Nuclear Instruments and Methods in Physics Research A 505 (2003) 552–555

A correction algorithm for detector nonuniformity incomputed tomography

Shaohua Sun*, Li Zhang, Wenhuan Gao, Zhiqiang Chen, Dan Xu

Department of Engineering Physics, Tsinghua University, Beijing 100084, PR China

Abstract

In actual computed tomography systems, detector nonuniformity can result in obvious concentric circles in

reconstructed images and it is very important to correct it. Although nonuniformity can be corrected by current

methods, residual nonuniformity still exists after correction. To resolve this problem, we have designed an effective

correction algorithm based on the geometrical relationships in tomographic data. The proposed method consists of two

steps, edge detection and correction. In the first step, detectors to be corrected are located using multiscale wavelet

analysis. In the second step correction is implemented for these detectors. This algorithm is implemented after ordinary

correction and can correct residual nonuniformity effectively. Promising results are achieved in our experiments.

r 2003 Elsevier Science B.V. All rights reserved.

Keywords: Nonuniformity correction; Computed tomography; Detector nonuniformity

1. Introduction

Detector nonuniformity can result in obviousartifacts in tomographic images [1,2]. To correctthis nonuniformity, some methods have beenproposed [2–5]. Although nonuniformity can belargely corrected by current methods [2–5], resi-dual nonuniformity still exists because of thenonlinear property of detector nonuniformity [6].In computed tomography (CT) systems, suchresidual nonuniformity can result in obviousconcentric circles in the reconstructed image andit is very important to correct it. To resolve thisproblem, we have designed an effective correctionalgorithm based on the geometrical relationships

in tomographic data. This method consists of twosteps, edge detection and correction. It is imple-mented after ordinary correction and can effec-tively correct for residual nonuniformity intomographic data. Promising results are achievedin our experiments.In Section 2, we will discuss this algorithm in

detail. Experimental results are given in Section 3and conclusions and discussion are given inSection 4. From the experiment, we can see thatresidual nonuniformity in tomographic data canbe corrected effectively using this method.

2. Correction algorithm

This method consists of two steps, edge detec-tion and correction. In the first step, we usemultiscale wavelet analysis to detect edges

*Corresponding author. Tel.: +86-10-6278-0909; fax: +86-

10-6278-8896.

E-mail address: sunshaohuath@yahoo.com.cn (S. Sun).

0168-9002/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0168-9002(03)01145-8

resulting from residual nonuniformity, whichspecifies detectors to be corrected. In the secondstep, correction is implemented for those detectorsaccording to the geometrical relationships given inSection 2.2.

2.1. Edge detection

In the first step, we do edge detection. Intomographic data, detector nonuniformity resultsin vertical edges. After ordinary correction, suchedges are not very obvious but still exist. Multi-scale wavelet analysis can distinguish subtle edgesfrom noise [7,8], so we use it to detect such edges.From the regions specified by detected edges, wecan identify detectors to be corrected. This stepcan be described as:

1. For each group of detector data, implementwavelet transform at multiple scales.

2. For every group of data, find all local maximumand minimum values in the wavelet coefficientsat every scale but disregard the maximum andminimum values whose amplitudes are largerthan a threshold. Those maximum and mini-mum values are normal edges in projectiondata.

3. At each scale, add the maxima and minimacorresponding to the same detector together. Ifthe sum is larger than an upper threshold, setthe detector as 1. If less than a down threshold,set the detector as �1. Otherwise, set thedetector as 0.

4. Then for each detector, add the 1s, �1s and 0sat each scale together. If the sum’s absolutevalue is larger than a threshold, then thisdetector is regarded as edge.

5. For each group of data, if neighboring detectorsare regarded as edges, set the most left one as 1and the most right one as �1 and otherdetectors as 0. Edges of abnormal detectorsare now thinned to one detector and every pairof 1 and �1 composes a narrow strip, whichshould be corrected.

After doing this step, we get edges of the stripswithin which are detectors to be corrected. For

every detector in those strips, we perform step twoto correct the residual nonuniformity.

2.2. Correction

In the second step, we implement correctionaccording to the following geometrical relation-ship in a narrow strip in the tomographic data. Wefirst deduce this relationship.As shown in Fig. 1, O is scanning center, rk1 is

the distance from O to the line from source todetector k1; mðx; yÞ is absorption coefficient ofpoint pðx; yÞ and mðk1; aÞ is the projection ofdetector k1 at angle a: So mðk1; aÞ is the integralof absorption coefficient along the line fromsource to detector k1 [9]. Add all mðk1; aÞ at everya; we get

Sumk1 ¼Xa

mðk1; aÞ ¼Z

S1

mðx; yÞ dx dy ð1Þ

where, S1 ¼ fpðx; yÞjjjpðx; yÞ � OjjXr1g: Sumk2 canbe got similarly. Because the strip is narrow, that isdetector k1 is close to detector k2; the area of thecircular ring DS can be written as 2 pr2ðr1 � r2Þapproximately. So

Sumk1 � Sumk2 ¼ZDS

mðx; yÞdx dy

E 2 p %mDSrk2 ðrk1 � rk2Þ ð2Þ

where, %mDS is the average absorption coefficientwithin DS:

Fig. 1. Geometrical relationship in tomographic data.

S. Sun et al. / Nuclear Instruments and Methods in Physics Research A 505 (2003) 552–555 553

As shown in Fig. 2, in a narrow strip whose leftedge is k1; right edge is k2; there is a detector k tobe corrected. Because the strip is narrow, detectork1 is close to detector k2; and %mDS1

; %mDS2and

%mDS1þDS2are approximately the same. Denote the

corrected value of mðk; aÞ as mðk; aÞ: Then usingthe relationship between detector k1 and detectork; we get a correction factor ak: Using therelationship between detector k2and detector k;we get another correction factor bk: Take ck as thearithmetic mean of ak and bk; then we get the finalcorrection factor ck to compute mðk; aÞ: Thewhole process can be written into four equations:

ðakSumk � Sumk1ÞðSumk2 � Sumk1 Þ

Eðrk1 � rkÞðrk1 � rk2Þ

ð3Þ

ðbkSumk � Sumk2ÞðSumk2 � Sumk1 Þ

Eðrk2 � rkÞðrk1 � rk2Þ

ð4Þ

ck ¼ akðrk � rk2 Þ=ðrk1 � rk2 Þ

þ bkðrk1 � rkÞ=ðrk1 � rk2Þ ð5Þ

mðk; aÞ ¼ ckmðk; aÞ: ð6Þ

For every detector in every strip, implement theprocess specified by Eqs. (3–6). After doing so, thecorrection process is completed.

3. Actual experiment

An experiment was done to verify this algo-rithm. Scanned object is a tyre whose radius isabout 1000mm. Total effective detector number is675. Effective scan number is 939. Acceleratorenergy is 6MeV.We first implemented a nonlinear correction

method [2] to precorrect the tomographic data.After precorrection, most nonuniformity wascorrected but residual nonuniformity still existedand we could still detect edges resulting fromresidual nonuniformity in the corrected tomo-graphic data. Fig. 3 illustrates such edges.Then, we implemented the proposed method to

correct the residual nonuniformity. In the firststep, edge detection, we used ‘db4’ waveletfunction to do multiscale analysis at the scales of4, 8, and 12. Fig. 4 illustrates 1s and –1s found inthis step.Then we performed the second step correction.

We first grouped neighboring 1s and �1s intopairs and disregarded the isolated ones. Then wecorrected every detector between those pairsaccording to Eqs. (3–6).Fig. 5 is the comparison of the reconstructed

images displayed in high contrast before and aftercorrection by this proposed method. Fig. 5a is theimage before correction. We can still noticeconcentric circles resulted from residual nonuni-formity. After correction, as shown in Fig. 5b,most concentric circles are corrected and satisfyingresults are obtained.

Fig. 2. Geometrical relationship within a narrow strip in

tomographic data.

Fig. 3. Edges resulted from residual nonuniformity.

S. Sun et al. / Nuclear Instruments and Methods in Physics Research A 505 (2003) 552–555554

4. Conclusions and discussion

In this paper we present a new correctionalgorithm to correct residual detector nonunifor-mity in CT system based on the geometricalrelationships in a narrow strip of tomographicdata. When this method is employed afterstandard correction methods, it can correct theresidual nonuniformity quite effectively.This method is most effective when used after

precorrection. If used directly to tomographicdata without any precorrection, it will be lesseffective because the strips specified by thedetected edges may not be narrow. In thederivation in Section 2.2, we assume that the stripis narrow. If this assumption is not satisfied, errorswill arise.

Acknowledgements

This work was supported by the NationalNatural Science Foundation of China (NSFC).Project number is 10135040.

References

[1] A.F. Milton, F.R. Barone, M.R. Kruer, Opt. Eng. 24 (1985)

855.

[2] Yinbing Liu, Studies on the operation & inspection and

image acquisition & processing of MCIS, Ph.D. Disserta-

tion, Tsinghua University, China, 2002.

[3] J.G. Harris, Y.-M. Chiang, IEEE Trans. Image Process. 8

(8) (1999) 1148.

[4] D.A. Scribneretal, Adaptive retina like preprocessing for

imaging detector arrays, Proceedings of the IEEE Interna-

tional Conference on Neural Networks, San Francisco, CA,

March 28–April 1, 1993, pp. 1955–1960.

[5] M. Schulz, L. Caldwell, Proc. SPIE 2470 (1995) 200.

[6] Liu Huitong, Wang Qi, Chen Sihai, Yi Xinjian, Analysis of

the residual error after nonuniformity correction for

infrared focal plane array, Conference Digest, 2000,

Infrared and Millimeter Waves, 25th International Con-

ference, Beijing, P.R. China, Sept. 2000, pp. 213–214.

[7] F. Yang, Engineering Analysis and Application of Wavelet

Transform, Science Press, Moscow, 1999.

[8] Xiaoping Guan, Zequn Guan, Edge detection of high

resolution remote sensing imagery using wavelet, Proceedings

of International Conferences on Info-Tech and Info-Net,

Beijing, P.R. China, Oct. 2001, Vol. 1, 2001, pp. 302–307.

[9] G.T. Herman, Image Reconstruction from Projections: the

Fundamentals of Computerized Tomography, Academic

Press, New York, 1980.

Fig. 4. 1s and �1s found by wavelet multiscale analysis.

Fig. 5. Reconstructed images before (left) and after (right)

corrected by proposed method.

S. Sun et al. / Nuclear Instruments and Methods in Physics Research A 505 (2003) 552–555 555