Functions defined on the sphere

This module contains two functions which can be used to convert functions defined on the sphere (given in the form of a HEALPix map) to an array of 12 cartesian images, where each of the cartesian images corresponds to a pullback of the given function with respect to one of the 12 HEALPix charts.

Since the HEALPix charts are area preserving, taking the sphere-based alpha-shearlet transform is the same as taking the cartesian alpha-shearlet transform of each of the 12 cartesian images and concatenating the results.

Reconstruction from these (possibly modified, e.g. thresholded) coefficients is straightforward: Reconstruct each of the 12 cartesian images, concatenate the results and push the result back to the sphere, using put_all_faces(), which again results in a HEALPix map.

SphereTransform.get_all_faces(Imag, nested=False)

This function maps a function defined on the sphere to an array of 12 cartesian images, where each of the cartesian images corresponds to a pullback of the given function with respect to one of the 12 HEALPix charts.

Required parameter

Parameters:Imag (numpy.ndarray) –

The function defined on the sphere, in the form of a one-dimensional numpy.ndarray (a HEALPix map).

Warning

If the HEALPix map Imag uses “NESTED” ordering, the parameter nested needs to be set to True.

Keyword parameter

Parameters:nested (bool) –

This parameter determines the ordering with which the different HEALPix pixels are stored in the array Imag; see http://healpix.jpl.nasa.gov/html/intronode4.htm for more details.

If nested is set to False, this signifies that Imag uses the “RING” ordering.

Return value

Returns:A 3-dimensional numpy.ndarray consisting of 12 cartesian images (2-dimensional arrays), where each image is a pullback of the input function Imag with respect to one of the 12 HEALPix charts.

SphereTransform.put_all_faces(CubeFace, nested=False)

Given 12 cartesian functions corresponding to the 12 HEALPix faces, this function pushes each of them back to the sphere using one of the HEALPix charts. The 12 different functions are pasted together to obtain a function defined on all of the sphere.

Required parameter

Parameters:CubeFace (numpy.ndarray) – A 3-dimensional numpy.ndarray consisting of 12 cartesian images (2-dimensional arrays) corresponding to the 12 HEALPix faces.

Keyword parameter

Parameters:nested (bool) –

This parameter determines the ordering with which the different HEALPix pixels are stored in the returned array. See http://healpix.jpl.nasa.gov/html/intronode4.htm for more details.

If nested is set to False, this signifies that the return value should use the “RING” ordering.

Return value

Returns:A one-dimensional numpy.ndarray (a HEALPix map) corresponding to the resulting “pasted” function.