Background

A geometry matrix is a linear model which predicts the brightnesses that line-integrated instruments (e.g. cameras, spectrometers) would measure when imaging a given distribution of emission.

The geometry matrix element can be derived as follows: To predict the measured brightness of the \(i\)’th line-of-sight \(b_i\) we multiply the emission \(\mathcal{E}(x, y, z)\) by a ‘sensitivity’ function \(\mathcal{S}_i (x, y, z)\) which describes what fraction of emission at any point in space is measured as brightness by the instrument, and intgrate this product over all space so that

\[b_i = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \mathcal{E}(x, y, z) \mathcal{S}_i (x, y, z) \,\mathrm{d}x \,\mathrm{d}y \,\mathrm{d}z .\]

Next, we assume that the emission function is toroidally symmetric, such that it depends only on the major radius \(R = \sqrt{x^2 + y^2}\) and \(z\). Additionally, we assume the emission can be expressed as a weighted-sum of 2D basis functions \(\phi_j (R,z)\) such that

\[\mathcal{E}(R, z) = \sum_{j} x_j \phi_j (R,z)\]

where \(x_j\) are the basis function weights. We may now re-write \(b_i\) as

\[b_i = \sum_j G_{ij} x_j\]

where \(G_{ij}\) is the geometry matrix element given by

\[G_{ij} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \phi_j (R,z) \mathcal{S}_i (x, y, z) \,\mathrm{d}x \,\mathrm{d}y \,\mathrm{d}z .\]

After calculating the geometry matrix \(\mathbf{G}\), the vector of brightness predictions \(\mathbf{b}\) can be obtained through a single matrix-vector product such that \(\mathbf{b = Gx}\).

However, defining a 3D sensitivity function for each line-of-sight is complicated, and computing the 3D integral for all elements of the geometry matrix is very expensive for instruments with a large number of lines-of-sight, such as a camera.

Instead, a further approximation is made, where we assume that each line-of-sight collects emission only along a single line \(\ell_i\). This allows \(G_{ij}\) to be re-written as a line-integral through the basis functions so that

\[G_{ij} = \int \phi_j (R,z) \,\mathrm{d}\ell_i.\]

Choice of basis functions

When using a triangular mesh to represent the solution of a tomography problem, the typical approach is to assume that the emission inside each triangle is constant. This is equivalent to zeroth-order interpolation, and leads to the following set of basis functions:

\[\begin{split}\phi_i (R,z) = \begin{cases} 1 & \quad \text{if point } (R,z) \text{ is inside triangle } i \\ 0 & \quad \text{otherwise} \end{cases}\end{split}\]

Tokamesh instead uses first-order (barycentric) interpolation to define the emission inside each triangle. In this approach, the basis-function weights \(x_i\) become the emissivity \(\mathcal{E}_i\) at each vertex. The emissivity inside a triangle made up of vertices \(i\), \(j\) and \(k\) is given by the plane defined by the three points \((R_i, z_i, \mathcal{E}_i), (R_j, z_j, \mathcal{E}_j), (R_k, z_k, \mathcal{E}_k)\).

This leads to the following ‘barycentric’ basis functions:

\[\begin{split}\phi_i (R,z) = \begin{cases} \lambda_i (R,z) & \quad \text{if point } (R,z) \text{ is inside a triangle containing vertex } i \\ 0 & \quad \text{otherwise} \end{cases}\end{split}\]

where \(\lambda_i (R,z)\) is the barycentric coordinate for vertex \(i\) given by

\[\lambda_i (R,z) = \frac{ (z_j - z_k)(R - R_k) + (R_k - R_j)(z - z_k) }{(z_j - z_k)(R_i - R_k) + (R_k - R_j)(z_i - z_k)}\]

and \((R_i, z_i), (R_j, z_j), (R_k, z_k)\) are the positions of vertices \(i\), \(j\) and \(k\) respectively.