computedmap.doc

Compute the space-concentration kernel of an arbitrary mask on the sphere.

Usage
-----
dij = ComputeDMap (dh_mask, lmax, [n, sampling, degrees])

Returns
-------
dij : float, dimension ( (lmax+1)**2, (lmax+1)**2 )
    The space-concentration kernel corresponding to the mask dh_mask.

Parameters
----------
dh_mask : integer, dimension (nin, sampling*nin)
    A Driscoll and Healy (1994) sampled grid describing the concentration region
    R. All elements should either be 1 (for inside the concentration region) or
    0 (for outside R).
lmax : integer
    The maximum spherical harmonic degree of the matrix dij.
n : optional, integer, default = nin
    The number of latitudinal samples in dh_mask. The effective spherical
    harmonic bandwidth of this grid is L=n/2-1.
degrees : integer, optional, dimension (lmax+1), default = 1
    List of degrees to use when computing the space-concentration kernel. Only
    those degrees where degrees[l] is non-zero will be employed.
sampling : optional, integer, default determined by dimensions of dh_mask
    For 1, dh_mask has n x n samples. For 2, dh_mask has n x 2n samples.

Description
-----------
ComputeDMap will calculate the space-concentration kernel for a generic mask
defined on the sphere. The input mask dh_mask must be sampled according to the
Driscoll and Healy (1994) sampling theorem with n samples in latitude, and
possess a value of 1 inside the concentration region, and 0 elsewhere. dh_mask
can either possess n samples in longitude (sampling=1) or 2n samples in
longitude (sampling=2). Given the approximate way in which the elements of dij
are calculated (see below), sampling=2 should be preferred. dij is symmetric,
and the elements are ordered according to the scheme described in
YilmIndexVector. See Simons et al. (2006) for further details. If the optional
vector degrees is specified, then the matrix will be computed only for elements
where degrees(l) is not zero.

The elements of DIJ are explicitly given by

Dlm,l'm' = 1/(4pi) Integral_R Ylm Yl'm' dOmega,

where R is the concentration region. In this routine, all values of l'm' are
calculated in a single spherical harmonic transform for a given value of lm
according to

Dl'm' = 1/(4pi) Integral_Omega F Yl'm' dOmega.

where

F = Ylm dh_mask.

The function F is in general not a polynomial, and thus the coefficients Dl'm'
should not be expected to be exact. For this reason, the effective spherical
harmonic degree of the input mask (L=n/2-1) should be greater than lmax. The
exact value of n should be chosen such that further increases in n do not alter
the returned eigenvalues. The routine prints out the fractional area of the mask
computed in the pixel domain divided by D(0,0) (the fractional area computed by
the spherical harmonic transforms), and the ratio of the two should be close to
1. Experience suggests that l should be about 5 times lmax.

References
----------
Driscoll, J.R. and D.M. Healy, Computing Fourier transforms and convolutions on
the 2-sphere, Adv. Appl. Math., 15, 202-250, 1994.

Simons, F.J., F.A. Dahlen, and M.A. Wieczorek, Spatiospectral concentration on a
sphere, SIAM Review, 48, 504-536, 2006.