Modified Phillips–Tikhonov regularization for plasma tomography Seung Hun Lee a , Junghee Kim a,1 , J.H. Lee b , Wonho Choe a, * a Department of Physics, Korea Advanced Institute of Science and Technology, 335 Gwahangro, Yuseong-gu, Daejeon 305-701, Republic of Korea b Research Institute of Basic Sciences, Chung Nam National University, Daehangro 79, Yuseong-gu, Daejeon 305-764, Republic of Korea article info Article history: Received 17 June 2009 Received in revised form 14 October 2009 Accepted 16 October 2009 Available online xxxx PACS: 52.70.m 52.70.La Keyword: Tokamak plasma tomography diagnostic KSTAR abstract The tomography has been used as a powerful diagnostic method for visualizing cross-sectional images in plasma research. Without any biasing data, the Phillips–Tikhonov (P–T) regularization method was attempted in this work to reconstruct cross-sectional phantom images of the plasma by minimizing the gradient between adjacent pixel data. A comparison of the tomographic reconstructions of the cross-sectional images similar to the tokamak plasmas by the P–T method and the maximum entropy (ME) method showed that the P–T method produced more accurate results. In addition, the P–T method was modified by adding an iteration procedure with a second-order correction to improve the accuracy of the reconstruction for noisy line-integrated data. Tomographic reconstructions in the presence of Gauss- ian-distributed random noise demonstrated that the modified P–T method with only several iterations significantly reduced the mean reconstruction error by (30–50)% of the original root-mean-square error caused by the random noise. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction The tomography has been a useful diagnostic for plasmas. In tokamak plasma research, for example, the soft X-ray tomography has been used for investigating plasma shape and physical activi- ties occurring inside the plasmas [1–3]. Independent, especially magnetics free, information on a plasma image is especially valu- able for the plasma core, where uncertainties of magnetic equilib- rium reconstructions are large. The X-ray tomography also allows access to the spatio-temporal transient phenomena due to MHD activities [4], which play an important role in plasma confinement. The problem consists of determining the distribution of the soft X- ray emissivity in a poloidal cross-section of the plasma from a number of line-integrated measurements [5]. A set of pinhole ‘cameras’, each having several X-ray detectors, is placed in a poloi- dal plane at the same toroidal position of the vacuum vessel. The inversion of the line-integrated data to obtain a two-dimen- sional emissivity distribution is in general an ill-posed problem [6]. In order to effectively reconstruct or restore an image from the spa- tially-limited experimentally-acquired data, several inversion algorithms have been developed over the years. In this paper, the Phillips–Tikhonov (P–T) regularization algorithm [7] with the second-order correction for better random noise treatment is attempted to reconstruct shaped tokamak plasma phantoms, and the results are compared with those by the maximum entropy (ME) algorithm. In Section 2, the modified P–T reconstruction algorithm is intro- duced, which was newly developed for KSTAR-like images. Section 3 describes numerical test results that were carried out to evaluate the performance of the inversion algorithm, and the results are compared with those by the ME algorithm. This is followed by dis- cussions and conclusions in Section 4. 2. Tomography reconstruction method 2.1. Phillips–Tikhonov regularization solution In the analysis, two assumptions are made to simplify the reconstruction problem: first, the cross-sectional image is discrete and is treated as a collection of two-dimensional pixels, and, sec- ond, each pixel has no drastically different value compared to its neighboring pixels. The measured line-integrated data represented as a column vector f (M  1 matrix) are related with the desired local pixel data represented as a column vector g (N  1 matrix) through the following equation: f ¼ Wg; ð1Þ where M and N are the number of detectors and the number of pix- els, respectively, of a tomography system. The weight matrix of the system W consists of w ij , which is the contribution factor of the jth pixel to the ith detector. It is an M  N matrix which has no inverse matrix. Since solving Eq. (1) is a highly ill-posed problem because 1567-1739/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2009.10.017 * Corresponding author. E-mail address: wchoe@kaist.ac.kr (Wonho Choe). 1 Present address: ITER Organization, CS 90 046, F-13067 Saint Paul lez Durance, France. Current Applied Physics xxx (2009) xxx–xxx Contents lists available at ScienceDirect Current Applied Physics journal homepage: www.elsevier.com/locate/cap ARTICLE IN PRESS Please cite this article in press as: S.H. Lee et al., Modified Phillips–Tikhonov regularization for plasma tomography, Curr. Appl. Phys. (2009), doi:10.1016/ j.cap.2009.10.017