ess.sans.beam_center_finder.cost

ess.sans.beam_center_finder.cost(xy, *args)

Cost function for determining how close the \(I(Q)\) curves are in all four quadrants. The cost is defined as

\[\text{cost} = \frac{\sum_{Q}\sum_{i=1}^{i=4} \overline{I}(Q)\left(I(Q)_{i} - \overline{I}(Q)\right)^2}{\sum_{Q}\overline{I}(Q)} ~,\]

where \(i\) represents the 4 quadrants and \(\overline{I}(Q)\) is the mean intensity of the 4 quadrants as a function of \(Q\). This is basically a weighted mean of the square of the differences between the \(I(Q)\) curves in the 4 quadrants with respect to the mean, and where the weights are \(\overline{I}(Q)\). We use a weighted mean, as opposed to relative (percentage) differences to give less importance to regions with low statistics which are potentially noisy and would contribute significantly to the computed cost.

Parameters

• xy (List[float]) – The x,y offsets in the plane normal to the beam.

• *args – Arguments passed to iofq_in_quadrants().

Returns

float – The sum of the residuals for \(I(Q)\) in the 4 quadrants, with respect to the mean \(I(Q)\) in all quadrants.

Notes

Mantid uses a different cost function. They compute the horizontal (Left - Right) and the vertical (Top - Bottom) costs, and require both to be below the tolerance. The costs are defined as

\[\text{cost} = \sum_{Q} \left(I(Q)_{\text{L,T}} - I(Q)_{\text{R,B}}\right)^2 ~.\]

Using absolute differences instead of a weighted mean is similar to our cost function in the way that it would give a lot of weight to even a small difference in a high-intensity region. However, it also means that an absolute difference of e.g. 2 in a high-intensity region would be weighted the same as a difference of 2 in a low-intensity region. It is also not documented why two separate costs are computed, instead of a single one. The Mantid implementation is available `here <https://github.com/mantidproject/mantid/blob/main/Framework/PythonInterface/plugins/algorithms/WorkflowAlgorithms/SANS/SANSBeamCentreFinder.py`_.