2.2. empix.annularly_average

annularly_average(input_signal, optional_params=None)

Average annularly a given input 2D hyperspy signal.

This Python function assumes that the input 2D hyperspy signal samples from a mathematical function \(F_{\mathbf{m}}\left(u_{x},u_{y}\right)\) which is piecewise continuous in \(u_{x}\) and \(u_{y}\), where \(u_{x}\) and \(u_{y}\) are the horizontal and vertical coordinates in the signal space of the input signal, and \(\mathbf{m}\) is a vector of integers representing the navigation indices of the input signal. The Python function approximates the annular average of \(F_{\mathbf{m}}\left(u_{x},u_{y}\right)\) given the input signal. We define the annular average of \(F_{\mathbf{m}}\left(u_{x},u_{y}\right)\) as

(2.2.1)\[\begin{split}&\overline{S}_{\mathbf{m}}\left(u_{r,i}\le U_{r}<u_{r,f} \left|0\le U_{\phi}<2\pi;c_{x},c_{y}\right.\right) \\&\quad=\frac{1}{\pi\left(u_{r,f}^{2}-u_{r,i}^{2}\right)} \int_{u_{r,i}}^{u_{r,f}}du_{r}\,\int_{0}^{2\pi}du_{\phi}\,u_{r} F_{\mathbf{m}}\left(c_{x}+u_{r}\cos\left(u_{\phi}\right), c_{y}+u_{r}\sin\left(u_{\phi}\right)\right),\end{split}\]

where \(\left(c_{x},c_{y}\right)\) is the reference point from which the radial distance \(u_r\) is defined for the annular averaging.

Parameters:

input_signalhyperspy._signals.signal2d.Signal2D | hyperspy._signals.complex_signal2d.ComplexSignal2D

The input hyperspy signal.

optional_paramsempix.OptionalAnnularAveragingParams | None, optional

The set of optional parameters. See the documentation for the class empix.OptionalAnnularAveragingParams for details. If optional_params is set to None, rather than an instance of empix.OptionalAnnularAveragingParams, then the default values of the optional parameters are chosen.

Returns:

output_signalhyperspy.signal.BaseSignal | hyperspy._signals.complex_signal.ComplexSignal

The output hyperspy signal that samples the annular average of the input signal input_signal. Note that the metadata of the input signal is copied over to the output signal, with the title being overwritten.