CHAPTER 3

Tomography

Inside yourself or outside, you never have to change what you see,only the way you see it.

– Thaddeus Golas

F or many years, the challenge of determining the internal structure of anobject—while not actually damaging the object itself—has attracted great in-terest. From the Greek word tomos meaning ’slice’ or ’section’, tomography seeksto recover the internal structure based on line-of-sight measurements of somequantity or property of the object. Figure 3.1 illustrates this concept. The the-ory of tomography is based on the work of Radon [Radon, 1917] which ob-tains an analytic formula for the inversion of the line-integral, or ‘Radon’ trans-form. An application for which tomography is perhaps most widely known isits use in medical imaging, where it is known as CT (computed tomography)or CAT scanning. Pioneering work by Hounsfield [Hounsfield, 1973], and Cor-mack [Cormack, 1963], [Cormack, 1964] was recognised with their joint awardof the Nobel Prize in Physiology or Medicine in 1979 “for the development ofcomputer assisted tomography”. Other applications are many and varied andinclude include earlier work in radio astronomy [Bracewell, 1956a], [Bracewell,1956b], more medical imaging [Mazziotta et al., 1986] including magnetic reso-nance imaging (MRI) [Damadian, 1971], [Lauterbur, 1973]†, supersonic gas flows[Morton, 1995] and seismology [Nolet, 1987].

The line-integrated measurements of the object under investigation can beobtained, depending on the nature of the object, by many varied techniques. Thechallenge is to invert the profiles created by the line-integrated measurements togive the correct, or at least the best approximation of, the internal structure of theobject. This chapter outlines the theory for the tomographic inversion techniquesused in this thesis, as well as considerations regarding their use.

Simulations demonstrating the performance of these procedures—applied withthe experimental configuration used in this thesis—are also presented.

† for which Paul Lauterbur jointly received the 2003 Nobel Prize in Physiology or Medicine.

3. Tomography 40

p

y(p, φ)

φ

(a) (b)

Figure 3.1: (a) Line integrals are used to construct a profile of an object. (b) Severalprofiles are used to build a cross-section of the object. Figure courtesy of G. Warr.

3.1 Theory

For two-dimensional tomography, the object or picture from which the line-integrated measurements are obtained can be represented by a function of twovariables. This is known as the picture function, denoted as P(r) and is definedto be zero outside the picture region.

The Radon transform of the picture function is defined as

P̌ = RP =∫

LP(r)dl (3.1)

where dl is the element along the line L and the symbols and geometry are givenin Figure 2.1.

In a typical experimental situation, P(r) might represent the emissivity in the(x, y) plane, L the line of sight and P̌ the measurement along that line.

It is convenient to re-write the Radon transform of (3.1) in terms of the impactparameter p and angle φ as

P̌(p, φ) =∫

P(r)δ(p − p̂.r)dr (3.2)

where the Dirac delta function δ is used to select the line L.Herman outlines two basic approaches for the reconstruction of the sampled

image [Herman, 1980]: the series expansion methods, including the arithmetic

3. Tomography 41

reconstruction technique (ART) and linear composition of orthogonal functions,both of which are utilised in this thesis. The other approach using Fourier andHilbert transform methods is not discussed here.

For the ith line-integrated measurement yi—associated with the line of sightspecified by (pi, φi) and with a total of I measurements—the Radon transform ofP for this line is

yi = RiP (3.3)and Ri is a functional. That is, it acts on a function to produce a real number.A collection of line integrals associated with a specific viewing position of theobject is known as a projection. Figure 3.1(a) shows a simple arrangement for aprojection using parallel lines with a constant φi. The entire set of measurementsy is known as the measurement vector and it is assumed that the set of lines isfixed and known.

The input data for the reconstructions are then samples of the Radon trans-form and the reconstruction itself becomes an approximation of the picture func-tion. This approximation can be made arbitrarily good by utilising an increasingnumber of projections [Hamaker et al., 1980]. For a finite set however, a theorem[Smith et al., 1977] states: A function P of compact support in R2 is uniquely de-termined by any infinite set, but by no finite set of its projections. Even thoughuniqueness is sacrificed in real applications, the challenge is to closely approxi-mate the real P with non-unique approximations. This can be done using a prioriinformation, knowledge before the fact, to help define the solution.

The series expansion methods discretise the picture function into basis pic-tures. A basis picture may represent, among other things, a spatial area (a ‘tra-ditional’ pixel) or a function. The estimate of the picture function is given by alinear combination of the basis pictures. If the jth basis picture is denoted as bj,with a total of J basis pictures, then the picture function estimate can be writtenas

P(r) = ∑j

xjbj (3.4)

where xj is the value (a co-efficient or weighting) for the basis picture and thecomplete set x is known as the image vector. Substituting (3.4) into (3.3) gives

yi � ∑j

Rijxj (3.5)

where Rij � Ribj. The matrix R, whose (i, j) element is Rij, is known as theresponse or projection matrix. The difference between the left and right handsides of (3.5) is given as ei and the aim of the reconstruction of the picture functionestimate is to minimise this error vector. Allowing for this discrepancy, (3.5) ismore succinctly written, in matrix form, as

y = Rx + e (3.6)

3. Tomography 42

The difference vector e may be due to errors in the measurements or an inabilityto adequately describe the picture function.

3.2 Methods of Reconstruction

3.2.1 Algebraic Reconstruction Techniques

These methods, often abbreviated as ART, were first proposed by [Gordonet al., 1970] and [Hounsfield, 1972] and though not actually any more algebraicthan other series expansion methods—the name is a historical accident—theyare iterative in nature. A sequence of image vectors x(0), x(1), x(2), ... converge toa final estimate image vector x∗. x(k+1) is derived from x(k) based on the relationin (3.5) and the iterative step can be written as a function αk such that

x(k+1) = αk(x(k), Rik , yik) (3.7)

with a total of K iterations. Here ik is used to denote (k mod I) + 1† such thatthe equations are used cyclically and Ri is the transpose of the ith row of R. ARTmethods differ from each other in the choice of the iterative step function αk.

The ART method primarily employed in this thesis uses an iterative-stepfunction of the form

αk = x(k) +

σ(yik − ∑j Rik jx(k)j )

J(3.8)

where σ is a relaxation factor to allow the solution to converge in a controlledway. This iterative-step function adds to the image vector a mean of the differ-ence between the measurement data and the response matrix-weighted sum ofthe current image vector†, modified by the relaxation factor. Other constraintsmay be imposed on the iterative step function such as ensuring the right-handside of (3.8) is positive or smoothing the image vector between iterations. In prac-tice, the computation is restricted to basis pictures which are directly associatedwith a given line-integrated measurement.

3.2.2 Linear Composition of Orthonormal Basis Functions

Recalling that the definition of a ‘basis picture’ (§3.1) equally refers to a func-tion as it does to a spatial area, this reconstruction method builds the picturefunction as a linear composition of carefully chosen basis functions. Some a prioriinformation about the general structure of the object is required so that a suitable

† That is, i0 = 1, i1 = 2, ..., iI−1 = I, iI = 1, iI+1 = 2, ...† The response matrix-weighted sum of an image vector is known as a re-projection.

3. Tomography 43

family of functions may be selected. If the magnetic field structure is transformedto flux co-ordinates—where the nested flux surfaces become simple circles andthe outermost has a radius of unity—a combination of radial and angular func-tions may be used to describe the picture function. Working in flux co-ordinateshas the additional advantage of being able to use any symmetry arguments, suchas constancy on a flux surface, which may help to further constrain the recon-struction.

The original Cormack inversion used Zernicke polynomials in the radial co-ordinate and Fourier functions in the angular co-ordinate. However for picturefunctions which are zero at their boundary a more suitable choice for the radialcomponent is Bessel functions [Nagayama, 1987].

For flux co-ordinates (r, θ), the Fourier-Bessel function is given as [Warr, 1998]

Fml(r, θ) =√

2|J′m(�ml)|

Jm(�mlr)eimθ (3.9)

where �ml are the roots of the Bessel functions (Jm(�ml) = 0) and scale r so thatJm(�mlr) is zero at r = 1. J′m = ddr Jm is the derivative and the normalising factor√

2/|J′m(�ml)| ensures orthonormality.The picture function is then given by

P(r, θ) =L

∑0

M

∑−M

χmlFml(r, θ) (3.10)

where χml is a complex co-efficient and the total number of functions used isgiven by

J = (2M + 1)(L + 1) (3.11)

If the object is real, the co-efficients must satisfy the Hermitian symmetry

χml = χ∗ml (3.12)

To determine the vector of co-efficients χ, (3.6) is re-written as

χ = R−1(y − e) (3.13)However since R is often ill-conditioned, singular value decomposition (SVD)[Press et al., 1986, pages 52–64]† is used to find the inverse which may be re-garded, in some sense, as the ‘best’ inverse. If R is an I × J matrix, it can bedecomposed into the product of an I × J column-orthogonal matrix U, a J × Jdiagonal matrix W with elements wj ≥ 0 and the transpose of a J × J orthogonalmatrix V,

R = UWVT. (3.14)† This reference contains a more comprehensive treatment of this method than the very brief

outline here.

3. Tomography 44

Since U and V are orthogonal matrices, the inverse of R is given simply by

R−1 = VW−1UT, (3.15)

where W−1 = diag(1/wj).The condition number C of the response matrix is obtained while using the

SVD method and is a measure of the robustness of the inversion against noise. Itis defined as

C = max(wj)min(wj)

(3.16)

The lower the condition number of a given response matrix the more reliable isits inverse. To improve the SVD solution, the wj below a certain threshold can beset to zero. In this case min(wj) refers to non-zero values.

Aikake Information Criterion

Since the number of basis pictures (functions) plays an important part in re-covering a reasonable picture function—too many can leave spurious artifactsand too few can fail to describe the function adequately—a useful techniquefor determining this number is the Aikake Information Criterion (AIC) [Akaike,1972, 1974]

In the context of this theory it is defined as

AIC = I ln ‖e‖2 + 2J (3.17)where ‖.‖ denotes the usual Euclidean vector norm and lower values of AICindicate a more suitable basis set. The AIC and the condition number of theresponse matrix are estimators of the overall suitability of a particular set of basisfunctions.

Coverage in Flux Space

The lines-of-sight (viewing chords) are mapped to flux space via a transformcalculated using the GOURDON code. The positions of the viewing chords(pi, φi) in real space for the ToMOSS system are shown in Figure 3.2 and Fig-ure 3.3 shows the coverage in flux space.

Inclusion of the Measured Spatial Response

The spatial response of each viewing chord of the system as been measured(see §5.2 for details) and is included in the calculation of the response matricesused in both ART and basis function reconstructions. The use of measured chordresponses, rather than simple line-integrals, should allow for improved recon-structions.

3. Tomography 45

θ0

viewing axis

Figure 3.2: A schematic of the designed configuration for the viewing chords, with chan-nel numbers shown. The (+) represent the location of the light-collection optics. The‘standard’ magnetic field configuration is also displayed.

3. Tomography 46

Figure 3.3: The lines-of-sight in magnetic flux space, mapped via a transformation calcu-lated using the GOURDON code. The lines-of-sight in real space are shown in Figure 3.2.

3. Tomography 47

3.3 Simulations of Emissivity

Simulations have been performed to analyse the performance of the ToMOSSsystem and to verify the tomographic inversion codes. A few of the simulationresults are discussed in this section. Using the rotational abilities of the light-collection system, a data set of a few hundred line-integrated measurements canbe used to reconstruct the picture function. The light collection system and itscapabilities are described in detail in §4.1 and §5.1. Figure 3.4 shows a few of theavailable viewing stations—a unique configuration of lines-of-sight obtained byrotation of the light-collection optics. The station number represents the angle ofanti-clockwise rotation from Viewing Station 0. Test picture functions, includinga ‘hat’ function bounded by the last closed flux surface and small width objects atdifferent locations in the viewing region, have been reconstructed using the ARTmethod.

Testing of the Fourier-Bessel basis function method included reconstructinga two-dimensional twin-Gaussian function. The basis function method does notneed to rotate the light collection system, but uses data from only a single view-ing station. Though the capacity to use many viewing stations in conjunctionwith the basis functions method certainly exists, requiring only a single view-ing station allows the system to capture the dynamic behaviour of the plasmawithout relying on shot-to-shot reproducibility.

3.3.1 ART Tests

Last Closed Flux Surface ‘Hat’ Function

A ‘hat’ function bounded by the last closed flux surface of the standard mag-netic field configuration is reconstructed using the ART method. The functionand reconstruction are shown in Figure 3.5). The viewing stations used were 0,9, 18, 27, 36, 45, 171, 180, 189 and 198 for a total of 460 line-of-sight measure-ments. A normalised, circular convolution kernel was used to smooth the imagebetween iterations. The overlaid points show the effective pixel spacing—theproduct of the smoothing kernel size and the actual pixel spacing—and the maskused to define pixels involved with the computation. Note that the mask extendsbeyond the function edge and does not bound the reconstruction. The recon-struction shows good uniformity across the main part of the image, droppingquickly with conformity to the ‘hat’ function boundary.

Smaller-Width Square Test Functions

Square test functions at several different locations in the viewing region arereconstructed to test the system’s ability to reconstruct smaller-scale structures.

3. Tomography 48

Figure 3.4: A sample representation of some of the viewing stations. The station numberis the number of degrees rotated anti-clockwise from Viewing Station 0. The (+) on thecircle circumference represent the position of the light-collecting optics.

3. Tomography 49

Figure 3.5: The ART reconstruction of a ‘hat’ function bounded by the computed lastclosed flux surface of the standard magnetic field configuration. Shown overlaid is themask and effective pixel spacing as well as the original function (solid black outline).Each colour shade represents a 5% level.

3. Tomography 50

A 3-point convolution kernel was used to smooth the image between iterations.The viewing stations were 0, 9, 18, 27, 36 and 45. The top-left reconstruction isof lower quality than the others due to the limited coverage of viewing chords inthat region (see Figure 3.4, top left). Conversely, more line-of-sight measurementsto the right side of the viewing region account for the good reconstructions foundin this area.

Figure 3.6: A compilation of the ART reconstructions of smaller-width ‘hat’ test func-tions. The original picture function is the overlaid square. The poorer reconstructionat the top-left is due to the lower coverage of lines-of-sight in that region. This can beovercome by rotating the viewing optics a full 180◦ from the original viewing station.

3.3.2 Fourier-Bessel Basis Function Test

Figure 3.7 shows a function used to test the basis function method—a two-dimensional twin-Gaussian.

The data set for the reconstruction—from Viewing Station 0 only (Figure 3.4,top left)—was created with 5% noise added to the projections of the test functionand is shown in Figure 3.8.

3. Tomography 51

Figure 3.7: Test object shown in flux space and real space, relative to H-1NF co-ordinates.The transformation to flux co-ordinates is for the standard magnetic field configuration.

3. Tomography 52

Figure 3.8: Projection data for the test image (Fig. 3.7) and its reconstruction (Fig. 3.10).The solid line is the projections of the original image and the diamonds are this datawith 5% noise added. The dashed line is the projections of the reconstruction which wasbased on the noisy data. The lines-of-sight associated with a given channel number areshown in Figure 3.2.

3. Tomography 53

The AIC and condition number C—both shown in Figure 3.9 for different val-ues of M and L—were used to choose the basis function set for the reconstruction.

M=2

M=1

M=0

Figure 3.9: The AIC () and the condition numbers of the response matrices (*) for thevarious mode bases. Degenerate values of a mode basis index are shown in colour. Onlyresponse matrices with a condition number of 40 or less are shown.

The mode basis index J (the number of basis functions) is determined by (3.11)and may contain degenerate values. For instance, J = 10 for both (M = 0, L = 9)and (M = 2, L = 1). Figure 3.9 indicates that although J = 12 (M = 1, L = 3) hasa lower condition number for its response matrix, its AIC is considerably higherthan J = 15 (M = 2, L = 2). Indeed the AIC for J = 18 (M = 1, L = 5) is alsomuch higher than for J = 15. A visual inspection of the test function confirmsthat a basis set that includes only up to M = 1 could not describe it adequately.The next most suitable candidate, according to the AIC, is J = 20 (M = 2, L = 3).Since there is little difference in the AIC (the goodness of fit of the reprojections)for J = 15 and J = 20, the basis set with the lower response matrix conditionnumber is chosen.

The reconstruction using (M = 2, L = 2) with the noisy data is shown inFigure 3.10. The reconstructed image reproduces the form of the twin-Gaussianwell, but is slightly less peaked and somewhat broader than the original.

3. Tomography 54

Figure 3.10: Reconstructed object shown in flux space and real space, with the recon-struction based on M = 2 angular modes and L = 2 radial modes. The measurementsused for the reconstruction contained 5% noise—see Figure 3.8.

3. Tomography 55

3.4 Ion Temperature Tomography

The ToMOSS system is in a unique position to perform tomographic recon-structions of the ion temperature, due to the time-domain nature of the Fourier-encoded interferogram and the good coverage of the plasma region.

The ion temperature Ti is not tomographically inverted in the same way asemissivity due to the nature of the line-integrated measurement. Recall from(2.24) that Ti is contained in an intensity-weighted measurement of the fringevisibility ζ. The method used in this thesis uses a constrained reconstruction ofthe image of I0(r)ζ(r)—created via the ART method with spatial basis pictures—divided by the reconstructed image of I0(r) to give the spatial function of

I0(r)ζ(r)I0(r)

≡ ζ(r) = exp[−Ti(r)

Tc

](3.18)

This function is then manipulated to give Ti(r).The method for constraining the reconstruction is derived from regularising

functions [Anton et al., 1996]. This method is restricted to spatial basis picturesand seeks to minimise the functional

E = ‖e‖ + αr f (x) (3.19)Here αr is the regularisation parameter and f the regularising function. αr is apositive number which determines the weighting between the closeness of thefit, represented by ‖e‖, and the requirements imposed on the reconstruction byf —for example, smoothness.

A first order regularising function seeks to minimise the gradient of the recon-structed image by setting αr = 1 and f = ‖x′‖2. A useful choice when constrain-ing the intensity-weighted fringe contrast image is to set f = ‖(I0(r)ζ(r))′‖2 andαr = 1/I0(r) [Michael, 2004]. Thus in regions of low light levels—where the noisein the contrast measurement is greater—the reconstructed image is constrainedthe most.

3.5 Reconstruction Constraints

One drawback of the bean-shaped magnetic field configuration of H-1NF isthe inability to obtain complete views of the plasma. This is offset somewhat,in the case of ART reconstructions, through the use of many viewing stations tocreate a sufficiently large data set. In the case of Fourier-Bessel basis functions,the confinement of the functions to flux surfaces provides the constraint.

For the ART method the number of pixels is kept high—to adequately samplethe spatial response of the viewing chords which are used—but the smoothingfactor is also kept relatively high to reduce the independence of the basis pictures.

3. Tomography 56

In the case of ion temperature reconstructions, areas which are below a cer-tain threshold in light level (typically 3–5% of the maximum) and are not wellconstrained by the data are removed from the final image.

In all methods, positivity is enforced.

3.6 Sine-Fitting Technique

The sine-fitting technique is useful when the picture function is a single point,or can be approximated as such. In this thesis, applications include determiningthe magnetic field axis using light-inducing electron beams (§5.1.4) and locatingthe in situ calibration light sources (described in §4.3.1). Through the use of manyviewing stations—in fact, a continuous rotation through 200◦ from Viewing Sta-tion 0—the line-integrated measurements create what is known as a sinogram, inresponse to the point illumination. Analysis of the sinogram reveals the positionof the point, as described below.

The point source may be considered as a picture function P(r) which is zeroeverywhere but the point r0 = (x0, y0) in the x − y plane. Thus the line-of-sightmeasurements will also be zero except when intersecting (x0, y0). The generalequation for a viewing chord, specified by (p, φ), is

y =− cos φ

sin φx +

psin φ

(3.20)

Substituting the fixed point (x0, y0), which has polar co-ordinates (p0, φ0), intothis equation gives

p = p0 cos(φ − φ0) (3.21)and this describes the trajectory of the point source in (p, φ) space.

A plot of the positions (p, φ) of viewing chords where they intersect the illumi-nation point is shown in Figure 3.11. The co-ordinates of the point illuminationare thus given by the amplitude p0 and phase φ0 of the fitted sine curve.

3. Tomography 57

Figure 3.11: A sine curve fitted to the impact parameter p and the impact angle φ ofviewing chords which intersect a light source which is considered to be point-like. Theposition of the point illumination is given, in polar co-ordinates, by the amplitude andphase of the fitted curve, as described in (3.21).

CHAPTER 3Tomography3.1 Theory3.2 Methods of Reconstruction3.2.1 Algebraic Reconstruction Techniques3.2.2 Linear Composition of Orthonormal Basis Functions

3.3 Simulations of Emissivity3.3.1 ART Tests3.3.2 Fourier-Bessel Basis Function Test

3.4 Ion Temperature Tomography3.5 Reconstruction Constraints3.6 Sine-Fitting Technique