Second Pass Through Photons

Once potential photon events have been located, the centroiding algorithm processes the list of potential events. If an event is determined to be a photon, then the resultant parameters are calculated and added to a list of detected photons. The flow of this algorithm is shown in this flowchart.

The algorithm starts by assessing what the potential photon’s overlap with other events is. This is done using the overlap image calculated in the firstpass. Here if the sum of the \(n \times n\) cluster is calculated. If the sum is equal to the cluster size, then the photon event is isolated. If the value is greater than that of the cluster size, then the photon event overlaps another event. If this value is greater than the value specified in the parameters (overlap_max), then the event is rejected.

Next, the pixel intensities in the \(n \times n\) box are sorted, highest first via a bubble sort . The background value is calculated by averaging the last pixel_photon_bgnd values in the sorted list. The integrated intensity of the photon is then calculated by summing up the first pixel_photon_num values is the sorted list after subtracting the average background value just calculated. The photon event is then filtered using the integrated intensity by the sum_min and sum_max.

After calculating the integrated intensity, the center of mass of the photon event is calculated using the equations:

\[C_x = \frac{\sum_i I_i x_i}{\sum_i I_i} \quad\text{and}\quad C_y = \frac{\sum_i I_i y_i}{\sum_i I_i}\]

where \(I_x\) and \(I_y\) are the intensity of the background corrected pixel intensities, \(x\) and \(y\) are the fractional pixel coordinates and the sum index \(i\) is over all the pixels in the cluster.

The pixel cluster is then fitted using a least squares algorithm in both 2D and 1D. The pixel cluster is fitted to the error function being the integral of a gaussian function. In 2D this function is:

\[ \begin{align}\begin{aligned}\DeclareMathOperator\erf{erf}\\I = B + I_0 \left( \erf \frac{\left( x - \frac{1}{2} - x_0 \right)}{\sqrt{2} \sigma} - \erf \frac{\left( x + \frac{1}{2} - x_0 \right)}{\sqrt{2} \sigma} \right)\cdot\left( \erf \frac{\left( y - \frac{1}{2} - y_0 \right)}{\sqrt{2} \sigma} - \erf \frac{\left( y + \frac{1}{2} - y_0 \right)}{\sqrt{2} \sigma} \right)\end{aligned}\end{align} \]

With the 1D version being:

\[ \begin{align}\begin{aligned}\DeclareMathOperator\erf{erf}\\I = B + I_0 \left( \erf \frac{\left( x - \frac{1}{2} - x_0 \right)}{\sqrt{2} \sigma} - \erf \frac{\left( x + \frac{1}{2} - x_0 \right)}{\sqrt{2} \sigma} \right)\end{aligned}\end{align} \]

where \(I\) is the intensity at pixel \(x\) (or \(x,y\) in 2D), \(B\) is the background and \(x_0\) and \(y_0\) are the center coordinates of the gaussian.

The fit parameters \(\sigma\) and \(x\) (or \(x,y\) in 2D), can be constrained by passing their constrains to the photon finding routine. These constraints are implemented in the least squares fitting using the \(\tanh\) function, of the form:

\[ \begin{align}\begin{aligned}\DeclareMathOperator\tanh{tanh}\\p = m + r \tanh p_f\end{aligned}\end{align} \]

where \(p_f\) is the parameter in the least squares fit, \(p\) is the parameter in the fitting function, \(m\) is the mean of the fitting parameter range and \(r\) is the range. This results in the fitting parameter being constrained in the range \((m - r) < p < (m + r)\).

The results of the fit are stored in the output table of photon parameters. The individual parameters are stored along with the computed error, calculated as the square root of the diagonal of the covarience matrix.

The standard error on the fit is also calculated, given a suitable fit. The standard error is computed by comparing the actual data with the model:

\[\mathrm{Std\,Err} = \sqrt{\frac{\sum_N \left( m - d \right)^2}{N}}\]

where \(N\) is the number of data points used in the fit, \(m\) is the model value and \(d\) is the (experimentally) measured value.

Second pass through events found in first pass