Single-Index Ordering Schemes

This package uses the ANSI Z80.28-2004 standard sequential ordering scheme where applicable, but provides several functions for converting between two other ordering methods, namely Noll and Fringe. The following methods are available:

• noll_to_j(noll::Int): converts Noll indices to ANSI standard indices;
• j_to_noll(j::Int): converts ANSI standard indices to Noll indices;
• standardize(noll::Noll): re-orders a Noll specified Zernike expansion coefficient vector according to the ANSI standard;
• fringe_to_j(fringe::Int): converts Fringe indices to ANSI standard indices; only indices 1:37 are valid;
• j_to_fringe(j::Int): converts ANSI standard indices to Fringe indices;
• standardize(fringe::Fringe): formats a Fringe specified Zernike expansion coefficient vector according to the ANSI standard;
• standardize(v_sub::Vector, orders::Vector{Tuple{Int, Int}}): pads a subset Zernike expansion coefficient vector to the full standard length up to n_max (1:j_max+1).

The Noll and Fringe types are used to wrap the input coefficient vectors for the standardize method arguments (e.g. standardize(Fringe(v::Vector{Float64}))). These can also be used to convert between the ordering schemes (e.g. Noll(fringe::Fringe), Fringe(s::Standard)).

The standardize fringe method expects unnormalized coefficients; the input coefficients will be re-ordered and normalized in line with the orthonormal standard. As Fringe is a 37 polynomial subset of the full set of Zernike polynomials any coefficients in the standard order missing a counterpart in the input vector will be set to zero.

For the standardize subset method the tuples in orders must be of the form (m, n) associated with the respective coefficients at each index in v_sub.

In addition, the functions get_j(m, n) & get_mn(j) allow you to convert between the single and double indices.