Plasma and Fusion Research: Rapid Communications

Volume 19, 1201014 (2024)

A Method to Analyze Plasma Images
Using Modified Fourier-Bessel Functions
Yuki NISHIMURA, Akihide FUJISAWA1) , Yoshihiko NAGASHIMA1) , Chanho MOON1) ,Takashi NISHIZAWA1) , Tatsuya KOBAYASHI1,2) , Akihiro SHIMIZU1,2) , Tokihiko TOKUZAWA1,2) ,Takeshi IDO1,2) , Daiki NISHIMURA and Taiki KOBAYASHIInterdisciplinary Graduate School of Engineering Science, Kyushu University,
6-1 Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan1)
Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan2)
National Institute for Fusion Science, Toki 509-5202, Japan(Received 11 January 2024 / Accepted 24 January 2024)
A new method is proposed to analyze plasma image of tomography using modified Fourier-Bessel Functions(FBF) instead of the original FBF analysis. The application of the method to an assumed plasma image shows thatthe diﬀerence between the original and the fitting image is improved considerably to that of FBF, owing to giving
a better fitting inside the plasma and eliminating the ghost values outside the plasma, without any disadvantages
in analysis of plasma structures and patterns.
c 2024 The Japan Society of Plasma Science and Nuclear Fusion Research⃝Keywords: plasma, turbulence, tomography, Fourier-Bessel function, Modified Fourier-Bessel Function
DOI: 10.1585/pfr.19.1201014
The entire cross-section image of local plasma emission is successfully reconstructed for turbulence studies
in PANTA (Plasma Assembly for Nonlinear Turbulence
Analysis), using tomography with an algorithm, Maximum
Likelihood Expectation Maximization [1]. The FourierBessel functions (FBF) expansions, amongst various methods [2], are used to characterize global structure of steadystate and coherent fluctuation patterns [3, 4]. However, theFBF images often show ghost values outside of the plasma,
giving a wrong and deceptive impression where no emission is expected. Moreover, the internal values of plasma,
which are practically accurate, may be aﬀected in the fitting procedure to try to eliminate the external value witha finite number of the bases. The paper proposes a simple method using modified FBFs to give a better fitting tosolve the problem.
In the analysis of structure and pattern, the tomography image of plasma is expanded or fitted with FBFs as∑ϵFBF (r, θ) =α0,n ϕ0,n (r, θ)n∑∑{}s+αm,n ϕcm,n (r, θ) + βm,n ϕm,n(r, θ) ,m(1)nswhere ϕcm,n (r, θ) and ϕm,n(r, θ) are basis functions andcs
ϕm,n (r, θ) = Jm (kmn r)cos(mθ), ϕm,n(r, θ) = Jm (kmn r)sin(mθ),
with Jm (r), kmn , and m being the m-th order Bessel function, n-th value to satisfy Jm (kmn L) = 0 and azimuthalmode number, respectively. The ghost values emerge outside the plasma in the fitting of a finite number of FBFbases.

author’s e-mail: nishimura.yuki@riam.kyushu-u.ac.jp

The modified bases are defined as
M−FBFFBFψm,n(r, θ) = M(r)ϕm,n(r, θ),

where the modification function M(r) is written as( r − r )}1{0M(r) =1 − tanh,
2
∆r(2)

(3)

where r0 and ∆r represent the modification parameters (position and width), respectively. The fitting using the modified functions is expected to keep the inside value almost
the same, however, eliminates the outside values of theplasma or remove the ghost values. In addition, the modification should provide the same advantageous if the original function is Zernike polynomials used in the Cormack
inversion [2], although the modified functions lose the orthogonal and complete properties that the original functions possess. Both expansion coeﬃcients of FBF and MFBF expansion are determined by minimizing the following residual using the least square fitting,∑ {}2χ2 =ϵobs (r j , θ j ) − ϵFIT (r j , θ j ) ,(4)jwhere ϵobs (r j , θ j ) is the tomography image of plasma emission, with ϵFIT (r j , θ j ) being the one expanded with FBF orM-FBF. In the case of M-FBF expansion, an extra process
is necessary to determine the position r0 and width ∆r.
A comparison between the fittings of FBF and MFBF is made for an assumed plasma image of tomography(or original image) shown in Fig. 1 (a). The radius of theplasma image and the square region that the tomography
covers are assumed to be 5 cm, similarly in PANTA, and1201014-1

c 2024 The Japan Society of Plasma⃝Science and Nuclear Fusion ResearchPlasma and Fusion Research: Rapid Communications

Volume 19, 1201014 (2024)

Fig. 3 (a) Comparison in radial integrated diﬀerence, (b) difference and similarity of azimuthal modal structure obtained with FBF and M-FBF, and the inset shows the reconstructed azimuthal patterns of m = 3 with FBFs andM-FBFs.

Fig. 1 (a) An assumed plasma image of tomography, (b) the differences along the white line in (a), (c) diﬀerence between the original and FBF images, and (d) diﬀerence
between the original and M-FBF. The circles in figuresindicate the plasma region.

Fig. 2 The residuals as a function of the width for the three casesof the position r0 = 3.6, 4.0 and 4.4 cm, with the residualsof (A) FBF and (B) simply tanh-multiplied FBF.

16 cm × 16 cm. Here, the image is assumed to have noemission outside of the plasma since no significant emission is observed there in standard operations. The otherfigures show the diﬀerences between the original and thefitted images using both 43 bases of FBFs and M-FBFs :the number of the bases is selected to give the minimum tothe Akaike information criterion [3, 4]. The figures showthat the diﬀerence of M-FBF is much smaller than that ofFBF, and the ghost values in FBF are eliminated in the MFBF. In addition, the oscillations seen in plasma edge ofFBF case disappear and turn smooth in the case of M-FBF.
In the presented M-FBF fitting in Fig. 1, the modification parameters are selected to minimize the residual,Eq. (4): the parameters are r0 = 4.0 cm and ∆r = 0.8 cm.
Figure 2 shows the residuals as a function of the width forthree positions, r0 = 3.6, 4.0 and 4.4 cm. The comparison
indicates that the residuals are much smaller than those ofFBF. For reference, the dashed line shows the residual between the original and the FBF image simply multiplied byM(r) with the position set as r0 = 5.0 cm. Even in this case,the residual is smaller than the original when the width issuﬃciently small (∆r < 0.65 cm) since it eliminates theghost values outside.
Figure 3 shows the details of the radially integrated
diﬀerence between the original and the fitting profiles using the FBF and M-FBF. Here, the radially integrated difference is defined as√∫(5)
D(r) ={ϵobs (r, θ) − ϵFIT (r, θ)}2 dS .
It is obvious that the radially integrated diﬀerence for MFBF is roughly a few times smaller than that of FBF, eveninside of the plasma. Although it is a matter of course,
in addition, the diﬀerence increases monotonically in theFBF case due to the ghost values, while that in the M-FBF
becomes constant outside the plasma. On the other hand,Fig. 3 (b) shows the modal diﬀerence and the pattern similarity between FBF and M-FBF for each azimuthal mode.
The modal diﬀerence δ, and similarity Cm , are defined asv∫t
{ϵ (r, θ) − ϵ M−FBF (r, θ)}2 dS
inside FBF∫δ=,(6)ϵ 2 (r, θ)dS
inside FBF∫

Cm = √∫ϵFBF (r, θ)ϵ M−FBF (r, θ)dS.
√∫22ϵFBF(r, θ)dS
ϵ M−FBF(r, θ)dS(7)As shown in Fig. 3 (b), the azimuthal mode patterns are almost the same; the modal diﬀerence is less than 1.4 % to
the maximum values of the modes until m = 3 and the similarity is almost 1 for every mode. Thus, in the analysis of
the modal pattern, both expansions should give the same
results within experimental errors.
Finally, a method is proposed to analyze plasma images using M-FBFs. The results demonstrate the advantages of the method: the residual of M-FBF case is improved considerably to that of FBF, owing to giving a better fitting inside the plasma and eliminating the ghost values outside the plasma, although both methods provide thesame capability for structure and modal pattern analysis.

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Plasma and Fusion Research: Rapid Communications

Volume 19, 1201014 (2024)

This work is supported by JSPS KAKENHI (Grant
Numbers. JP22H00120, JP22K03576 and JP21K13898).

(2016).
[2] A.M. Cormack, J. Appl. Phys. 35, 2908 (1964).
[3] K. Yamasaki et al., Rev. Sci. Instrum. 88, 093507 (2017).
[4] Y. Nagayama et al., J. Appl. Phys. 62, 2702 (1987).

[1] A. Fujisawa et al., Plasma Phys. Control. Fusion 58, 025005

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