flat_generic.jl

const FlatField{P,T,M} = Union{FlatS0{P,T,M},FlatS2{P,T,M},FlatS02{P,T,M}}

const FlatFieldMap{P,T,M} = Union{FlatMap{P,T,M},FlatS2Map{P,T,M},FlatS02Map{P,T,M}}
const FlatFieldFourier{P,T,M} = Union{FlatFourier{P,T,M},FlatS2{P,T,M},FlatS02Fourier{P,T,M}}

### pretty printing
@show_datatype show_datatype(io::IO, t::Type{F}) where {N,θ,∂mode,T,M,F<:FlatField{Flat{N,θ,∂mode},T,M}} =
    print(io, "$(pretty_type_name(F)){$(N)×$(N) map, $(θ)′ pixels, $(∂mode.name.name), $(M.name.name){$(M.parameters[1])}}")
for F in (:FlatMap, :FlatFourier, 
          :FlatQUMap, :FlatQUFourier, :FlatEBMap, :FlatEBFourier, 
          :FlatIQUMap, :FlatIQUFourier, :FlatIEBMap, :FlatIEBFourier)
    @eval pretty_type_name(::Type{<:$F}) = $(string(F))
end

### field info
@generated fieldinfo(::Type{P},::Type{T}=Float32,::Type{M}=Matrix) where {Nside,θpix,∂mode,P<:Flat{Nside,θpix,∂mode},T,M} = 
    (;FlatInfo(T,basetype(M),Val(θpix),Val(Nside))..., ∂mode=∂mode)
@generated fieldinfo(::Type{F}) where {P<:Flat,T,M,F<:FlatField{P,T,M}} = 
    (;fieldinfo(P,T,M)..., @namedtuple(P,M,B=basis(F),S=spin(F))...)
fieldinfo(::F) where {F<:FlatField} = fieldinfo(F)

### promotion & conversion
# note: we don't need to promote the eltype T here since that will be
# automatically handled in broadcasting
function promote(f1::F1, f2::F2) where {T1,θ1,N1,∂mode1,F1<:FlatField{Flat{N1,θ1,∂mode1},T1},T2,θ2,N2,∂mode2,F2<:FlatField{Flat{θ2,N2,∂mode2},T2}}
    B     = promote_type(basis(F1),basis(F2))
    ∂mode = promote_type(∂mode1,∂mode2)
    B(∂mode(f1)), B(∂mode(f2))
end
(::Type{∂mode})(f::F) where {∂mode<:∂modes,N,θ,F<:FlatS0{<:Flat{N,θ}}} = basetype(F){Flat{N,θ,∂mode}}(fieldvalues(f)...)
(::Type{∂mode})(f::FieldTuple{B}) where {∂mode<:∂modes,B} = FieldTuple{B}(map(∂mode,f.fs))

### basis-like definitions
LenseBasis(::Type{<:FlatS0})  =    Map
LenseBasis(::Type{<:FlatS2})  =  QUMap
LenseBasis(::Type{<:FlatS02}) = IQUMap
DerivBasis(::Type{<:FlatS0{<:Flat{<:Any,<:Any,fourier∂}}})  =    Fourier
DerivBasis(::Type{<:FlatS2{<:Flat{<:Any,<:Any,fourier∂}}})  =  QUFourier
DerivBasis(::Type{<:FlatS02{<:Flat{<:Any,<:Any,fourier∂}}}) = IQUFourier
DerivBasis(::Type{<:FlatS0{<:Flat{<:Any,<:Any,map∂}}})      =    Map
DerivBasis(::Type{<:FlatS2{<:Flat{<:Any,<:Any,map∂}}})      =  QUMap
DerivBasis(::Type{<:FlatS02{<:Flat{<:Any,<:Any,map∂}}})     = IQUMap

### derivatives

## Fourier-space
# the use of @generated here is for memoization so that we don't have do the
# `prefactor * im * ...` each time. actually, since for the first two cases
# these are only 1-d arrays, that's a completely unnecessary optimization, but
# without the @generated it triggers an order-dependenant compilation bug which
# e.g. slows down LenseFlow by a factor of ~4 so we gotta keep it for now ¯\_(ツ)_/¯
@generated broadcastable(::Type{F}, ::∇diag{1,<:Any,prefactor}) where {P,T,M,prefactor,F<:FlatFourier{P,T,M}} = 
    basetype(M)(@. prefactor * im * $fieldinfo(F).k')
@generated broadcastable(::Type{F}, ::∇diag{2,<:Any,prefactor}) where {N,P<:Flat{N},T,M,prefactor,F<:FlatFourier{P,T,M}} = 
    basetype(M)(@. prefactor * im * $fieldinfo(F).k[1:N÷2+1])
@generated broadcastable(::Type{F}, ::∇²diag) where {N,P<:Flat{N},T,M,F<:FlatFourier{P,T,M}} =
    basetype(M)(@. $fieldinfo(F).k'^2 + $fieldinfo(F).k[1:N÷2+1]^2)

## Map-space
function copyto!(f′::F, bc::Broadcasted{<:Any,<:Any,typeof(*),Tuple{∇diag{coord,covariant,prefactor},F}}) where {coord,covariant,prefactor,T,θ,N,F<:FlatMap{Flat{N,θ,map∂},T}}
    f = bc.args[2]
    n,m = size(f.Ix)
    α = 2 * prefactor * fieldinfo(f).Δx
    if coord==1
        @inbounds for j=2:m-1
            @simd for i=1:n
                f′[i,j] = (f[i,j+1] - f[i,j-1])/α
            end
        end
        @inbounds for i=1:n
            f′[i,1] = (f[i,2]-f[i,end])/α
            f′[i,end] = (f[i,1]-f[i,end-1])/α
        end
    elseif coord==2
        @inbounds for j=1:n
            @simd for i=2:m-1
                f′[i,j] = (f[i+1,j] - f[i-1,j])/α
            end
            f′[1,j] = (f[2,j]-f[end,j])/α
            f′[end,j] = (f[1,j]-f[end-1,j])/α
        end
    end
    f′
end



### bandpass
HarmonicBasis(::Type{<:FlatS0}) = Fourier
HarmonicBasis(::Type{<:FlatQU}) = QUFourier
HarmonicBasis(::Type{<:FlatEB}) = EBFourier
broadcastable(::Type{F}, bp::BandPass) where {P,T,F<:FlatFourier{P,T}} = Cℓ_to_2D(P,T,bp.Wℓ)
    

### logdets
logdet(L::Diagonal{<:Complex,<:FlatFourier})   = real(sum_kbn(nan2zero.(log.(unfold(L.diag.Il)))))
logdet(L::Diagonal{<:Real,<:FlatMap})          = real(sum_kbn(nan2zero.(log.(complex(L.diag.Ix)))))
logdet(L::Diagonal{<:Complex,<:FlatEBFourier}) = real(sum_kbn(nan2zero.(log.(unfold(L.diag.El)))) + sum_kbn(nan2zero.(log.(unfold(L.diag.Bl)))))
### traces
tr(L::Diagonal{<:Complex,<:FlatFourier})   = real(sum_kbn(unfold(L.diag.Il)))
tr(L::Diagonal{<:Real,<:FlatMap})          = real(sum_kbn(complex(L.diag.Ix)))
tr(L::Diagonal{<:Complex,<:FlatEBFourier}) = real(sum_kbn(unfold(L.diag.El)) + sum_kbn(unfold(L.diag.Bl)))


### misc
Cℓ_to_Cov(f::FlatField{P,T}, args...) where {P,T} = Cℓ_to_Cov(P,T,spin(f),args...)

function pixwin(f::FlatField) 
    @unpack θpix,P,T,k = fieldinfo(f)
    Diagonal(FlatFourier{P,T}((pixwin.(θpix,k) .* pixwin.(θpix,k'))[1:end÷2+1,:]))
end