Synthesis Adjoint Wavelet Transform

[CosmoMatt]

Translate adjoint of the synthesis wavelet transform from s2let (c) to s2wav (base python).

Difficulty:

• High

Background:

• The wavelet transform can be performed as either analysis or synthesis, both with corresponding adjoint operators. Here we consider the adjoint of the synthesis transform which should map from a signal on the sphere to wavelet coefficients.

S2let File to translate:

• can be found here ignore the function on this line as it is no longer necessary.

S2Wav File location:

• should be put in the main package directory here

Notes:

• These functions will require some helper functions from ssht and so3 You should be able to access the necessary functions from the python versions of these packages for the time being. Look here for SSHT and here for so3 functions.

[CosmoMatt]

It is important that these transforms support both real and complex fields, however these two implementations are extremely similar and should (in the end) be refactored into a single function. Hence the sub-issues will be as follows:

• Implement real transform
• Implement complex transform
• Refactor real + complex transform into a single function call (i.e. use parameter dict as a switch)

Furthermore it is important that everything is thoroughly tested so we must include:

• Regression test against s2let (i.e. s2wav(x) ~ s2let(x) )
• Round-trip error test (i.e. x ~ synthesis( analysis ( x ) ) )
• Adjoint dot test