Synchronous scanning mode of industrial computed tomography for multiple objects test

2.1 Imaging model

In this section, we introduce the geometry of our proposed CT scanning mode. As shown in Fig. 2, the detectors are placed on a circular, N is the number of detectors, λ represents the angle size between adjacent detector-cells. S _o is the distance from X-ray source to the detectors. There are n turntables linearly arranged within the FOV. The line passing through all the rotating centers of multi-turntables is perpendicular to the central X-ray of the fan-beam. R represents the distance from the line of the rotating centers to X-ray source. Thus, the global coordinate fixed on the rotating centers of multi-turntables can be written as σ : = o ; x, y}, where o is original point. The moving coordinate can be defined as σ _s  : = o _s  ; x _s , y _s } whose origin o _s is fixed at X-ray source. For simplicity, we can further define σ _i  : = o _i  ; x _i , y _i } as local coordinate of the ith turntable, whose origin o _i is fixed at the center of ith turntable. X _I is the distance from o _i to o, γ _i is the angle between oo _s and o _i o _s . Let ω denotes the turntables rotated angle related to the original position.

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Fig.2

CT geometry of the novel CT scanning mode based on synchronously rotating multi-turntables.

The geometry model of the ith turntable is shown in Fig. 3. The radius of the ith turntable is r _i . α _i denotes the angle of fan-beam. R _i is the distance between the source and the center of the ith turntable, i _o  → i _m is the detector array corresponding to the FOV of the ith turntable. Therefore, the scanning mode of the ith turntable can be expressed as

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Fig.3

CT geometry of the ith turntable.

S i = S o(1) R i = R cos ( γ i )(2) m = α i λ(3) i o = N 2 + γ i - α i / 2 λ(4) i m = i o + m(5)

As shown in Fig. 4, we assume the object is a compact support function f (x _i , y _i ) within the radius r _i . σ i ′ : = o i ′ ; x i ′ , y i ′ } is defined as the rotation coordinate relatively to the original position. Therefore, the projections p (s, θ) from the object f (x _i , y _i ) can be written as

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Fig.4

Scan geometry of the ith turntable in the full FOV.

p ( s , θ ) = ∫ - ∞ + ∞ ∫ - ∞ + ∞ f ( x i , y i ) δ ( x i cos θ + y i sin θ - s ) d x d y(6)

θ is the angle between the ray and the rotation coordinate, s is the distance from the origin to a specified X ray.

The parallel beam reconstruction is defined as [12]: f ( x i , y i ) = ∫ 0 π ∫ - ∞ + ∞ p ( s , θ ) h ( x i cos θ + y i sin θ - s ) d s d θ(7)

where h ( s ) = ∫ - ∞ + ∞ | ϖ | exp ( i 2 π ϖ s ) d s is the ramp filter. Obviously, the view of each turntable is independent of one another, which means we only need the projection data over the range of [- r _i , r _i ] for the ith turntable. Accordingly, f (x _i , y _i ) is also written as f ( x i , y i ) = 1 2 ∫ 0 2 π ∫ - r i r i p ( s , θ ) h ( x i cos θ + y i sin θ - s ) d s d θ(8)

Equation (8) is the parallel-beam FBP reconstruction formula for the ith turntable. Now, let g (γ, ω) denote the fan-beam projection, γ is the angle between a fixed ray and the center ray of fan-beam, ω represents the rotated angle related to the original position of all turntables. Thus, we can get the expression of θ and s: { θ = ω + γ - γ i s = R i · sin ( γ - γ i )(9)

Furthermore:

f ( x i , y i ) = 1 2 ∫ 0 2 π ∫ γ i - a i / 2 γ i + a i / 2 g ( γ , ω ) h [ x i cos ( ω + γ - γ i ) + y i sin ( ω + γ - γ i ) - R i · sin ( γ - γ i ) ] | J | d γ d ω(10)

where γ ∈ [γ _i  - a _i /2, γ _i  + a _i /2] depends on the FOV of ith turntable. J = R _i  · cos(γ - γ _i ) represents the Jacobi factor. For simplicity, Equation (10) equals to f ( r i , φ i ) = ∫ 0 π ∫ γ i - a i / 2 γ i + a i / 2 g ( γ , ω ) [ γ ′ - γ - γ i L sin ( γ ′ - γ - γ i ) ] 2 h [ L sin ( γ ′ - γ - γ i ) ] R i · cos ( γ - γ i ) d γ d ω(11)

where, L = r i 2 cos 2 ( ω - φ i ) + [ R i + r i sin ( ω - φ i ) ] 2 .

According to the characteristic of filter function h (·), we can get h [ L sin ( γ ′ - γ - γ i ) ] = [ γ ′ - γ - γ i L sin ( γ ′ - γ - γ i ) ] 2 h ( γ ′ - γ - γ i ) .(12)

From the above analysis, we only discuss the ith turntable in specified coordinate. To realize the global coordinates, we should change the local coordinate σ _i  : = o _i  ; x _i , y _i } to global coordinate σ : = o ; x, y} under the relation that

{ r i = r 2 + x i 2 + 2 rx i cos φ φ i = arcsin [ r sin φ r 2 + x i 2 + 2 rx i cos φ ]

From the above, Equation (11) can be simplified to f ( r i , φ i ) = 1 2 ∫ 0 2 π R i L 2 ∫ γ i - a i / 2 γ i + a i / 2 cos ( γ - γ i ) g ( γ , ω ) h ( γ ′ - γ - γ i ) ( γ ′ - γ - γ i ) 2 sin 2 ( γ ′ - γ - γ i ) d γ d ω .(13) Equation (13) illustrates the fan beam FBP reconstruction algorithm for ith turntable. On the basis of this, we can reconstruct each object successively from corresponding projections. The procedure of reconstruction algorithm of our proposed CT scanning mode is listed as follows:

Step 1. Looping the parameter i. Determine if the reconstructed point is covered by the FOV of ith turntable. If it cannot be satisfied, the point would be arranged to zero.

Step 2. Pre-weighting g ^a  = g (γ, ω) · cos(γ - γ _i )

Step 3. Filtering g b = g a * ( ( γ ′ - γ - γ i ) 2 sin 2 ( γ ′ - γ - γ i ) h ( γ ′ - γ - γ i ) )

Step 4. Back-projection of filtering data f ( r , φ ) = 1 2 ∫ 0 2 π R i L 2 g b d γ d ω

3.1 Numerical simulation

As shown in Fig. 5, five testing phantoms with different sizes are fixed on five turntables from left to right respectively. The size of each phantom is shown in Table 1. Other scanning parameters are shown in Table 2. The pixel size of the reconstructed image is 0 . 5 × 0 . 5mm². We took 720 views for data acquisition.

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Fig.5

Original phantoms for numerical simulation.

In the process of scanning, each tested object rotates synchronously around the center of its corresponding turntable, respectively. Figure. 6 shows the sinogram of projections. We can observe that the original tested objects are covered by the range of X-rays, and that the projections of each detected phantom do not overlap each other (i.e., the projections from specified object are confined to designated range).

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Fig.6

Sinogram of projections.

The reconstructed images by our proposed FBP algorithm are displayed in Fig. 7, and the root mean square error (RMSE) of each phantom is displayed in Table 3. From the results, we know that the novel CT scanning method is feasible and the proposed FBP algorithm can be applied.

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Fig.7

The reconstructed images. The display window is [0 1].

As shown in Fig. 8, we have constructed a real CT system with our proposed scanning mode and FBP reconstruction algorithm. This system consists of a 4 MeV X-ray source and CdWO₄ detectors with 486 channels. In addition, 9 synchronous-rotating turntables with 200 mm diameter and 210 mm intervals are arranged linearly on top of a turntable with 600 mm diameter that can be used for conventional CT scan. While it is performed with conventional CT scan, the radius of FOV is 300 mm. R _o is the distance from source to the center of the turntable. As it is performed on the novel CT scanning mode, the distance between the source and the central line of turntables is R, which is adjustable form 1100 to 1755 mm. As show in Fig. 9, with different R, only 2 to 3 turntables are within the fan beam. Correspondingly, 9 turntables are divided into 3 to 5 groups to be tested, which means the time for objects loading is decreased to improve test efficiency. The experiment scanning parameters are listed in Table 4.

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Fig.8

The illustration of experimental CT setup. 9 synchronous rotating turntables with 200 mm diameter and 210 mm intervals are arranged linearly on top of a turntable with 600 mm diameter.

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Fig.9

Scan geometry of the experiment CT system.

From the parameters for experimental parameters we can see that completing the scanning of 3 to 5 groups objects just need 5–8 minutes, while the system just need activate once because multiply objects are loaded on turntables simultaneously. Compared with the mode of testing objects successively, which needs approximately 30 minutes in general, the test efficiency has been improvedsignificantly.

Two sets of multi-object CT slice images are displayed in Fig. 10. The images with sharp edges are free of artifacts. In fact, the system has been applied to test large quantity small complex metal parts. The test efficiency and image quality can meet the requirements.

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Fig.10

Two sets of multi-object CT slice images. (a) displays the CT image with 2 tested metal parts and (b) displays the CT image with 3 different specimen.