CPBASEX

pBASEX Abel Inversion without Polar Rebinning... in MATLAB and Python. A broader introductionto Abel inversion and the pBASEX algorithm is available in README.pdf.

Quick feature list:

• Implementation of the pBASEX Abel inversion algorithm by direct sampling of the basis functions on a Cartesian grid rather than rebinning the data into polar coordinates and introducing some error.
• Use of singular value decomposition to speed up the least-squares fitting step.
• L2-regularized (also fast) or pixel weighted (slower) fitting available.
• Parallelized and efficient Abel transformed basis function calculations.

Installation

For MATLAB, simply add CPBASEX/pbasex-MATLAB/ to the MATLAB path.

For Python, a pip installation is recommended. From the command line (perhaps using a virtual environment), simply install the PyPi CPBASEX distribution:

$ pip3 install pbasex

This will install the Python version as well as several dependencies:

• NumPy, a popular Python package for numerical programming.
• H5Py, an HDF5 file storage implementation.
• Cython, C-extensions for Python, required by the H5Py package.
• Quadrant, a small package to deal with folding and unfolding images to and from the image quadrants.

Pre-installing Cython and H5Py will make the installation process much faster, as opposed to installation using pip dependencies:

$ pip3 install cython
$ pip3 install h5py
$ pip3 install pbasex

The dill is also required to parallelize the Abel transformed basis function sampling step:

$ pip3 install dill

Running the code

The pBASEX algorithm works in two steps. First, the Abel transformed basis functions must be sampled. Next, a least-squares fit inverts measured data. Sample code for the first (examples/save_gData.m, examples/save_gData.py) and the second (examples/sample_pbasex.m, examples/sample_pbasex.py) is available. The first step must only be run once, with the results being saved locally. The second step can be run with these results for any amount of new raw image data.

In save_gData, several key parameters that can be changed are:

• the number of radial pixels (quadrant.resizeFolded can also be used to appropriately resize a folded image)
• the ratio of radial pixels to radial basis functions
• the number of angular basis functions
• the functional form of the radial basis functions.

In sample_pbasex, the steps required to fold an image and invert it are shown. Various forms of the result are also plotted. This requires the Matplotlib library, which can also easily be installed with pip:

$ pip3 install matplotlib

Code wish list

• Propagating experimental errors (as mentioned by GitHub user phockett) through the inversion procedure. Although implementation optimizations hide this fact, all calculations here can be represented as matrix multiplications, such that error propagation can be implemented as Var(y) = A * Var(x) * A', where y = A * x. I have this partially implemented locally, although there it is a bit difficult to test for accuracy. Also, if Var(x) is a full covariance matrix, calculations get prohibitively large, so I have only tried using diagonal covariance matrices (no coupled errors).
• Allowing for odd angular momentum contributions, accounting for non-isotropic target ensembles. This would need a bit of rework since the single quadrant picture is no longer accurate.
• Rework the Python code into an object-oriented architecture (similar to cart2polar).