26aE10P	 
直線プラズマにおける乱流解明を目指した特異値分解法を用いたトモグラフィアルゴリズムの開発
Development of SVD Algorithm for Turbulence Tomography in Linear Plasma
柴田遼介	 1,藤澤彰英	 2,3,永島芳彦	 2,3,濱本直紀 1,福永史樹 1,江藤健太 1,稲垣滋 2,3	 ,糟谷直宏 2,3,	 
小菅佑輔 5,佐々木真 2,3,山田琢磨 3,4,伊藤早苗 2,3,伊藤公孝 3,6	 
1
Ryosuke SHIBATA , Akihide Fujisawa2,3, Yoshihiko NAGASHIMA2,3,Naoki HAMAMOTO1,Fumiki FUKUNAGA1 et al.
1 九大総理工,2 九大応力研,3 九大極限プラズマ研究連携セ,4 九大基幹教育院 5 九大高等研究院,6 核融合研
1 Interdisciplinary Graduate School of Engineering Sciences Kyushu Univ.,
2 RIAM Kyushu Univ.,3 RCPT kyushu Univ.,4 FAS kyushu Univ., 5 IAS kyushu Univ.,6 NIFS
Plasma turbulence is one of the important issues to bestudied for realizing a nuclear fusion reactor. It is because
plasma turbulence degrades the plasma confinement
property greatly. At present, for further understanding of theplasma turbulence, it is necessary to observe the turbulence
not only in local region of plasma but also over the entire
plasma. Tomography is one of possible diagnostics to makesuch measurements in sufficient spatial and temporal
resolution.
We have constructed a prototype of such tomographysystem to measure plasma local turbulence for a linear
cylindrical device, Plasma Assembly for Nonlinear
Turbulence Analysis (PANTA) in Kyushu University. We
are now developing a tomography algorithm using Singular
Value Decomposition (SVD) in addition to the alreadydeveloped ones, Maximum Likelihood-Expectation Method(MLEM), Algebraic Reconstruction Technique (ART),
Fourier-Bessel series and Cormack expansion.
The purpose of our tomography is to obtain plasma local
emissivity from a set of the line-integrated values of theemission. The relation between the line-integrated
values  𝑔  ! and local emission 𝜀! is expressed as𝑔 = ℎ𝜀(1)
where h is a matrix composed of the areas of theline-of-sights included in the grids as is shown in Fig.1.
The emission can be obtained with a least square method tominimize the modified 𝑥 ! function, with assuming ∥  ∥being the norm,                    𝛷 =∥ 𝑔 − ℎ𝜀 ∥! /𝑀 +⋎∥ 𝐶𝜀 ∥!
(2)where ⋎  is the regularization parameter and C is selected tobe the identify matrix here. The solution is expressed as  𝜀 = (ℎ! ℎ + 𝑀𝛶𝐶 ! 𝐶)!! ℎ! 𝑔(3)The solution can be rewritten using the SVD, which
factorizes a matrix to three elemental products, andℎ𝐶 !! = 𝑈𝑊𝑉 ! where U and V represent orthogonal
matrices, and W represents diagonal matrix composed ofsingular values.
!∙!!
𝜀= !  𝐶 !! 𝑣!   , 𝑤! (𝛾) = 1/(1 + 𝑀𝛾𝜎!!! )            (4)!!! 𝑤! (𝛶)detectors are assumed to belocated at 16 different
azimuthal angle positions with
101 channels for eachdirection, respectively. In theoriginal image, there is a peakon the left-top to examine
whethertheazimuthallyasymmetric structure can bereconstructed or not. The
results show that the sharp Fig.1. Schematic view oftomography condition
peak can be reconstructed on
both methods. In this case, thetotal error of the SVD is smaller than that of the ML-EM.
In summary, tomography algorithm based on SVD isbeing developed, and its application to an assumed emission
data well reproduces the original image well. As a futurework, the best regularization parameter Υ also should beselected using GCV criteria [1,2], and comparisons between
the SVD and other already developed algorithms should bemade in aspects of reproducibility, computing speed, and soon. In this conference, we report the results of tomographyreconstruction using the SVD algorithm, and its realapplication on the experimental data obtained in PANTA.
(a) Original

Residual error	 (b)SVD(d)(c)ML-EM(e)!!

where 𝜎! represents i-th singular value, 𝑢!   and 𝑣! are i-th
column vector of matrices U and V.	  Figure 2 shows an example of the reconstructed images
using SVD method and MLEM for comparison, together
with residual error and MAE (Mean Absolute Error). The
definition of MAE is! !×!
𝑀𝐴𝐸 =(5)!!! 𝜀! − 𝜀! 	  	  	 !

where 𝜀 represents the original value. Note that here we set
Υ = 10!! . The assumption for the calculation is that theplasma is observed with totally 1616 detectors. TheMAE (10!! )

2.66

2.77

Fig.2. Reconstructed images and residual error by means of SVD and MLEM
(a) Original image, (b) reconstructed image by SVD, (c)reconstructed image byML-EM, (d)residual error image for (b), and residual error for (c), respectively.

References[1] N. Iwama Computed Tomography for Plasma
Measurement (1992)[2] N. Iwama and S. Ohdachi Numerical Methods ofTomography for Plasma Diagnostics (2006)