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API Reference

Functions

SphericalHarmonicTransforms.synthesize! — Function
synthesize!(mapbuf::AbstractArray{<:Real}, ringpix::AbstractRingPixelization, alms::AbstractMatrix{<:Complex})

Performs the spherical harmonic synthesis transform of the harmonic coefficients alms for a real-valued map mapbuf, pixelized as described by ringpix.

See also synthesize, analyze, analyze!

Extended help

The synthesis transform is implemented such that a real-valued function $f(θ, ϕ)$ is generated from complex-valued spherical harmonic coefficients $a_{ℓm}$ as

\[ f(θ, ϕ) = \sum_{ℓ=0}^{ℓ_\mathrm{max}} \sum_{m=0}^{ℓ} a_{ℓm} Y_{ℓm}(θ, ϕ)\]

where the $a_{ℓm}$s matrix contains only the non-negative orders ($m ≥ 0$) since the negative orders are not independent when $f(θ, ϕ)$ is real.

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SphericalHarmonicTransforms.analyze — Function
alms = analyze(ringpix::AbstractRingPixelization, mapbuf::AbstractArray{<:Real}, lmax::Integer, mmax::Integer = lmax)

Performs the spherical harmonic analysis transform of the real-valued map mapbuf, pixelized as described by ringpix, to the corresponding complex-valued spherical harmonic coefficients alms.

See also analyze!, synthesize, synthesize!

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SphericalHarmonicTransforms.analyze! — Function
analyze!(alms::AbstractMatrix{<:Complex}, ringpix::AbstractRingPixelization, mapbuf::AbstractAray{<:Real})

Performs the discrete spherical harmonic analysis transform of the real-valued map mapbuf, pixelized as described by ringpix, to the corresponding complex-valued spherical harmonic coefficients alms.

See also analyze, synthesize, synthesize!

Extended help

The analysis transform is implemented such that a real-valued function $f(θ, ϕ)$ is analyzed for its corresponding complex-valued spherical harmonic coefficients $a_{ℓm}$ as

\[ a_{ℓm} = \sum_{ℓ=0}^{ℓ_\mathrm{max}} \sum_{m=0}^ℓ f(θ, ϕ) \overline{Y_{ℓm}}(θ, ϕ) ΔΩ\]

where the overline indicates complex conjugation, $ΔΩ$ is the integration element (pixel area) on the sphere, and the $a_{ℓm}$s matrix contains only the non-negative orders ($m ≥ 0$) since the negative orders are not independent when $f(θ, ϕ)$ is real.

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Ring-based Pixelizations

Pixelizations

SphericalHarmonicTransforms.RingPixelization — Type
RingPixelization{R} <: AbstractRingPixelization{R}

A pixelization of the unit sphere described by an arbitrary vector of Rings.

Examples

julia> ringpix = RingPixelization(ECPPixelization(5, 10))
5-ring RingPixelization{Float64} with 50 pixels
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SphericalHarmonicTransforms.ECPPixelization — Type
ECPPixelization{R} <: AbstractRingPixelization{R}

An Equidistant Cylindrical Projection pixelization in column-major order, with dimensions nθ × nϕ in the colatitude/azimuth directions, respectively.

See also RingPixelization

Examples

julia> ECPPixelization(250, 500)
250×500 ECPPixelization{Float64}

julia> ECPPixelization{Float32}(250, 500)
250×500 ECPPixelization{Float32}

Extended help

The described pixelization covers the entire sphere $[0, π] × [0, 2π]$ with uniformly-sized pixels in coordinate space of size $Δθ = π/n_θ$ by $Δϕ = 2π/n_ϕ$. The pixel centers take on coordinates

\[ (θ_j, ϕ_k) = \left( π \frac{2j + 1}{2n_θ}, 2π \frac{2k + 1}{2n_ϕ} \right)\]

for integers $j ∈ \{0, …, n_θ - 1\}$ and $k ∈ \{0, …, n_ϕ - 1\}$. On the sphere, the pixels cover a solid angle $ΔΩ_{jk} ≈ \sin(θ_j) Δθ Δϕ$, i.e. the pixels' physical size approaches zero towards the poles.

See also equidistant cylindrical projection

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Abstract Pixelization Interface

The following functions and types are considered part of the public interface, though they are unlikely to be used unless you are implementing a new pixelization.

SphericalHarmonicTransforms.Ring — Type
Ring{R<:Real}

Parameters required to describe an isolatitude ring on the sphere compatible with spherical harmonic transforms.

Fields

  • offset::Tuple{Int, Int}: Offsets (1-indexed) to the first pixel of the isolatitude ring(s) within a map array. The first offset in the tuple corresponds to the colatitude $θ$ given by .cosθ, while the second offset is for the ring mirrored over the equator (colatitude $π - θ$). Either offset may be 0 to indicate the ring is not present (such as for the equator, but also partial sphere coverage such as transforms on bands).
  • stride::Int: Stride between pixels within the ring.
  • nϕ::Int: The number of equispaced pixels in the isolatitude ring.
  • cosθ::R: Cosine of the colatitude of the ring, i.e. $\cos(θ)$.
  • ϕ₀_π::R: Azimuth of the first pixel in the isolatitude ring $ϕ₀$ divided by $π$.
  • ΔΩ::R: Solid angle area of pixels in the ring, roughly $\sin(θ) × Δθ × Δϕ$.
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SphericalHarmonicTransforms.buffer — Function
mapbuf = buffer(ringpix::AbstractRingPixelization, ::Type{T})

Allocates an array mapbuf with element type T appropriate for containing a map in the pixelization scheme described by ringpix.

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