j T b

Then you can write (5) as: (9) [ v 12 T [ v 11 − v 12 ] T ] b = 0

Suppose you take n images of the template plane and get n images. (10) Vb = 0 n is the matrix of 2 n × 6

If n ≥ 3, you can get the unique solution b and matrix B, you can get: (11) r 1 = λ A − 1 h 1 r 2 = λ A − 1 h 2 r 3 = r 1 r 2 t = λ A − 1 h 3

Get the internal and external parameters of the camera for optimization: (12) Σ i = 1 n Σ j = 1 m ‖ m i j − m ^ ( A , R i , t i , M j ) ‖ 2

Where m(A,k₁,k₂,k₃,R_i,T_i,M_j) represents the coordinate point at which the jth point is projected onto the i-th image according to equation (4).

As shown in Figure 8, AB is a one-dimensional calibration with a length of L[11].

Figure 8.

(13) ‖ A − B ‖ = L

Since the ratio of the line segments is known, B points can be calculated in the case where points A and C are known.

Then, according to the formula (2-1), there is a following formula, where Z_A Z_B, Z_C is the depth of the corresponding point on the one-dimensional calibration object. (14) Z C = Z A λ A a + Z B λ B b

Where λ_B = BC/AB, λ_B = AC/AB Representation ratio.

According to (13): (15) ‖ K − 1 ( Z B b − Z A a ) ‖ = L

According to (14): (16) Z A 2 h T K − T K − 1 h = L 2

The parameters of the camera can be obtained by (16). According to the theory of higher geometry, the projection of the absolute quadratic curve on the image plane is actually described.