Applied Radiation and Isotopes 55 (2001) 793–797

Geometrical rectification for images from a mobile largecontainer inspection system

Hui Jin*, Jian-Ping Cheng, Zhi-Qiang Chen, Li Zhang

Department of Engineering Physics, Tsinghua University, Beijing 100084, People’s Republic of China

Received 20 November 2000; received in revised form 13 February 2001; accepted 14 February 2001

Abstract

The mobile large container inspection system (MLCIS) is a more flexible X-ray radiographic system thanconventional fixed systems. Ideally, the X-ray detectors should be arranged along an arc with the source at the center.However, detectors of the MLCIS are arranged in an L-shaped frame and result in a factual image which is distorted

compared with the ideal image. We propose a geometrical rectification algorithm to convert the factual image to theideal image. We choose the dimension ratios of identical objects shown on the factual and processed images with ouralgorithm as a measure, and the results is desirable. # 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Geometrical rectification; Mobile large container inspection system; X-ray imaging; Radiography

1. Introduction

Many countries now show more and more concern on cargo inspection for purpose of national, social andeconomical safety. The large container inspection system (LCIS), which is an X-ray radiographic system, is widely used

(Wang, 1996). Details of cargo in containers including commercial contrabands, arms and drugs can be displayedbefore custom inspectors, and the small dose as well as the narrow X-ray spectrum enables safe inspection for mostmaterials including food (World Health Organization, 1990) and normal optical films. Therefore the LCIS system has

exerted strong power on anti-smuggling. However, most LCISs in use now are fixed systems, which need complexconcrete construction to shield the relatively high energy ray and spacious land for building (Wang, 1998). This kind ofsystem is not convenient enough for most container docks today which are busy and congested.Now a new kind of system}mobile large container inspection system (MLCIS) has been developed. This system is

much more flexible compared with the traditional fixed system. Differences and similarities between the two are shownas following (Jin, 1999):(1) In a mobile system, the container is static while the X-ray source (a linear particle accelerator) and detectors are

fixed on a distantly controlled truck; in a fixed system, the X-ray source and detectors are fixed in an indoor space withthick shielding walls around, and the container moves through the system.(2) In a mobile system, the detectors are arranged in an L-shaped frame which is in a plane vertical to the direction of

the relative moving; while in fixed system, detectors are arranged in a linear frame approximating a small arc.(3) X-ray energy in a mobile system is about 2–3MeV, less than that in a fixed system which is about 8–9MeV.(4) In both systems, each time the source releases an X-ray pulse, a vertical line of the image is formed.

*Corresponding author. Presently at Harvard University.

E-mail address: huijin@fas.harvard.edu (H. Jin).

0969-8043/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 9 6 9 - 8 0 4 3 ( 0 1 ) 0 0 0 7 3 - 2

(5) In both systems, as the container and source-detector plane move relatively with a uniform speed and the source

releases X-ray pulses at intervals of equal time, the whole scanning image is formed.MLCIS’s main characteristics are shown in Fig. 1.Because the source can be treated as a point, ideally the detectors should be arranged in an arc with the source at its

center. Therefore, there are geometrical distortions in the factual MLCIS image from an ideal image due to the L-

shaped detector frame. Thus geometrical rectification is necessary (Jin, 1999).

2. Algorithm of geometrical rectification

Assume there are n factual detectors in one arm of the L-shaped frame with the serial number i (i ¼ 0; 1,. . .,n�1), andthere are N imaginary detectors in the ideal arc. We should create a transmission array BitN[n], so that i stands for afactual detector in the L-shaped frame arm and BitN[i] stands for an imaginary detector on the ideal arc, respectively.

The two detectors are of the same size and receive the same X-ray beam. If we ignore the attenuation of the X-ray whenit passes the air, the outputs of the two detectors should be the same, and thus the gray levels of their correspondingpixels should also be the same. Because of the relative moving, a detector results in gray level data for a horizontal line

in the ultimate factual image. Thus we can obtain gray level of each pixel in horizontal line numbered with BitN[i] in theimaginary image (i.e. the ideal image) by copying the gray level of each pixel in the NO i horizontal line in the factualimage. In the end, we can apply interpolation to obtain the gray levels for the other horizontal lines in the imaginaryimage.

The geometrical model is shown in Fig 2.O: the source, also the center of the ideal arc.s: the intersection point of the two arms of the L-shaped frame.

L: the line section from O to s, also the radius of the ideal arc, its length is already known.X: one arm of the frame, its length is already known. We define top edge point as the intersection point of top edge of

a detector and X, and bottom edge point as the intersection point of bottom edge of a detector and X.

Dxi: distance between top edge point and bottom edge point of a detector.lpi: the line section from the top edge point of a detector and O.y: the angle between L and X, already known.a: the planar angle of X to O, already known.

Dai: the planar angle of Dxi to O.#aai :the angle between lpi and L for a detector.#xxi :distance between top edge point of a detector and s.

D: the thickness of each detector, also the vertical distance between top edge and bottom edge of a detector, alreadyknown.Because the length of the imaginary arc is more than X, N is greater than n. We can secure the rationality for creating

BitN[n] and interpolation.The mathematical analysis to create BitN[n] is shown as following: According to law of cosines,

lpi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ #xx2i � 2L #xxi cos y:

qð1Þ

Fig. 1. The main components in MLCIS.

H. Jin et al. / Applied Radiation and Isotopes 55 (2001) 793–797794

Then according to law of sines,

#xxisin #aai

¼L

sinðp� #aai � yÞ

#xxi ¼L sin #aai

sinðp� #aai � yÞð2Þ

and

Dxisin Dai

¼lpi

sinðp� #aai � y� DaiÞ

Because Dai is very small, then sin Dai � Dai, therefore

DxiDai

�Dxi

sinDai

¼lpi

sinðp� #aai � y� DaiÞ

¼lpi

sinð#aai þ yþ DaiÞ

¼lpi

sinð#aai þ yÞcos Dai þ cosð#aai þ yÞ sinDai

¼lpi

sinð#aai þ yÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðsin DaiÞ

2q

þ cosð#aai þ yÞ sinDai

�lpi

sinð#aai þ yÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Da2i

pþ cosð#aai þ yÞDai

:

Thus

lpiDai ¼Dxi sinð#aai þ yÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Da2i

qþ Dai Dxi cosð#aai þ yÞ

Dai lpi � Dxi cosð#aai þ yÞ½ � ¼Dxi sinð#aai þ yÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Da2i

qDa2i lpi � Dxi cosð#aai þ yÞ½ �2¼ ð1� Da2i Þsin

2ð#aai þ yÞDx2i

Da2i ¼sin2ð#aai þ yÞDx2i

lpi � Dxi cosð#aai þ yÞ½ �2þsin2ð#aai þ yÞDx2i: ð3Þ

Fig. 2. Geometrical model for MLCIS.

H. Jin et al. / Applied Radiation and Isotopes 55 (2001) 793–797 795

Because

Dxi �D

sinð#aai þ Dai=2þ yÞ�

Dsinð#aai þ yÞ

ð4Þ

we can obtain the following from (1), (3) and (4)

Dai �sinð#aai þ yÞ

Dsinð#aai þ yÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L2 þ #xx2i � 2L #xxicos yq

�D

sinð#aai þ yÞcosð#aai þ yÞ

� �2þsin2ð#aai þ yÞ

Dsinð#aai þ yÞ

� �2s ;

Dai �Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L2 þ #xx2i � 2L #xxicos yq

�D

sinð#aai þ yÞcosð#aai þ yÞ

� �2þD2

s : ð5Þ

In formula (4) and (5), the unknown parameters are #aai, Dai, #xxi, Dxi, and in Fig. 2, we can find the relations#aaiþ1 ¼ #aai þ Dai; #xxiþ1 ¼ #xxi þ Dxi, so we can calculate as following:

1. #aa0 ¼ 0; #xx0 ¼ 02. From formula (4), #aai ! Dxi3. From formula (5), #aai; #xxi ! Dai4. #aaiþ1 ¼ #aai þ Dai; #xxiþ1 ¼ #xxi þ Dxi5. If (iþ 1 >n), stop; Else, i ¼ i þ 1, turn to step 2.

Then, we can obtain BitN[n] by

BitN½i� ¼ð#aai þ Dai=2ÞL

D: ð6Þ

In the end, we can perform two-dimensional linear interpolation or two-dimensional cubic interpolation to finish thewhole ideal image.

3. Results

In a practical MLCIS system, a standardized 8 f 8 f 40 f (2438mm 2438mm 12190mm) cargo container madeof iron plate is used. The length of the vertical arm is 3482mm, composed of 320 detectors, sizes of which are10mm 10mm. And the length of the horizontal arm is 2097mm, composed of 112 detectors. The X-ray source is a

linear electron accelerator with 2.5MeV ray energy. Diameter of the target area is less than 1.5mm. Distance betweenthe source and the vertical arm is 6000mm and distance between the source and the horizontal arm is 2680mm. Theradius of the imaginary arc is 6571mm. And the container is placed just inside the radiation area [2]wholly.

See Fig. 3. Fig. 3(a) is a part of the factual image and Fig. 3(b) is the corresponding part of the transformed image.We measure the four CRT heights tagged with white circles from top to bottom in the two images, respectively. InFig. 3(a), the heights are 52,51,44,43 in pixels. Because the CRTs are of identical sizes, Fig. 3(a) is obviously distorted.

In Fig. 3(b), the heights are 52,51,49,48, the results in Fig. 3(b) are more approximate to the real data in the ideal arc.The effect of the rectification is apparent. In fact, even in the ideal image, sizes of identical objects placed in a straightline cannot be the same due to the differences between line and arc, so the results approximating arithmetical

progression are reasonable. And such gradual small changes do not interfere much with the perception of human eyes.

4. Conclusion

We propose an algorithm of geometrical rectification for MLCIS Images. With it, we can convert the factual image to

an image under ideal conditions. The results show the algorithm is effective and the dimension distortions of the objectsare rectified to normal scales.

H. Jin et al. / Applied Radiation and Isotopes 55 (2001) 793–797796

References

Wang, J., 1996. Large container radiographic inspection system. Mod. Phy. Knowledge V 8 (5), 20 (in Chinese).

World Health Organization, 1990. Food safety aspects relating to the application of X-ray surveillance equipment: Memorandum from

a WHO meeting. Bull. World Health Organization, 68 (3), 297–301.

Wang, J., Radiograph theory and general design parameter of Large Container Inspection System. Senior conference for progressing

of industrial radiology imaging and biomedical imaging, Beijing, 1998 (in Chinese).

Jin, H., Research on images of mobile container inspection system. Project Paper, Tsinghua University, 1999 (in Chinese).

Fig. 3. Experiment on practical MLCIS system. L=6571mm, vertical y is 65.98in angle, and horizontal y is 24.18; vertical a is 31.78 inangle, and horizontal a is 10.48; vertical n is 320, and horizontal n is 112. (a) The factual image. Heights of the four CRTs tagged with

white circles are 52,51,44,43 pixels from top to bottom. (b) The image after geometrical rectification. Heights of the four CRTs tagged

with white circles are 52,51,49,48 pixels from top to bottom.

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