Normal operator

This operator implements a lazy normal operator implementing:

\[\begin{equation} (\mathbf{A})^*\mathbf{A} \end{equation}\]

for some operator $\mathbf{A}$:

using FFTW
fop = op = FFTOp(ComplexF32, shape = (N, N))
nop = NormalOp(eltype(fop), parent = fop)
isapprox(nop * vec(image), vec(image))

true

And we can again access our original operator:

typeof(nop.parent)

LinearOperatorFFTWExt.FFTOpImpl{ComplexF32, Vector{ComplexF32}, FFTW.cFFTWPlan{ComplexF32, -1, true, 2, Tuple{Int64, Int64}}, FFTW.cFFTWPlan{ComplexF32, 1, true, 2, Tuple{Int64, Int64}}}

LinearOperatorCollection also provides an opinionated normalOperator function which tries to optimize the resulting normal operator. As an example consider the normal operator of a weighted fourier operator:

weights = Float32.(collect(range(0, 1, length = N*N)))
wop = WeightingOp(weights)
pop = ProdOp(wop, fop)
nop = normalOperator(pop)
typeof(nop.parent) == typeof(pop)

false

Note that the parent was changed. This is because the normal operator was optimized by initially computing the weights:

\[\begin{equation} \tilde{\mathbf{W}} = \mathbf{W}^*\mathbf{W} \end{equation}\]

and then applying the following each iteration:

\[\begin{equation} \mathbf{A}^*\tilde{\mathbf{W}}\mathbf{A} \end{equation}\]

Other operators can define different optimization strategies for the normalOperator method.

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