Optical boundary reconstruction of tokamak plasmas forfeedback control of plasma position and shapeCitation for published version (APA):Hommen, G., Baar, de, M. R., Nuij, P. W. J. M., McArdle, G., Akers, R., & Steinbuch, M. (2010). Opticalboundary reconstruction of tokamak plasmas for feedback control of plasma position and shape. Review ofScientific Instruments, 81(11), 113504-1/13. https://doi.org/10.1063/1.3499219

DOI:10.1063/1.3499219

Document status and date:Published: 01/01/2010

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Optical boundary reconstruction of tokamak plasmas for feedback controlof plasma position and shape

G. Hommen,1,2 M. de Baar,1,2 P. Nuij,1 G. McArdle,3 R. Akers,3 and M. Steinbuch1

1Control Systems Technology Group, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven,The Netherlands2FOM Institute for Plasma Physics “Rijnhuizen,” Association EURATOM-FOM, Trilateral Euregio Cluster,P.O. Box 1207, 3430 BE Nieuwegein, The Netherlands3EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB,United Kingdom

�Received 22 June 2010; accepted 13 September 2010; published online 10 November 2010�

A new diagnostic is developed to reconstruct the plasma boundary using visible wavelength images.Exploiting the plasma’s edge localized and toroidally symmetric emission profile, a new coordinatetransform is presented to reconstruct the plasma boundary from a poloidal view image. The plasmaboundary reconstruction is implemented in MATLAB and applied to camera images of Mega-AmpereSpherical Tokamak discharges. The optically reconstructed plasma boundaries are compared tomagnetic reconstructions from the offline reconstruction code EFIT, showing very good qualitativeand quantitative agreement. Average errors are within 2 cm and correlation is high. In the currentsoftware implementation, plasma boundary reconstruction from a single image takes 3 ms. Theapplicability and system requirements of the new optical boundary reconstruction, called OFIT, foruse in both feedback control of plasma position and shape and in offline reconstruction tools arediscussed. © 2010 American Institute of Physics. �doi:10.1063/1.3499219�

I. MEASUREMENT AND CONTROL OF TOKAMAKPLASMA POSITION AND SHAPE

Reconstruction of the tokamak plasma boundary is usedfor both real-time control applications and offline analysistools. Control of plasma position and shape in tokamaks isused to optimize plasma performance, access specific physicsregimes, and protect first-wall components.1–4 Offline recon-struction tools such as EFIT �Ref. 5� are used for postdis-charge analysis using a wide range of diagnostics and wouldgreatly benefit from the determination of an accurate plasmaboundary.

In present day tokamaks, magnetic measurements from aset of coils located around the vacuum vessel are used tomeasure flux variations. After integration, the fluxes are in-terpreted in terms of the plasma centroid position and veloc-ity, the plasma boundary as defined by the last closed fluxsurface �LCFS�, and the location of the x-point�s� and thestrike points. The full magnetic topology of flux surfacescannot be determined in detail without the use of additional,internal diagnostics such as polarimetry or the motionalStark effect.6 Simultaneous control of the current and pres-sure profile in tokamaks relies on the real-time identificationof the full magnetic topology, the reconstruction of whichwould greatly benefit from an unambiguous plasma bound-ary.

There are a number of limitations to magnetic boundaryreconstruction. Firstly, the integration of the time derivativeof the fluxes can lead to drifts in the flux measurements ontime-scales that are relevant for a reactor.7 The reconstruc-tion of the flux surfaces must take into account the passiveconducting structures of the tokamak. Often these structures,

such as the iron core in the Joint European Torus, exhibitnonlinear behavior which hampers the interpretation of mag-netic signals.8 During startup or in optimized shear dis-charges, the flux distribution is far from equilibrium andthere are strong transients. The reconstruction of the plasmaboundary using only external magnetic measurements isproblematic in such dynamic situations. Diagnostics coilsmeasuring poloidal fields are very sensitive to alignment;small misalignments can cause large disturbances from themuch stronger toroidal field.

To address these problems, an alternative plasma bound-ary diagnostic is developed using high speed camera imagesof the visible light emitted by the plasma edge. The cameradata do not suffer from drift, are intrinsically robust againstthe aforementioned transients, and provide an independentmeasurement of the optical plasma boundary. The exact re-lation between optical plasma boundary and the magneticplasma boundary, however, remains to be investigated. Thecaptured images provide information not only on the plasmaboundary but also show filament structures, edge localizedmodes �ELMs�,9 distinct confinement regimes such asL-mode and H-mode, and hot spots on plasma-facing com-ponents �PFCs�. The diagnostic could be extended to provideinformation on these plasma edge specific phenomena withhigh accuracy relative to the plasma edge and create an op-portunity for intelligent, integrated sensing and control of notonly the plasma position and shape, but also the resultingheat-fluxes to plasma-facing components. A previous workon optical plasma boundary reconstruction10 for offline ap-plications has already shown promising results, although theapplied reconstruction was only valid on the equatorial mid-plane and produced significant errors elsewhere.

REVIEW OF SCIENTIFIC INSTRUMENTS 81, 113504 �2010�

0034-6748/2010/81�11�/113504/13/$30.00 © 2010 American Institute of Physics81, 113504-1

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This paper introduces an optical plasma boundary recon-struction and, using camera images from the Mega-AmpereSpherical Tokamak �MAST�,11 compares the obtained shapeof the radiative shell to EFIT reconstructions of the magneticplasma boundary. The feasibility of the developed plasmaboundary diagnostic for use in a feedback control loop onMAST is assessed, as well as possible applications in offlineanalysis tools.

The paper is organized as follows. A new coordinatetransform for the reconstruction of the plasma boundaryfrom a poloidal view image is presented and synthetic recon-structions are shown to illustrate the properties of this trans-form in Sec. II. A MATLAB implementation of the new diag-nostic is then applied to poloidal view images of MASTdischarges and the resulting plasma boundaries are comparedto magnetic reconstructions from EFIT in Sec. III. SectionIV discusses the usability of the new diagnostic in feedbackcontrol systems of plasma position and shape and highlightsa number of applications in offline analysis tools. The mainresults are summarized in Sec. V.

II. PLASMA BOUNDARY RECONSTRUCTION USINGTWO-DIMENSIONAL-VISIBLE IMAGING

Figure 1 shows camera images of MAST plasmas inwhich the plasma edge is clearly visible. As only the plasmaedge emits a significant amount of light from a thin shell, forthe purpose of real-time plasma boundary reconstruction, theplasma edge is treated as a thin, toroidally symmetric emis-sive surface Sp that is the boundary of the plasma volume�p.

Every pixel of an image is a line integrated measurementof the light emitted by the plasma. Pixels corresponding to aline of sight that is tangent to the emissive surface Sp willshow a peak in intensity compared to neighboring sightlinesthat either cross the plasma surface or miss it altogether.

Figure 2 shows the plasma emission in the tokamak mid-plane �u ,w� and the resulting pixel intensities for horizontalsightlines. The plasma edge in image can now be identifiedas lines of peak intensity, corresponding to the set of pixelspe that have a line of sight le tangent to the plasma surfaceSp.

A. Coordinate transform: From image to tokamakcoordinates

For the plasma shape reconstruction, a new coordinatetransform is derived to reconstruct the toroidally symmetricplasma surface Sp from the set of sightlines le from a singlecamera tangent to the plasma surface Sp. The problem isdefined in two coordinate systems, a Cartesian system�u ,v ,w� and the cylindrical tokamak coordinate system�R ,Z ,��, where the origin and the v and Z axes of bothcoordinate systems coincide and the point �0,0 ,�� in toka-mak coordinates is the center of the tokamak. A projectionplane is defined as the vertical �u ,v ,0� plane. Figure 3 showsthe two coordinate systems and the projection plane. Giventhe camera location �uc ,vc ,wc� and knowledge of the projec-tion properties of the optics, the trajectories of all sightlines lare known and each has a projection �u ,v� on the projectionplane. In the ideal case of a perfect pinhole camera with itsoptical axis normal to the projection surface, the projection�u ,v� of sightlines on the projection surface is an affine scal-ing of the pixel coordinates �x ,y�. Any type of projection can

FIG. 1. Images of MAST shot no. 23344, showing �a� L-mode, �b� H-mode,and �c� ELM plasmas, respectively.

0 100 200 3000

0.2

0.4

0.6

0.8

1line integrated intensityplasma emission

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

(a) (b)

FIG. 2. �Color online� �a� Emissionprofile of the plasma in the equatorialplane and �b� resulting line integratedintensity for horizontal sightlines par-allel to the w-axis.

FIG. 3. �Color online� Cartesian and tokamak coordinates, a circularplasma, and the projection surface on the �u ,v� plane.

113504-2 Hommen et al. Rev. Sci. Instrum. 81, 113504 �2010�

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be accounted for with a suitable projection function. �u ,v�= f�x ,y�.

The sightline le of an edge pixel pe is tangent to theplasma surface but the point of tangency along the sightlineis not directly known. To find this point, the tangency isexpressed as a vector equation. The vectors introduced here-after are drawn in Fig. 4. Figure 5 also depicts the coordinatetransform, but in this image the projection plane is shiftedbackward �u ,v ,−1.5� for another look at the situation. Theprojection plane can be moved along the w axis arbitrarily,however, choosing �u ,v ,0� provides the mathematically sim-plest solution.

The plasma surface is known to be toroidally symmetric,such that at each point �R ,Z ,�� on Sp the surface is spanned

by two vectors in Cartesian coordinates �u ,v ,w�

Vt = �− sin���0

cos����, Vc = �

dR

dZcos���

1

dR

dZsin��� � ,

being the toroidal unit vector and the local conicity. Addi-tionally, the sightline vector of le is

Vl = �ue − uc

ve − vc

− wc� ,

where �ue ,ve� is the projection of le on the projection sur-face. This set of vectors does not uniquely specify the tan-gent point along a sightline, as R and Z are unknown and Vt

and Vl are not multiples. To obtain an additional vector, theset of tangent sightlines is treated as a curved sightsurface.Locally, this surface is spanned by the sightline le and thedirection of the plasma edge on the projection plane at�ue ,ve�. This allows introduction of the edge vector Ve as thedirection along the plasma edge in the projection plane

Ve = �due

dve

1

0� ,

such that the sightsurface along a sightline is spanned by Vl

and Ve. The toroidal angle �e of the tangent point on thesightline can now be obtained from the vector equation

Vt = �− sin��e�0

cos��e�� = �Vl + �Ve = ��ue − uc

ve − vc

− wc� + ��

due

dve

1

0� .

�1�

By substitution of � and �, this leads to

tan��e� =

�ue − uc� −due

dve�ve − vc�

wc. �2�

Given the sightline le and the toroidal angle �e of thetangent point, �Re ,Ze ,�e� can be found as well as the conic-ity dRe /dZe of the plasma surface at �Re ,Ze ,��. In Fig. 4, aperspective view of the plasma surface and the projectionplane shows how the points Re and Ze are defined by le and�e.

Expressing Re and Ze in terms of ue, ve, and �e

Re =uewc

wc cos �e + �ue − uc�sin �e, �3�

Ze = vc + �ve − vc��wc − Re sin �e�

wc, �4�

completes the coordinate transform. Important to note is thederivative term due /dve in Eq. �2�. This term must be deter-

FIG. 4. �Color� Perspective view of lines on projection plane �red�, sightlinele �thick black line�, tokamak coordinates R and Z �blue lines�, and tangentpoint on plasma surface �blue circle�. Also shown are the vectors used in thecoordinate transform in green.

FIG. 5. �Color� Similar to Fig. 4 but with the projection plane shifted back-ward for clarity.

113504-3 Hommen et al. Rev. Sci. Instrum. 81, 113504 �2010�

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mined either as a numerical derivative between edge pointspe or as the derivative of a smooth function fitted to the edgepoints.

Given the set of tangent sightlines le, a reconstructionof the toroidally symmetric plasma surface Sp can now bemade. The result is generally presented as a poloidal crosssection of the plasma surface.

1. Deviations from toroidal symmetry: Ripple

The assumption of toroidal symmetry of the plasma sur-face may not hold exactly; in tokamaks with a significanttoroidal field ripple, caused by the limited number of toroidalfield coils, the plasma boundary may be somewhat deformed.

It is assumed that a ripple in toroidal field results in aperiodic disturbance of the major radius of the �outer� fluxsurfaces of the form

R = R0 + �R�R0,Z�sin�N�� . �5�

Depending on the magnitude and frequency of theripple, three distinct regimes can be identified. When theperiodic deformation of the plasma boundary is small, in theorder of the thickness of the emissive layer, and the fre-quency of the ripple is high, for example, more than tenperiods per rotation, ripple is expected to broaden the edge inthe image of the plasma. This will result in a lower accuracyof the plasma boundary reconstruction.

When the deformation of the plasma boundary is muchbigger than the thickness of the emissive layer, multipleplasma edges can show up in the image. This regime is,however, not observed in any tokamak, but can be seen instellarators, which are strongly nonaxisymmetric. In this casethe toroidal surface reconstruction cannot be applied.

Limited deviations from toroidal symmetry with limitedspatial frequency can result in small displacements of theplasma edge in the image when compared to the toroidallysymmetric case. This regime can be handled by the coordi-nate transform. A toroidally symmetric plasma surface isspanned at any point by two vectors: the toroidal unit vectorVt and the local conicity Vc, as introduced earlier. A rippledplasma surface, however, will have a different orientation inthe horizontal plane and is not spanned by the toroidal unitvector Vt. To define the horizontal orientation of a rippledplasma surface, the angle � is introduced and the toroidalunit vector is replaced by the ripple vector Vripple

Vripple = �− sin���0

cos���� ,

where � is the angle between the ripple vector Vripple and thew-axis. Applying Vripple in Eqs. �1� and �2�, Eq. �2� nowprovides the horizontal orientation of the plasma surface atthe point of tangency. Using the knowledge of the rippledtoroidal field, the horizontal orientation of the plasma surface� can be determined at any point �R ,Z ,�� and thus alsoalong a sightline. Figure 6 shows the angle � along a mid-plane sightline for an arbitrary rippled plasma of the form

� = � + f rippleR2 sin�N�� . �6�

As Eq. �2� determines the horizontal orientation of theplasma surface at the point of tangency, the point of tangencyon the sightline can now be found by matching the result ofEq. �2� to the point on the sightline with the same orientation�. Once this point is found, the rippled major radius R can bemapped back to the nominal radius R0 using the angle � ofthe tangent point.

A prerequisite is that � is monotonically increasing ordecreasing along the sightline. This limits the maximummagnitude and spatial frequency of ripple that can behandled by the toroidal surface reconstruction.

B. Inverse coordinate transform: From tokamak toimage coordinates

The inverse toroidal surface reconstruction allows toconstruct the projection of the plasma edges on the projec-tion plane �u ,v ,0� for a given plasma boundary and cameralocation. The approach is the same as for the transform fromprojection plane to plasma surface. At the point �R ,Z ,��, theplasma surface Sp is spanned by the toroidal unit vector andthe conicity dR /dZ of the surface

Vt = �− sin���0

cos����, Vc = �

dRe

dZecos���

1

dRe

dZesin��� � .

For a given camera location �uc ,vc ,wc�, the sightline tangentto �Re ,Ze ,�� is sought. The direction of this sightline is ex-pressed as the vector

Vl = �ue − uc

ve − vc

− wc� .

Using Eqs. �3� and �4�, ue−uc and ve−vc can be expressed interms of Re, Ze, �e, uc, vc, and wc,

ue − uc = wcRe cos��e� − uc

wc − Re sin��e�, �7�

0.2 0.4 0.6 0.8 1 1.2−0.5

0

0.5

1

1.5

coordinate along sightline

surf

ace

orienta

tion

α

FIG. 6. �Color online� Horizontal surface orientation along sightline.

113504-4 Hommen et al. Rev. Sci. Instrum. 81, 113504 �2010�

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ve − vc = wcZe − vc

wc − Re sin��e�, �8�

in which ue, ve, and �e are unknown. Now, the sightlinevector Vl can be spanned by Vt and Vc

Vl = �Vt + �Vc, �9�

which, after rewriting, results in the expression used to find�e

uc cos��e� + wc sin��e� = Re − �Ze − vc�dRe

dZe,

which can be further simplified to

�uc2 + wc

2 sin�e + arctan uc

wc�� = Re − �Ze − vc�

dRe

dZe,

�10�

such that Eqs. �7�–�10� form the complete inverse transform.The inverse coordinate transform can be used to derive

the limits of the plasma dimensions that can be reconstructedfrom the image given the limits of view in the projectionplane and the location of the camera. The expression for thetoroidal angle �e of the tangent point on a sightline tangentto the plasma surface at �R ,Z�, Eq. �10�, has a solution onlywhen

�uc2 + wc

2 Re − �Ze − vc�dRe

dZe , �11�

implying that the camera must be located outside a cone thatfits around the plasma surface at �Re ,Ze ,dRe /dZe� to be ableto see this part of the plasma surface. Also taking into ac-count the limited viewing angle of the camera, the projectionpoints �ue ,ve� obtained from the inverse transform must lieinside the area of the projection plane viewed by the camera.The existence of the transform, i.e., the visibility in the cam-era image of the plasma edge at �Re ,Ze ,dRe /dZe�, thus de-pends on the camera location and the maximum viewingangle of the optics. This knowledge can be helpful in deter-mining the optimum camera location for an optical plasmaboundary diagnostic.

For the special case of a circular plasma cross section,the newly derived coordinate transform is mathematicallyvalidated in Appendix A.

C. Example of shape reconstruction from an imageand image synthesis from a plasma shape

To illustrate the mechanism of the derived coordinatetransform, two examples are treated: the synthesis of a cam-era image showing a circular plasma and the plasma shapereconstruction from a double null diverter �DND� plasmaimage.

For the image synthesis, a circular plasma cross sectionwith a major radius R0=2.25 m and a minor radius r=0.4 m is used, similar to plasmas generated in the ToreSupra tokamak12 with reduced minor radius. The camera lo-cation is chosen to be �uc ,vc ,wc�= �−2,−0.3,2.5�; a locationthat provides a poloidal view from 30 cm below equatorialheight, viewing the plasma around toroidal angle �=−.

The plasma shape in the poloidal plane �R ,Z� can nowbe expressed in terms of the poloidal angle �

Re = R0 + r cos��� ,

Ze = r sin��� , �12�

which allows writing dR /dZ as

dRe

dZe=

dRe

d�dZe

d��−1

=− r sin���r cos���

= − tan��� . �13�

Using this description of the plasma edge and applying thetransform derived in Sec. II B results in the projection of theplasma edge on the projection plane. In this case, the projec-tion plane is treated directly as the image, effectively takingthe camera projection properties out of the problem. Figure7�a� shows the resulting plasma edges in the projectionplane.

The plasma edge in the projection plane is not a circle.Instead, the inner and outer halves of the plasma circumfer-ence result in two separate curves. For the top and bottomof the circular plasma shape, �� /2 and ��3 /2, thecondition of Eq. �11� is not met for the given cameralocation and no projection exists. More generally, Eq. �11�shows any horizontal part of the plasma circumference,where �dRe /dZe�−1=0 is only visible when Ze=vc. In prac-tice, the top and bottom of a circular plasma circumferencetherefore typically do not appear in the image.

The projection of the inner plasma edge in Fig. 7�a� isdiscontinuous, the projection around the midplane is ap-proximately circular while the top, and bottom of the plasmaappear as two tails stretching out around the torus. This ef-fect also appears in camera images of circular plasmas suchas the Tore Supra camera image shown in Fig. 8. In Fig. 9,

−3 −2 −1−1

−0.5

0

0.5

1

u [m]

v[m

]

projection plane

1.8 2 2.2 2.4 2.6 2.8

−0.4

−0.2

0

0.2

0.4

0.6

R [m]

Z[m

]

poloidal plane

(a) (b)

FIG. 7. �a� Projection of circular plasma with minor radius r=0.4. �b�Plasma circumference, gray where no projection exists for the applied cam-era position.

FIG. 8. �Color online� Image of detached circular plasma in Tore Supra.

113504-5 Hommen et al. Rev. Sci. Instrum. 81, 113504 �2010�

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the sightlines tangent to the inner plasma surface are drawn.Around the equatorial plane, the toroidal angle of tangency�e is approximately constant, resulting in a projection thatis circular. The strong rise in �e near the top and bottom ofthe plasma surface, however, results in projection frompoints around the back half of the torus. The top and bottomof the plasma surface are near-horizontal and are at alarge horizontal distance from the camera, meaning the term�Z−vc�dR /dZ in Eq. �8� is large and, as a result, �e will belarge.

For the shape reconstruction of the DND plasma from acamera image, the projected plasma edges are presented astwo curves on the �u ,v ,0� projection plane as shown in Fig.10�a�. The camera dependent projection from pixel coordi-nates to projection surface is not regarded. For simplicity, theprojected plasma edges are described by two second orderpolynomials of the form

u = a0 + a2v2, �14�

such that

du

dv= 2a2v . �15�

The coefficients of the inner and outer edge are chosensuch that the two lines cross at v= �1.4 m and DND plasma

image is obtained. Applying the coordinate transform pre-sented in Sec. II A to these curves results in the poloidalplasma cross section as shown in Fig. 10�b�.

The coordinate transform results in different deforma-tions of the resulting inner and outer plasma edges becauseof the opposite signs of the du /dv terms. While both curvesin the projection plane are defined in the region −1.7 v 1.7, a different region is spanned in the Z coordinate.

The transform of two points at �ue ,ve� with differentvalues of du /dv results in two different points �Re ,Ze� in thepoloidal plane. The intersections of the curves in the projec-tion plane therefore do not correspond to the intersections ofthe curves in the poloidal plane, implying that the crossing ofthe plasma edges in a camera image does not relate to theplasma x-point! The sightline corresponding to the intersec-tion point on the projection plane is tangent to both the innerand the outer plasma surface but at different distances fromthe camera, thus at different points �Re ,Ze�. To illustrate this,the trajectory of this sightline in the poloidal plane is drawnin black in Fig. 10�b�. This results in a curve that is tangentto the inner and the outer plasma edge, but at differentpoints.

The optically reconstructed x-points are shown as thered crosses in Fig. 10�b�. A point in the poloidal cross section�R ,Z� corresponds to a circle in tokamak coordinates�R ,Z ,��. The projections of these x-point circles are drawnin the projection plane, shown in red in Fig. 10�a�. The re-sulting circles are tangent to both plasma edges in the pro-jection plane. Concluding, tokamak plasma x-points are notdirectly visible in an image of the plasma but their locationcan be derived from the reconstructed plasma surfaces in thepoloidal plane.

When drawing the trajectory of the tangent sightlines le

in the poloidal plane, horizontal sightlines have a straight,horizontal trajectory, while vertically inclined sightlinesshow a parabola trajectory. These trajectories can be treatedas upper bounds on the R coordinate of the plasma edge.Nearly horizontal sightlines are therefore desired for opti-mum resolving power, as they impose only a very local up-per bound on R. Figure 11 shows the top half of the DNDplasma shape in the poloidal cross section with the trajecto-ries of the tangent sightlines used to reconstruct the plasmashape. The resolving power of the reconstruction is highestaround the equatorial plane, where sightlines are nearly hori-zontal.

III. EXPERIMENTAL RESULTS ON MAST

To test the optical plasma boundary reconstruction, orOFIT, camera images from MAST discharges providing a fullview of the plasma are used. The optical reconstruction pro-cedure is described and the results are compared to plasmaboundaries derived from magnetic measurements by the EFIT

�Ref. 5� code. As the two forms of boundary reconstructionare based on different physical properties of the plasma, anexact agreement is not to be expected. The comparison of thetwo reconstructions may indicate a possible relation betweenthe magnetic and emissive properties of the plasma edge.

FIG. 9. �Color online� Tangent sightlines around the inner edge of a circularcross section plasma in Tore Supra geometry.

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2projection plane

u [m]

v[m

]

0.5 1 1.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

R [m]

Z[m

]

Poloidal plane

(a) (b)

FIG. 10. �Color� �a� Projection plane, showing the image of plasma edge,x-point �red circles�, and intersection of edges �black cross�. �b� Poloidalcross section, showing the plasma shape, x-points �red crosses�, and projec-tion of sightlines corresponding to the intersection of edges in the projectionplane �black lines�.

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A. Camera calibration

Figure 1 shows three examples of images captured fromMAST shot no. 23344 by a Photron13 Ultima APX-RS cam-era located at equatorial port 10. The images provide a viewof the entire plasma, including the divertor region and strikepoints. The images are 8-bit grayscale and have a resolutionof 512�528 pixels. The projection properties of the optics,i.e., the line of sight of each pixel, must be known in order touse the image as a quantitative spatial measurement. Using acalibration procedure based on the visibility of the centercolumn and the poloidal field coils and a simple cameramodel, the required parameters are estimated. In Appendix Bthe calibration procedure is elaborated. The resulting fit ofthe tokamak structures on the image is shown in Fig. 12�a�.

During the discharge, significant horizontal movement ofthe tokamak center in the images is observed. To compensatethis disturbance, the horizontal coordinate of the tokamakcenter is determined for each image so that the reconstructedplasma surface does not show this horizontal shift. Figure12�b� shows the evolution of the tokamak center horizontalpixel coordinate x0 during discharge no. 23344. The horizon-

tal tokamak center pixel coordinate x0 is then used to definethe horizontal camera coordinate uc to be used in the recon-struction equations. At a distance of wc=2.08 m from thecamera, a 1 pixel displacement in x0 corresponds to a changein uc of 7 mm.

B. Edge detection

The full view images of MAST shown in Fig. 1 can betreated as two poloidal view images in the left and right halfof the image. In each of the two poloidal images, the plasmaedge has the form of two lines; an inner and an outer plasmaedge, intersecting at the top and bottom of the image in thecase of a dual null divertor plasma. To locate the edges,regions of interest �ROIs� are appointed in the image havingthe shape of a curved rectangle around the expected plasmaedges. In the ROIs, the image is resampled and filtered toform rectangular pixel grids. On each row of this grid, themaximum intensity is identified as the plasma edge. The de-gree of smoothing is optimized to reduce sensitivity to noiseand filaments while maintaining resolving power. AdaptiveROIs can be used to cater for different phases of the dis-charge with significantly different plasma shapes. The detec-tion of the outer plasma edge in the divertor regions is ham-pered by the presence of other, sharper edges in this part ofthe image and the low contrast of the plasma edge. There-fore, the ROI chosen for the outer edge is limited to thecentral part of the outer edge up to the intersection with theinner plasma edge. Figure 13 illustrates the detection of theouter plasma edge.

To improve the robustness of the detected edge, whichmay contain spurious edge points, a piecewise polynomialfunction is fitted through the edge coordinates, such that theeffect of noise on individual edge points is reduced but theoverall shape of the plasma edge is retained. The here ap-plied fit is formed of a fourth order polynomial around theequatorial plane and two second order polynomials for thetop and bottom of the edges, joined with continuous firstderivative. The use of these functions also allows extrapola-

0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

R [m]

Z[m

]

Poloidal plane

FIG. 11. �Color online� Top half of plasma shape in poloidal plane, showingprojections of tangent sightlines used to reconstruct the plasma shape.

FIG. 12. �Color online� �a� Tokamak structures in image. �b� Tokamak cen-ter horizontal pixel coordinate during discharge.

FIG. 13. �Color online� Detection of the right outer plasma edge, showing�a� region of interest, detected edge points, and piecewise polynomial fit. �b�Resampled and filtered pixel grid with maximum value pixels on each rowmarked.

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tion of the plasma edges in regions of the image where edgedetection is problematic, such as the divertor region for theouter plasma edge.

During ELMs, such as shown in Fig. 1�c�, large fila-ments show up and the plasma edge is less localized. Theplasma edge can still be identified during an ELM but thenoise and uncertainty are momentarily higher.

C. Optical plasma shape reconstruction: OFIT

Given the two edges in the image and the projectionproperties of the camera, the edges are projected onto theprojection surface �u ,v� and plasma surface reconstruction isperformed using the coordinate transform described in Sec.II A. This procedure is abbreviated as the OFIT reconstruc-tion. Because of the toroidal symmetry of the plasma, thetwo halves of the image effectively provide two measure-ments of the plasma shape. However, as the camera is notpointed exactly at the tokamak center, the two half imagesare not symmetric. Only after reconstruction of the plasmashape from both half images will two similar plasma shapesbe obtained. Figure 14 shows a camera image of a DNDplasma from MAST discharge no. 23344 and the correspond-ing OFIT reconstruction from both halves of the image.

As described in Sec. II C, the inner and outer plasmaedges span a different height after reconstruction. The outerplasma edge is defined only in a limited domain, approxi-mately −0.8 Z 0.8, because of the limited visibility of theouter edge in the divertor region. To extend the outer plasmaedge beyond this range, an extrapolation using the fittedpiecewise polynomial functions is used, also shown in Fig.14. This extrapolation allows making an estimate of thex-point locations of the DND plasma, as the outer plasmaedge now intersects the inner plasma edge in the poloidalplane. However, this extrapolation introduces extra uncer-tainty and noise to the detected x-point locations.

During the initial phase of a discharge, the plasma is inlimiter configuration before forming an upper and lowerx-point. An x-point plasma can be optically recognized by itsso called strikes: the legs that extend from the x-points out-

ward to the divertor targets. Figure 15 shows MAST plasmasin lower single null divertor �SND� and DND configurations.The distinct feature of the DND plasma in the top half of theimage is the outer strike: the inner plasma edge extendingbeyond the outer edge toward the divertor. The top end of theSND plasma has the same qualitative shape as the circularplasma in Fig. 7�a�. The transition from limiter to x-pointconfiguration in the images, however, is not sharp. Figure 16shows three images of the transition from limiter to x-pointplasma in the first 0.1 s of MAST discharge no. 23344. Asthe magnetic topology evolves from limiter to x-point con-figuration, strikes gradually become visible. Because of thefinite thickness of the emission layer, there is no discretetransition in the images between the limiter and the x-pointconfiguration. Additionally, the emission of light is not asstrongly concentrated on the LCFS in this early phase of thedischarge, further amplifying this effect. Discriminating be-tween x-point and limiter plasmas during this transition onthe basis of the images is therefore a matter of choosing anarbitrary threshold.

Because OFIT does not yet cater for the reconstruction oflimiter or SND plasmas, only the part of the discharge that isoptically recognized as a DND plasma is reconstructed. InMAST discharge no. 23344, this holds for t�0.1 s.

Based on the toroidal symmetry of the plasma, compar-ing plasma shape reconstructions from the left and righthalves of the image can provide insight into the effects oferrors in camera calibration and uncertainties introduced inedge detection and reconstruction. To compare the two re-constructions, a set of 13 shape descriptors is used: the innerand outer plasma edge radii at Z=0.5 and Z=−0.5, the mini-mum and maximum plasma radius, the R and Z coordinatesof both x-points, the elongation �, and the upper and lowertriangularities �UP and �LOW.

FIG. 14. �Color� Plasma shape reconstruction in MAST discharge no.23344. Reconstructions from both the left �blue� and right �red� halves of theimage are shown. Extrapolated plasma edges are shown as the dashed-dottedlines.

FIG. 15. Images of MAST H-mode plasmas in SND �left� and DND �right�configurations.

FIG. 16. Transition from limiter to x-point plasma during the first 0.1 s ofMAST discharge no. 23344.

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Two measures of errors are used for each set of signalss1 and s2 on the interval �t1 , t2�

�1 =�t1

t2s1 − s2dt

t2 − t1, �16�

�2 =��t1

t2�s1 − s2 − �1�2dt

t2 − t1, �17�

such that �1 is a measure of average constant offset and �2 isa measure of offset-corrected variance between the signals.The value of �2 is treated as an uncertainty band. When, fora set of two signals, �1��2, there exists a significant con-stant offset. In comparing the left and right OFIT reconstruc-tions, this may indicate errors in the camera calibration.

For each of the shape descriptors, the error measuresbetween OFIT from both half images are given in Table I. Fora selection of the shape descriptors, the signals from bothoptical reconstructions are also plotted in Fig. 17. The optical

reconstructions show upper and lower triangularity �UP and�LOW of around 0.36 for most of the discharge. The defini-tions of elongation and triangularity of Beghi and Cenedese6

are used.For almost all of the 13 shape descriptors �2��1, indi-

cating good agreement within the uncertainty band that isapproximated by �2. Only the outer lower plasma radiusRo,−0.5 shows an offset that is larger than the uncertaintyband, indicating there may be a small error in the calibrationparameters or a shortcoming in the camera model. Arguably,the tokamak construction tolerances may be such that theplasma is not exactly toroidally symmetric.

The effect of extrapolation of the outer plasma edges onthe reconstructed x-point locations is obvious from the largeruncertainties present in these shape quantities. This uncer-tainty could be greatly reduced by placing additional cam-eras at approximately x-point height that provide a near-horizontal view of the plasma edge in the x-point region,resulting in a reconstruction of the x-points similar in accu-racy to the Rmax signal from the equatorial camera.

D. Comparison to EFIT reconstructions

Figure 18 shows EFIT and OFIT reconstructions of equi-libriums from MAST shot no. 23344. Table II shows theerror measures �Eqs. �16� and �17�� of the shape descriptorsbetween EFIT and OFIT. In this case, the average of the OFIT

TABLE I. Deviations of shape descriptors between reconstructions from leftand right half image.

�1 �2

Rmin 4.6 mm 6 mmRx,upper �1.9 mm 24 mmZx,upper �12 mm 27 mmRo,0.5 �2.0 mm 5.6 mmRi,0.5 6.1 mm 8.4 mm�UP 0.0097 0.044� 0.0022 0.033Rmax 0.6 mm 4.8 mmRx,lower 1.5 mm 15 mmZx,lower �6.6 mm 19 mmRo−0.5 �6.2 mm 5.0 mmRi−0.5 1.4 mm 5.9 mm�LOW 0.0032 0.029

0.1 0.2 0.31.25

1.3

1.35

1.4

Rmax

OFIT left OFIT right EFIT

0.1 0.2 0.31.8

2

2.2

2.4

2.6Elongation

κ

time [s]

0.1 0.2 0.30.9

1

1.1

1.2

1.3

1.4

Zx,upper

0.1 0.2 0.30.4

0.5

0.6

0.7

0.8

Rx,upper

0.1 0.2 0.3

−1.3

−1.2

−1.1

−1

−0.9

Zx,lower

time [s]0.1 0.2 0.3

0.45

0.5

0.55

0.6

0.65

0.7

Rx,lower

time [s]

FIG. 17. �Color� Time traces of shape descriptors from OFIT and EFIT.

0.20.40.60.8 1 1.2−1.5

−1

−0.5

0

0.5

1

1.5

Z[m

]

R [m]

t=125ms

0.20.40.60.8 1 1.2−1.5

−1

−0.5

0

0.5

1

1.5

R [m]

t=200ms

0.20.40.60.8 1 1.2−1.5

−1

−0.5

0

0.5

1

1.5

R [m]

t=348ms

EFIT OFIT

FIG. 18. �Color online� Plasma edge cross sections from EFIT and OFIT forthree MAST shot no. 23344 equilibriums.

TABLE II. Deviations of shape descriptors between OFIT and EFIT.

�1 �2

Rmin �8.7 mm 14 mmRx,upper 17 mm 17 mmZx,upper �3.3 mm 18 mmRo0.5 �14 mm 7.6 mmRi0.5 �7.8 mm 12 mm�UP �0.044 0.049� 0.064 0.082Rmax �14 mm 4.1 mmRx,lower 8.1 mm 14 mmZx,lower �19 mm 18 mmRo−0.5 �9.2 mm 11 mmRi−0.5 �19 mm 11 mm�LOW �0.017 0.051

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reconstructions from the left and right half images is used tocompare to the EFIT reconstruction. A significant offset of 14mm is present in the Rmax signal, also clearly visible in Fig.17. An offset of similar magnitude is present in the Ro,0.5 andRo,−0.5 signals. The EFIT reconstruction relies on optical con-straints for the midplane outer plasma edge. This constraintis applied such that the LCFS at the outer edge is placed 1cm outboard of the optically detected peak of Dalpha emis-sion, partially explaining the Rmax offset.14

The EFIT reconstructions show a limiter plasma untilt=0.125 s before switching to a DND configuration,whereas the images and therefore OFIT show a DND plasmafrom before t=0.1 s. EFIT does, however, detect x-pointsfor t 0.125 s, albeit outside of the LCFS, whose locationsagree well with those found by OFIT, as can be seen in Fig.17. For t�0.125 s, the EFIT reconstruction alternates be-tween a DND and SND plasma, where the lower x-point issometimes placed just outside of the LCFS when it is locatedon a marginally different magnetic flux surface. These smallexcursions of the x-points over the flux surfaces are not ap-parent in the images and thus in the OFIT reconstructions.

The EFIT reconstructions show a triangularity of approxi-mately �=0.4 during most of the shot versus �=0.36 forOFIT. The difference in triangularity between EFIT and OFIT

results from the differences in x-point locations and Rmax

signals. The optical reconstruction detects the x-points at alarger radius R but shows a smaller Rmax, resulting in a lowertriangularity.

The elongation signals from EFIT and OFIT shown inFig. 17 have a significant offset only between t=0.1 s andt=0.125 s. In this part of the discharge, EFIT detects a limiterconfiguration and places both x-points outside the LFCS,meaning the top and bottom extremes of the LCFS are closerand the resulting elongation is lower. The dips in the EFIT

signal for t�0.125 s occur when a SND plasma is detected.The elongations found by OFIT are always based on thex-points of the plasma and agree to within 3% of the DNDequilibriums from EFIT.

IV. APPLICATIONS AND DISCUSSION

The agreement in shape and the correlation in time be-tween the EFIT and OFIT plasma shape reconstructions sug-gest the magnetic and emissive properties of the plasma edgeare closely related and the optical boundary reconstructionprovides a measurement of the plasma boundary that is validwithin the uncertainty bounds.

Visible accessibility of the plasma edge is required forthe optical shape reconstruction. In many tokamaks, thevacuum vessel fits tightly around the plasma. The lens of thecamera must then also be placed close to the plasma to havean unobscured view of the plasma. This camera location,however, means that only a fraction of the plasma edge isvisible in the camera image, as explained in Sec. II A. Toperform a reconstruction of the full plasma shape, multiplecameras will then be required, each viewing a part of theplasma circumference. The MAST tokamak is unique in thisrespect; its open design allowing a full view of the plasmausing only a single camera.

To optimize the resolution and sensitivity of the recon-struction, a configuration with nearly horizontal sightlines isdesired. The MAST images used in this paper are made usinga wide angle lens viewing the top and bottom of the plasmaat large angles to the horizontal plane. This setup limits theaccuracy of the reconstruction around the x-points. Extracameras at approximately x-point height can greatly improvethe accuracy of the x-point localization. During some MASTdischarges, a low framerate camera has been placed in theupper and lower ports, the so called DIVCAM. The resultingimages provide a nearly horizontal view of the strikes andx-point. Preliminary results of boundary reconstructions us-ing DIVCAM images suggest that the reconstruction of theouter strike from the midplane camera is accurate while thereconstruction of the inner strike, based on an extrapolationof the outer plasma edge, has a systematic offset of around5 cm at the strike point.

The MAST images used in this paper are intensity mapsof visible light; no optical wavelength filtering other than thecamera’s inherent sensitivity is applied. Using wavelengthspecific filters may provide clearer images and result in amore reliable reconstruction. Previous research on the DIII-Dtokamak has shown distinct emission footprints around theLCFS for different characteristic wavelengths.15

A. OFIT for real-time control

In this case study, all processing is done in MATLAB

functions. A single reconstruction of the plasma shape inMAST, including image processing and centering, curve fit-ting, coordinate transform, and x-point detection takes 3 mson a desktop computer with an Intel Core 2 6600 centralprocessing unit. Using both halves of the image to effectivelyobtain two reconstructions takes 4.5–5 ms.

Real-time optical detection of the plasma midplane outerradius for feedback control is already routinely applied atMAST with a cycle time of 0.5 ms. An implementation ofthe optical shape reconstruction in a real-time environmentsuch as currently used in MAST �Ref. 16� can provide areal-time diagnostic usable for feedback control. Currentlyavailable real-time cameras and frame grabbers can provideseveral thousand full resolution frames per second.13,17 Thecycle time of this diagnostic would therefore depend stronglyon the processing power of the applied system. The 3 msprocessing time of the shape reconstruction implemented inMATLAB can be seen as the minimum achievable speed forthe real-time diagnostic.

As tokamaks grow, both PF-coil inductive time constantsand plasma skin times lengthen, effectively lowering thesampling requirements for a real-time diagnostic. A real-timecontrol application of OFIT therefore is expected to be fea-sible in the near future.

To operate OFIT in a control loop of plasma position andshape, the behavior of the diagnostic must be known in allplasma regimes and the quality of the boundary reconstruc-tion must be sufficient. Different plasmas of various shapes,densities, confinement modes, and turbulence levels will re-sult in very different images and will often require modifica-tions to the image processing and the camera shutter timingto optimally identify the plasma edge. The different images

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obtained in different plasma regimes can result in a varyingaccuracy of the reconstructed plasma shape. For example, ahigh density H-mode discharge will produce a bright imagewith very sharp plasma edges and is very well suited foroptical boundary reconstruction. On the other hand, anL-mode plasma with strong turbulence will show a muchfuzzier image and will result in a more uncertain boundaryreconstruction. The diagnostic is required to detect and adaptto different plasma regimes and to not only provide a plasmashape but also an indication of the current regime and thecorresponding reliability of the boundary reconstruction.

For this research, OFIT was developed to explore the pos-sibilities of optical plasma boundary reconstruction and wastuned to a small number of MAST DND discharges. Morework is needed to obtain a plasma shape diagnostic that isflexible, robust, and automatically adapts to different plasmaregimes.

B. OFIT as a scientific diagnostic

Currently, only indirect �magnetic� or local �MAST lin-ear HOMER camera,16 plasma boundary reflectometry,18 orThomson scattering19� boundary diagnostics are available intokamak operation. OFIT uniquely provides a direct and in-dependent measurement of the entire plasma boundary.

Different plasma boundary diagnostics measure differentphysical properties of the plasma. Magnetic reconstructionsgenerally provide the plasma boundary as the last closed�poloidal-� flux surface. Reflectometry provides the locationof an arbitrary density surface. Thomson scattering provideslocal electron density and temperature profiles. OFIT recon-structs the boundary from the light emitted from the vicinityof the plasma boundary.

To relate these measurements, a detailed model of theplasma boundary is required. However, the availability ofthose measurements also aids in developing such a plasmaedge model. Ultimately, given enough understanding of thedetails of plasma edge physics, different boundary diagnos-tics are equivalent. In this case, a strong argument for install-ing the required cameras for OFIT boundary reconstruction isthe simplicity and independence of the diagnostic that canpotentially reconstruct the entire plasma boundary. OFIT canthen provide a strong, global constraint to equilibrium recon-struction codes such as EFIT and diagnostics such as motionalStark effect arrays.

V. CONCLUSIONS

An optical plasma shape diagnostic, OFIT, is developedand applied to images of MAST discharges. Comparison ofthe optically reconstructed plasma shapes with reconstruc-tions available from EFIT show spatial agreement within 19mm and strong temporal correlation. The processing time fora single MAST plasma shape reconstruction in its currentMATLAB implementation is 3 ms and scales with processingpower of the applied system. The speed of the algorithmallows application in real-time environments such as feed-back control of the plasma shape. Further development isneeded, however, to obtain the desired robustness and accu-racy in all plasma regimes. Specifically, a robust and adap-

tive edge detection procedure is required to interpret the im-ages from plasmas in different phases of the discharge andwith different emission regimes. Some plasmas may not pro-vide enough edge localization to reach the accuracy requiredfor control of the plasma boundary.

An important prerequisite for the new diagnostic is theoptical accessibility to the plasma edge. Multiple cameraswill be required in tokamaks with tight fitting vacuum ves-sels, where a single camera cannot view all of the plasmacircumference. In open tokamaks such as MAST, applyingmultiple cameras placed at different heights will also resultin a more accurate and reliable reconstruction by providing amore horizontal view of the plasma edges far from the mid-plane. The use of divertor cameras in MAST to obtain betterx-point and strike-point localization is currently being ex-plored.

Having detected in the image the shape of the plasmaedge, other visible features such as filaments and hot spotscan also be detected and their orientation can be related tothe local plasma edge. A single diagnostic can then provideinformation about plasma shape, PFC loads, and ELM activ-ity with low relative error, paving the way to intelligent sens-ing and integrated control of the plasma edge.

The coordinate transform derived for the optical plasmashape reconstruction can find applications in various analysistools for line of sight measurements by providing the loca-tion of the plasma edge in the coordinates of the diagnosticand is already being used at MAST.

ACKNOWLEDGMENTS

This work, supported by NWO and the European Com-munities under the contract of the Association EURATOM/FOM, was carried out within the framework of the EuropeanFusion Programme. The views and opinions expressed hereindo not necessarily reflect those of the European Commission.The authors would like to thank Nicola Bertelli, Michel vanHinsberg, and the colleagues at the TU/e DCT-lab for fruitfuldiscussions and support. Equipe Tore Supra is kindly ac-knowledged for making Fig. 8 available.

APPENDIX A: SYSTEM OF A CIRCULAR PLASMAIMAGE

The newly derived coordinate transform is mathemati-cally validated for a circular tokamak plasma. To do so, apoloidal view image of a circular plasma is synthesized usinga different approach to the coordinate transform. Again, it isassumed that sightlines tangent to the plasma surface willresult in edge points in the image.

A point on the circular plasma surface and the normalvector Vn to this surface are expressed in Cartesian coordi-nates �u ,v ,w� in terms of the toroidal and poloidal angles �and �

�ut

vt

wt� = �cos����r cos��� + R0�

r sin���sin����r cos��� + R0�

� ,

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Vn = �cos���cos���sin���

sin���cos���� .

The sightline vector from a camera at �uc ,vc ,wc� is

Vl = � ut − uc

vt − vc

wt − wc� = �cos����r cos��� + R0� − uc

r sin��� − vc

sin����r cos��� + R0� − wc� .

For the sightline to be tangent to the plasma surface, thedot product of the surface normal and the sightline vectorsmust equal 0.

Vn · Vl = 0.

This allows, for a given poloidal angle �, to find thetoroidal angle � of the tangent point on the plasma surface.Solving for � results in

uc cos��� + wc sin��� = R0 +r − vc sin���

cos���,

which can be further simplified to

�uc2 + wc

2 sin� + arctan uc

wc�� = R0 +

r − vc sin���cos���

.

Substituting into the right-hand-side of Eq. �10�, the R,Z, and dR /dZ coordinates of a circular plasma

R = R0 + r cos��� ,

Z = r sin��� ,

dR

dZ= − tan��� ,

result in the same equation. The coordinate transform istherefore validated for the special case of a circular plasma.

Given the poloidal angle � the tangent point can now befound. The projection of this point is obtained by drawing aline from the camera through the tangent point and taking theintersection of this line with the projection surface.

APPENDIX B: CAMERA CALIBRATION

To use the camera images as a quantitative measurement,the projection properties of the optics, i.e., the line of sight ofeach pixel, must be known. A model of the camera projectionis presented and the parameters are estimated using a cali-bration procedure based on the visibility of the center col-umn and the poloidal field coils and on the knowledge of thecharge coupled device �CCD� dimensions and the focallength of the lens. The result will be the projection function

�u,v� = fp�x,y� ,

where x and y are pixel coordinates in the image and u and vare coordinates on the �u ,v� projection plane.

The pinhole camera model with its optical axis normal tothe �u ,v� projection plane is used, providing an affine scalingfrom pixel coordinates to projection plane. A first order bar-reling correction accounts for lens distortion. The requiredcamera parameters are then the camera location �uc ,vc ,wc�,

the barreling factor KB, the scaling factor Kscale, and the lo-cation of the optical center in the image �xoc ,yoc�.

The barreling correction assumes a first order barrelingmodel20 of the form

xbc − xoc = �x − xoc��1 + Kb��x − xoc�2 + �y − yoc�2� ,

ybc − yoc = �y − yoc��1 + Kb��x − xoc�2 + �y − yoc�2� ,

where �x ,y� is the pixel coordinate in the image and �xbc ,ybc�is the barreling-corrected pixel coordinate, corresponding tothe image taken with an ideal linear lens.

With the camera’s optical axis parallel to the w-axis, theprojection coordinates �u ,v� are then calculated from thebarreling-corrected pixel coordinates �xbc ,ybc� by the projec-tion functions

u − uc = wcKscale,x�xbc − xoc� ,

v − vc = wcKscale,y�ybc − yoc� .

The camera coordinates uc and vc are derived from thelocation of the tokamak center in the image �x0 ,y0� using thesame equations

uc = − wcKscale,x�x0 − xoc� ,

vc = − wcKscale,y�y0 − yoc� .

The vertical tokamak center coordinate y0 is fixed whilethe horizontal coordinate x0 is updated for every frame basedon the location of the inner plasma edges.

Given the focal distance of the applied lens and the di-mensions of the pixels on the CCD chip, the scaling factorKscale can be determined as

Kscale =lp

f,

where lp is the size of a pixel on the CCD and f is the focallength of the lens. It is assumed that the CCD is at a distancef from the lens.

The calibration parameters used to draw the PF-coiledges in Fig. 12�a� are given in Table III, where lp and f arecamera specifications and the other parameters are tunedmanually to obtain the best fit of the coil edges on the image.

1 H. Weisen, Y. Martin, J.-M. Moret, and the TCV Team, Nucl. Fusion 42,L5 �2002�.

2 J.-M. Moret M. Anton, S. Barry, R. Behn, G. Besson, F. Biihlmann, A.Burri, R. Chavan, M. Corboz, C. Descbenaux, M. J. Dutch, B. P. Duval,D. Fasel, A. Favre, S. Franke, A. Hirt, F. Hofmann, C. Hollenstein, P.-F.Isoz, B. Joye, J. B. Lister, X. Llobet, J. X. Magnin, P. Mandrin, B. Mar-letaz, P. Marmillod, Y. Martin, J.-M. Mayor, J. Moravec, C. Nieswand, P.J. Paris, A. Perez, Z. A. Pietrzyk, V. Piffl, R. A. Pitts, A. Pochelon, O.Sauter, W. van Toledo, G. Tonetti, M. Q. Tran, F. Troyon, D. J. Ward, and

TABLE III. Camera calibration parameters.

wc 2.08 mxoc 252 pxyoc 276.5 pxKB 4�10−4

lp 17 �mf 5 mmy0 276.5 px

113504-12 Hommen et al. Rev. Sci. Instrum. 81, 113504 �2010�

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9 A. Kirk, N. Ben Ayed, G. Counsell, B. Dudson, T. Eich, A. Herrmann, B.Koch, R. Martin, A. Meakins, S. Saarelma, R. Scannell, S. Tallents, M.Walsh, H. R. Wilson, and the MAST Team, Plasma Phys. Controlled Fu-sion 48, B433 �2006�.

10 D. P. Kostomarov, A. A. Lukianitsa, F. S. Zaitsev, R. J. Akers, L. C.Appel, D. Taylor, and V. V. Zlobin, Proceedings of the 32nd EPS Confer-

ence on Plasma Physics, Tarragona, Spain, 27 June–1 July 2005 �ECA,2005�, Vol. 29C, p. 1.092.

11 See http://www.fusion.org.uk/mast/index.html for Mega-Ampere Spheri-cal Tokamak.

12 See http://www-fusion-magnetique.cea.fr/gb/cea/ts/ts.htm for Tore Supratokamak.

13 See http://www.photron.com/ for Photron high-speed cameras.14 J. Storrs �private communication�.15 M. Groth, M. E. Fenstermacher, G. D. Porter, R. L. Boivin, N. H. Brooks,

W. P. West, P. C. Stangeby, and D. G. Whyte, presented at the Proceedingsof the 44th Annual Meeting Division of Plasma Physics, Orlando, FL,11–15 November 2002.

16 J. Storrs, J. Dowling, G. Counsell, and G. McArdle, Fusion Eng. Des. 81,1841 �2006�.

17 See http://www.visionresearch.com/ for Vision Research high speed cam-eras.

18 C. E. Kessel and N. L. Bretz, Nucl. Fusion 39, 445 �1999�.19 H. Murmann and M. Huang, Plasma Phys. Controlled Fusion 27, 103

�1985�.20 D. C. Brown, Photogramm. Eng. 7, 444 �1966�.

113504-13 Hommen et al. Rev. Sci. Instrum. 81, 113504 �2010�

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