AutoCorrelationResampling

This package provides means to resample autocorrelation functions.

Autocorrelation function of one variable is a Laurent polynomial on ℝ in the form $s(x) = f(x)f(x^{-1})$ where $f(x)$ is a usual polynomial on ℝ.

This package provides resampling of autocorrelation function in the sense that it changes the degree of $s(x)$ while maintaining the form $f(x)f(x^{-1})$.

Technically, this package works with autocorrelation function of three variables $s(x,y,z)$ in the form of three-dimensional arrays and rescales them along the third variable (or axis), i.e. an array of shape (x, y, z) becomes an array of shape (x, y, nz) where n is a resampling factor.

Why is it needed?

This package is an attempt to solve the problem of reconstruction of a porous media when only a fraction of information about original media is available (e.g. you have to reconstruct 3D cube from a stack of 2D slices taken along z axis). If those slices are evenly sampled (i.e. you have each n-th slice), you can do the following:

1. Stack the slices in 3D array. This array will have a length along z axis reduced by n times.
2. Calculate autocorrelation function (also known as two-point function) for the reduced array.
3. Upsample it by n times using this package
4. Reconstruct original 3D image (you can use PhaseRec.jl package for it).

How to use?

Resampling is done by ac_resample function which takes an autocorrelation array and a resample ratio. An optional low-pass filter can be passed to ac_resample and by default is obtained with filter_coeffs function. These filters are not unique, you can play with them supplying different argument initial to filter_coeffs.