Usage

• Usage
• • Getting Started
  • Performing Transforms

Getting Started

This library is not registered in Julia's General registry, so the package must be installed either by cloning it directly:

(@v1.7) pkg> add https://github.com/jmert/SphericalHarmonicTransforms.jl

or by making use of my personal registry

(@v1.7) pkg> registry add https://github.com/jmert/Registry.jl

(@v1.7) pkg> add SphericalHarmonicTransforms

After installing, just load like any other Julia package:

julia> using SphericalHarmonicTransforms

SphericalHarmonicTransforms exports only a couple of functions from its public interface: pairs of functions for spherical harmonic transform synthesis (synthesize/synthesize!) and analysis (analyze/analyze!).

To access other portions of the public API, the bindings must be referred to by full name or imported into the local scope.

julia> import SphericalHarmonicTransforms as SHT

Performing Transforms

The only built-in pixelization is for an equidistant cylindrical projection (ECP), represented by an instance of an ECPPixelization structure. The constructor takes at least the number of pixels in the colatitude and azimuth directions, and optionally an element type as a type parameter which defaults to Float64. (The type parameter is used to determine the element type of newly allocated output arrays.)

julia> pix = SHT.ECPPixelization(50, 100) # == SHT.ECPPixelization{Float64}(50, 100)50×100 ECPPixelization{Float64}

The object pix describes an array which corresponds to a column-major ordered Float64 matrix of size 50 × 100 that covers the sphere with pixels that are spaced uniformly in latitude and longitude.

With the pixelization described, a set of spherical harmonic coefficients $a_{ℓm}$ can be synthesized to form a map using synthesize. For example, we can draw the real part of the quadrupole moment $a_{20}$:

julia> alm = zeros(ComplexF64, 3, 3); alm[3, 1] = 1;  # 1-indexed array for 0-indexed quantity
julia> quad = synthesize(pix, alm)50×100 Matrix{Float64}:
 0.62985   0.62985   0.62985   0.62985   …  0.62985   0.62985   0.62985
 0.622403  0.622403  0.622403  0.622403     0.622403  0.622403  0.622403
 0.607629  0.607629  0.607629  0.607629     0.607629  0.607629  0.607629
 0.585758  0.585758  0.585758  0.585758     0.585758  0.585758  0.585758
 0.557137  0.557137  0.557137  0.557137     0.557137  0.557137  0.557137
 0.522216  0.522216  0.522216  0.522216  …  0.522216  0.522216  0.522216
 0.481546  0.481546  0.481546  0.481546     0.481546  0.481546  0.481546
 0.43577   0.43577   0.43577   0.43577      0.43577   0.43577   0.43577
 0.385607  0.385607  0.385607  0.385607     0.385607  0.385607  0.385607
 0.331851  0.331851  0.331851  0.331851     0.331851  0.331851  0.331851
 ⋮                                       ⋱
 0.385607  0.385607  0.385607  0.385607     0.385607  0.385607  0.385607
 0.43577   0.43577   0.43577   0.43577      0.43577   0.43577   0.43577
 0.481546  0.481546  0.481546  0.481546     0.481546  0.481546  0.481546
 0.522216  0.522216  0.522216  0.522216     0.522216  0.522216  0.522216
 0.557137  0.557137  0.557137  0.557137  …  0.557137  0.557137  0.557137
 0.585758  0.585758  0.585758  0.585758     0.585758  0.585758  0.585758
 0.607629  0.607629  0.607629  0.607629     0.607629  0.607629  0.607629
 0.622403  0.622403  0.622403  0.622403     0.622403  0.622403  0.622403
 0.62985   0.62985   0.62985   0.62985      0.62985   0.62985   0.62985

and likewise we can (approximately) analyze the rendered map back to spherical harmonic coefficients with analyze:

julia> analyze(pix, quad, #=lmax=# 5)6×6 Matrix{ComplexF64}:
 0.000368242+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
         0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
     1.00082+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
         0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
  0.00110856+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
         0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im