aare/docs/src/Interpolation.rst

Interpolation

The Interpolation class implements the η-interpolation method. This interpolation technique is based on charge sharing: for detected photon hits (e.g. clusters), it refines the estimated photon hit using information from neighboring pixels.

The method relies on the so-called η-functions, which describe the relationship between the energy measured in the central cluster pixel (the initially estimated photon hit) and the energies measured in its neighboring pixels. Depending on how much energy each neighboring pixel receives relative to the central pixel, the estimated photon hit is shifted toward that neighbor by a certain offset to the actual photon hit position in the pixel (x, y).

The mapping between the η values and the corresponding spatial photon position (x, y) can be viewed as an optimal transport problem.

One can readily compute the probability distribution P_η of the η values by forming a 2D histogram. However, the probability distribution P_x, y of the true photon positions is generally unknown unless the detector is illuminated uniformly (i.e. under flat-field conditions). In a flat-field, the photon positions are uniformly distributed.

With this assumption, the problem reduces to determining a transport map T : (η_x, η_y) → (x, y), that pushes forward the distribution of (η_x, η_y) to the known uniform distribution of photon positions of a flatfield.

The map T is given by:

$$\begin{aligned} \begin{align*} T_1: & F_{x}^{-1} F_{\eta_x|\eta_y} \\ T_2: & F_{y}^{-1} F_{\eta_y|\eta_x}, \end{align*} \end{aligned}$$

where F_ηx|ηy and F_ηy|ηx are the conditional cumulative distribution functions e.g. F_ηx|ηy(η_x^′, η_y^′) = P_ηx, ηy(η_x ≤ η_x^′|η_y = η_y^′). And F_x and F_y are the cumulative distribution functions of x and y. Note as x and y are uniformly distributed F_x and F_y are the identity functions. The map T thus simplifies to

$$\begin{aligned} \begin{align*} T_1: & F_{\eta_x|\eta_y} \\ T_2: & F_{\eta_y|\eta_x}. \end{align*} \end{aligned}$$

Note that for the implementation P_η is not only a distribution of η_x, η_y but also of the estimated photon energy e. The energy level correlates slightly with the z-depth. Higher z-depth leads to more charge sharing and a different η distribution. Thus we create a mapping T for each energy level.

η-Functions:

aare::Eta2

Note

The corner value c is only relevant when one uses calculate_eta_2 or calculate_full_eta2. Otherwise its default value is cTopLeft.

Supported are the following η-functions:

η-Function on 2x2 Clusters:

$$\begin{equation*} {\color{blue}{\eta_x}} = \frac{Q_{1,1}}{Q_{1,0} + Q_{1,1}} \quad \quad {\color{green}{\eta_y}} = \frac{Q_{1,1}}{Q_{0,1} + Q_{1,1}} \end{equation*}$$

The η values can range between 0,1. Note they only range between 0,1 because the position of the center pixel (red) can change. If the center pixel is in the bottom left pixel η_x will be close to zero. If the center pixel is in the bottom right pixel η_y will be close to 1.

One can apply this η not only on 2x2 clusters but on clusters with any size. Then the 2x2 subcluster with maximum energy is chosen and the η function applied on the subcluster.

aare::calculate_eta2(const ClusterVector<ClusterType>&)

aare::calculate_eta2(const Cluster<T, ClusterSizeX, ClusterSizeY, CoordType>&)

Full η-Function on 2x2 Clusters:

$$\begin{equation*} {\color{blue}{\eta_x}} = \frac{Q_{0,1} + Q_{1,1}}{\sum_i^{1}\sum_j^{1}Q_{i,j}} \quad \quad {\textcolor{green}{\eta_y}} = \frac{Q_{1,0} + Q_{1,1}}{\sum_i^{1}\sum_j^{1}Q_{i,j}} \end{equation*}$$

The η values can range between 0,1. Note they only range between 0,1 because the position of the center pixel (red) can change. If the center pixel is in the bottom left pixel η_x will be close to zero. If the center pixel is in the bottom right pixel η_y will be close to 1.

aare::calculate_full_eta2(const ClusterVector<ClusterType>&)

aare::calculate_full_eta2(const Cluster<T, ClusterSizeX, ClusterSizeY, CoordType>&)

Full η-Function on 3x3 Clusters:

$$\begin{equation*} {\color{blue}{\eta_x}} = \frac{\sum_{i=0}^{2} Q_{i,2} - \sum_{i=0}^{2} Q_{i,0}}{\sum_{i=0}^{2}\sum_{j=0}^{2} Q_{i,j}} \quad \quad {\color{green}{\eta_y}} = \frac{\sum_{j=0}^{2} Q_{2,j} - \sum_{j=0}^{2} Q_{0,j}}{\sum_{i=0}^{2}\sum_{j=0}^{2} Q_{i,j}} \end{equation*}$$

The η values can range between -0.5,0.5.

aare::calculate_eta3(const ClusterVector<ClusterType>&)

aare::calculate_eta3(const Cluster<T, ClusterSizeX, ClusterSizeY, CoordType>&)

Cross η-Function on 3x3 Clusters:

$$\begin{equation*} {\color{blue}{\eta_x}} = \frac{Q_{1,2} - Q_{1,0}}{Q_{1,0} + Q_{1,1} + Q_{1,2}} \quad \quad {\color{green}{\eta_y}} = \frac{Q_{0,2} - Q_{0,1}}{Q_{0,1} + Q_{1,1} + Q_{2,1}} \end{equation*}$$

The η values can range between -0.5,0.5.

aare::calculate_cross_eta3(const ClusterVector<ClusterType>&)

aare::calculate_cross_eta3(const Cluster<T, ClusterSizeX, ClusterSizeY, CoordType>&)

Interpolation class:

Warning

The interpolation might lead to erroneous photon positions for clusters at the borders of a frame. Make sure to filter out such cases.

Warning

Make sure to use the same η-function during interpolation as given by the joint η-distribution passed to the constructor.

Note

Make sure to use reasonable energy bins, when constructing the joint distribution. If data is too sparse for a given energy the interpolation will lead to erroneous results.

aare::Interpolator