The Tomography Problem

The pinhole camera does not directly measure the local emissivity, but only a signal integrated over the volume of sight (VOS). The goal of tomography is thus to reveal the best estimate of the local spatial distribution of the emissivity. The power P_i incident on the detector i is given by

P_i =

Z Z Z VOS

Ωi(r)

4π G(r)_dV, (4.1)

4. Tomographic Reconstruction of the Local Emissivity

where the integration is done over the whole volume observed by the detector; Ωi(r) denotes the solid angle of the cone from the exposed surface of the detector and G(r) is the emitted power density in W/m³, i.e. the spectral power density integrated over the sensitivity range of the detector. The brightnessb_i (in units W/m²) can be expressed via theetendue, defined as the product of the solid angle Ωi and the area A perpendicular to the axis of the cone Ωi

b_i=P_i 4π hAΩii =

Z Z

T_i(R,z)G(R,z,ϕ)dRdz. (4.2) The coordinates R and z correspond to the horizontal and vertical directions in the poloidal tokamak cross-section, ϕ represents the toroidal angle and h·i repre-sents the average value over the diode surface. Since the toroidal extend of the VOS is small, a variation of the emissivityG in the toroidal direction will be ignored. The operatorT_ithen introduces a mapping of the 2D emissivityGto a discrete measurement b_i. This operator is defined as follows:

T_i(R,z) =

Z Ωi(R,z,ϕ)

hAΩii Rdϕ (4.3)

and describes the geometric properties of the diagnostics. The equation (4.2) for b_i represents the Fredholm integral equations of the first kind, which will be solved by a discretization on a rectangular grid with the total number n of volume elements (pixels). Thus, the problem can be rewritten as a set of m linear equations for m

detectors

b_i =

n X j=1

T_ijg_j i∈1,. . .,m (4.4) T∈R^m,n denotes a geometry matrix, defining the contribution of a pixel j to the mea-surement i and g_j is the local emissivity at the j-th pixel in units of W/m³. The measured brightnessbˆi is subject to experimental errors. Therefore, bˆi=bi+ξi, where ξ_i represents a random error with zero mean and a variance of σ²_i.

4.1.1. Tikhonov Regularization

The system of equations obtained by discretization of the Fredholm integral equations is ill-posed and for the plasma tomography also usually significantly under-determined (m n). A common method to find a unique and sensible solution is the Tikhonov-Philips regularization [43] in the general form, which searches for a minimum of

4.1. The Tomography Problem

the functional Λ(g) given by

Λ(g) = (Tg−bˆ)^TΣ⁻¹(Tg−bˆ) +λO(g)_{. (4.5)} The first term stands for the data fidelity, i.e. the residuum weighted by the expected covariance matrix Σ, O(g) denotes a regularization functional and λ is a positive regularization parameter balancing the strength of a priori constraints with respect to the goodness of fit. For the sake of simplicity, we will further assume that T andb are already weighted by a square root of the covariance matrix

T←Σ⁻¹²T b←Σ⁻¹²b.

In other words, we are removing the ill-posedness of the problem by an intentional biasing of the solution. The biasing is unavoidable and any other approach to solve the tomography is hiding this regularization as well. Using the optimal choice of priori knowledge and a detailed model of the diagnostic, the bias can be minimized.

A priori knowledge about the emissivity profiles is imposed by the operatorO(g)_, which is often some kind of a roughness penalty and a boundary constraint. The regu-larization operator is typically a quadratic form O(g) = g^TH(g)g, with a symmetric and positive semi-definite operator H∈R^n,n which can be a function of g. A common boundary constraint is zero emissivity at the borders, enforced by adding a sufficiently large positive value to the diagonal points of H corresponding to the pixels outside of the boundary and this is forcing these pixels during reconstruction to zero.

The minimum of Eq.(4.5) can be now expressed as a quadratic optimization problem for a constant positive definite matrix H

Λ(g) =||Tg−b||ˆ ²₂+λg^THg (4.6) with the minimum reached at

g= (T^TT+λH)⁻¹T^Tbˆ. (4.7) A direct solving of this system of linear system of equations is possible. However, due to the large dimensionality of the matrices (n×n), the high computational complexity O(n³) of the inversion, and the required number of inversions to find a proper λ this procedure is highly impractical. Much more efficient algorithms to solve Eq. (4.7) and methods to estimate the optimal value ofλ, have been developed within this thesis and more details can be found in the Appendix A. As a result the linear system of equations is typically solved with a complexity O(n), allowing for large number of pixelsn and a fast inversions.

4. Tomographic Reconstruction of the Local Emissivity

4.1.2. Minimum Fisher Information

We have not specified the H operator, yet. The most common regularization opera-tors are the identity operator, suppressing the Euclidean norm of the solution, and the Laplace operator, reducing the curvature of the emissivity profile [44, 45]. Neverthe-less, the special features of the SXR profiles, like the peaked distribution of the SXR ra-diation, the sharp gradients, and the large dynamic range, make the nonlinear Minimum Fisher Information regularization (MFI) [46] an ideal candidate for the regularization functional. This functional can be expressed in the following form:

O_MFI(G) =

ZZ dS 1

G(∇_uG)^TJ₂(∇_vG)_{, (4.8)} where u(x,y) _and v(x,y) are two locally orthogonal vector fields and J₂ ∈ R^2,2 represents a matrix of ones. Ifu is parallel with the Cartesian coordinates, the method is called isotropic MFI and if u is locally tangential to the magnetic flux surfaces, theanisotropic MFI regularization is obtained. More details about the implementation of the anisotropic MFI can be found in [47]. The regularization operatorH, representing a linearized and discretized functional (4.8), is given by:

H^(k) = ^X

`∈{u,v}

B^T_{`</sub>W<sup>(k)</sup>B<sub>`}, (4.9)

whereB_{`</sub>denotes a discretized gradient operator∇<sub>`}andW_ij^(k) is a weight matrix defined as the inverse ofg^(k)

W_ij^(k+1) =δ_ij/max{g_j^(k),ε}.

The MFI regularization must be solved iteratively, because the weight matrixW depends on the emissivity g.