2.1

Geometry

The system is composed of n_D linear detectors, denoted D_l, l = 1, …, n_D, and an unique x-ray source denoted S. We denote (O, x, y, z) the world coordinate system centred at the origin O. An illustration of such a system is given in the Fig. 1. Furthermore, we introduce the detector coordinate system associated to the l^th detector (O_l, u_l, v_l,w_l) where v_l is the image axis, w_l is the axis perpendicular to v_l pointing towards the source and u_l = v_l ×w_l. The origin of the system is the point O_l which is the orthogonal projection of the source S on the detector D_l. The real v_l is the coordinate of O_l along the linear detector, along v_l, relative to the pixel 0 of D_l.

Figure 1:

A two linear detector system.

2.2

Single detector calibration

We start by introducing the single linear detector system calibration method derived from Horaud et al.⁶

Pinhole linear camera model Let’s consider a point (X, Y,Z) expressed in the world coordinates system and its projection v on the linear detector D. To be seen, the point has to belong to the viewing plane Π which is defined in the Eq. (3) using three real parameters p, q and r.

Moreover, we adjust the pinhole camera model presented in the Eq. (2) to the system using a 1D detector. The rotation matrix R ∈ ℝ^3×3 and the translation vector t ∈ ℝ^3×1 are representing the rigid transformation from the world to the source coordinates. The calibration matrix K_2D ∈ ℝ^3×4 defined in the Eq. (1) becomes K_1D ∈ ℝ^2×4.

The pinhole linear camera model is given by:

Within the plane Π, i.e. using Eq. (3), we can rewrite the Eq. (5) such that the calibration problem is reduced to the estimation of five real parameters n₁, n₂, n₃, n₄, n₅ from :

and the three parameters p, q, r from Eq. (3).

Calibration object The calibration object is made of four co-planar lines. Three of them are parallel and the last one is oblique. For example, we can define these lines within the plane Z = 0 by the following equations.

The intersections of these lines with the plane Π and their projections on D are used to solve the calibration problem. A key idea introduced by Horaud et al. is to use the intersections of (L₁), (L₂) and (L₃) with the plane Π and their projections on D to estimate the intersection point of (L₄) and Π using a projective invariant : the cross-ratio.¹ Therefore, by translating this object several times along the y and/or z axes, we can acquire enough data to solve the calibration problem.

Calibration problem We denote and the sets of known data related to the parallel and oblique lines, respectively, with i = 1, …, n_P and j = 1, …, n_O, where n_P ≥ 5 and n_O ≥ 3 are the numbers of parallel and oblique lines, respectively. In Fig. 2 we show a calibration object containing the minimal number of eight calibration lines. Using (6) and the parallels lines set of data, we can estimate the parameters n₁, n₂, n₃, n₄, n₅ by solving a system of n_P equations in the least square sense. Likewise, we estimate the viewing plane parameters p, q, r by solving a system of n_O equations based on the Eq. (3).

Figure 2:

The minimal calibration object.

Calibration parameters estimation The intrinsic and extrinsic parameters can easily be extracted from n₁, n₂, n₃, n₄ and n₅ using p, q and r (see⁶). Then, the source position (X_S, Y_S,Z_S) can be estimated by solving the linear system (8) where v* and v^⋄ are two different detector pixels. Indeed, the source belongs to all the backprojection lines and the viewing plane Π.

2.3

Multi-detector calibration

The first obvious idea to calibrate a multiple linear detector system would be to use the previous method on each detector. However, all subsystems share the same source. In this section, we present four different methods exploiting this property.

In the following, as in section 2.1, the index l refers to the detector D_l, l = 1, …, n_D.

Method 1 (M1) We use the previous method on each subsystem to calibrate the system. We then agregate the equations (8) associated to each detector D_l, l = 1, …, n_D for estimating an unique source position. The agregated system of equations (9) is composed of 3n_D equations and can be easily solved by least squares.

Method 2 (M2) From the combination of the equations (3) and (9), we get a system of (n_O+3)n_D non-linear equations (10) in the parameters p_l, q_l, r_l, X_S, Y_S and Z_S, l = 1, …, n_D.

This system can be solved using the Gauss-Newton algorithm. We suggest to initialize the algorithm with the results of M1.

Method 3 (M3) Similarly to M2, by combining the equations (6) and (9), we build a system of (n_P + 3)n_D non-linear equations (11) in the parameters n_l,1, n_l,2, n_l,3, n_l,4, n_l,5, X_S, Y_S and Z_S, l = 1, …, n_D.

We use the same numerical method as for M2 for solving Eq. (11).

Method 4 (M4) The last method combines the equations of M2 and M3. Consequently, the system of (n_P + n_O + 3)n_D equations (12) to solve is non-linear in the parameters p_l, q_l, r_l, n_l,1, n_l,2, n_l,3, n_l,4, n_l,5, X_S, Y_S and Z_S, l = 1, …, n_D.

The resolution can be done using the Gauss-Newton algorithm as for M2 and M3.

2.4

Calibration object

The use of sufficient four co-planar lines calibration objects presented in the section 2.2 could be enough to generate the data required to solve the calibration problem. Nevertheless, we present in this section several improvements.

Minimal calibration object The first improvement is to construct a minimal calibration object composed of 5 parallel lines and 3 oblique lines. The parallel lines are shared among the oblique lines such that 3 groups of 3 parallel plus one oblique line, as in section 2.2, can be provided. The 5 parallel lines are defined by the equations (13).

The three oblique lines are defined by the equations (14).

The minimal calibration object is illustrated in the Fig. 2. One can notice that the group of lines in the plane Y = 2ε doesn’t result from a translation as suggested in the section 2.2. The cross-ratio can be adapted in this plane. Besides, we remark that the lines are positioned such that their projections can be spanned all over the detectors using an adequate value of ε.

Adapted object First, we observed that adding to the minimal calibration object the parallel line (L₉) defined in the Eq. (15) improved significantly the accuracy of the calibration.

Then, another improvement is the positioning of the oblique lines relatively to each detector D_l, l = 1, …, nD. The lines are placed such that the intersections of the oblique lines and the viewing planes are at equal distance of the two closest co-planar parallel lines. We denote these lines (L_6,l), (L_7,l) and (L_8,l), l = 1, …, n_D. The lines (L_6,l) and (L_7,l) are positioned in the plane Z = 0 between the lines (L₁) and (L₂), and the lines (L₃) and (L₉), respectively. They are inclined at a fixed angle λ. The lines (L_8,l) are positioned in the plane Y = 2ε, between the lines (L₃) and (L₄). The inclinations of these lines are specific to each detector. Nevertheless, the position of the lines (L_8,l) doesn’t impact the results as much as the two others obliques lines. Thus, every way of positioning the lines can be used providing that the intersection of the lines and the viewing planes are at . An example of such a calibration object is given in the Fig. 3.

Figure 3:

(a) Vertical plane and (b) horizontal plane of the object used for the 8-detectors system calibration.