New storage format for harmonic RGSpaces?
I'm wondering whether we can apply the hermitian/anti-hermitian separation trick that we employ for spherical harmonic coefficients in the LMSpace to harmonic RGSpaces as well.
The symmetries are nearly the same: on the sphere we have
a_l,m = (-1)**m *conj(a_l,-m)
in a (multidimensional) RGSpace it is
f_k = conj(f_-k)
So in principle we could store the Fourier coefficients of a real-valued field in another real-valued field with exactly the same dimensions.
A 1D example:
Instead of storing the complex values
[ f0 f1 f2 f(-2) f(-1) ]
we would simply store
[ f0.re x*f1.re x*f1.im x*f2.re x*f2.im ], where x=sqrt(2.)
The coefficients at negative frequencies can be recovered trivially, and f0.im is 0 by symmetry. In analogy to the spherical case, sqrt(2.) is needed to conserve the dot products.
For arrays of even length:
[ f0 f1 f(-2) f(-1) ]
becomes
[ f0.re x*f1.re x*f1.im x*f2.re ]
(f2.im is zero in this case)
In multi-D, this treatment has to be applied to exactly one dimension; in analogy with FFTW, I suggest to use the last one.
Adopting such a scheme would have several significant advantages:
- memory requirements go down
- FFTs of real data will be faster
- the input data type of an FFToperator can always be the same as the output data type
- the transform of a real field will by construction always be exactly hermitian, and operators cannot destroy the hermitianity by accident
- minimizers won't be able to explore degrees of freedom that do not really exist in a given problem
Of course there will be drawbacks as well:
- whenever Fourier coefficients in the traditional format are required, conversions need to be run, which are quick but not absolutely free. (I don't think that this will be needed often.)
- the get_distance_array method of RGSpace will have to be adjusted.
- probably more stuff I'm missing at the moment.
I know that this is a crazy-looking idea, but especially in the light of the numerical problems we are currently encountering with the reconstruction of radio data, I think it should be investigated.