Transform Methods

The numerical Abel transform is computationally intensive, and a basic numerical integration of the analytical equations does not reliably converge. Consequently, numerous algorithms have been developed in order to approximate the Abel transform in a reliable and efficient manner. So far, PyAbel includes the following transform methods:

  1. * The basex method of Dribinski and co-workers, which uses a Gaussian basis set to provide a quick, robust transform. This is one of the de facto standard methods in photoelectron/photoion spectroscopy.
  2. The hansenlaw recursive method of Hansen and Law, which provides an extremely fast transform with low centerline noise.
  3. The direct numerical integration of the analytical Abel transform equations, which is implemented in Cython for efficiency. In general, while the forward Abel transform is useful, the inverse Abel transform requires very fine sampling of features (lots of pixels in the image) for good convergence to the analytical result, and is included mainly for completeness and for comparison purposes. For the inverse Abel transform, other methods are generally more reliable.
  4. * The three_point method of Dasch and co-workers, which provides a fast and robust transform by exploiting the observation that underlying radial distribution is primarily determined from changes in the line-of-sight projection data in the neighborhood of each radial data point. This technique works very well in cases where the real difference between adjacent projections is much greater than the noise in the projections (i.e. where the raw data is not oversampled).
  5. * The two_point method is also well described by Dasch. It is a simpler approximation to the three point transform. Computationally, very efficient in Python.
  6. * The onion_peeling onion-peeling deconvolution method described by Dash is one of the simpler, and faster inversion methods. The article states the onion-peeling deconvolution is similar to the two point Abel. Both methods have less smoothing than the other methods examined by Dasch.
  7. The onion_bordas onion-peeling method of Bordas et al. is based on the MatLab code of Rallis and Wells et al. The article claims “the method works properly only in the limit of large electrostatic energy to initial kinetic energy ratio and gives qualitatively the same results as a standard inversion method”.
  8. * The linbasex 1D-spherical basis method of Gerber et al. evaluates 1D projections of velocity-map images in terms of 1D projections of spherical functions. The results produce directly the coefficients of the involved spherical functions, making the reconstruction of sliced Newton spheres obsolete.
  9. (Planned implementation) The Fourier–Hankel method, which is computationally efficient, but contains significant centerline noise and is known to introduce artifacts.
  10. (Planned implementation) The POP (polar onion peeling) method. POP projects the image onto a basis set of Legendre polynomial-based functions, which can greatly reduce the noise in the reconstruction. However, this method only applies to images that contain features at constant radii. I.e., it works for the spherical shells seen in photoelectron/ion spectra, but not for flames.

* Methods marked with an asterisk require the generation of basis sets. The first time each method is run for a specific image size, a basis set must be generated, which can take several seconds or minutes. However, this basis set is saved to disk (generally to the current directory) and can be reused, making subsequent transforms very efficient. Users who are transforming numerous images using these methods will want to keep this in mind and specify the directory containing the basis sets.

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