Removing the mask -- reconstructing a scalar field on the sphere from a masked field

TL;DR

A spectral approach to reconstructing a scalar field on the sphere, given only information about a masked version of the field together with precise information about the (smooth) mask, which is highly satisfactory in the absence of noise and in the presence of moderate noise.

Abstract

The paper analyses a spectral approach to reconstructing a scalar field on the sphere, given only information about a masked version of the field together with precise information about the (smooth) mask. The theory is developed for a general mask, and later specialised to the case of an axially symmetric mask. Numerical experiments are given for the case of an axial mask motivated by the cosmic microwave background, assuming that the underlying field is a realisation of a Gaussian random field with an artificial angular power spectrum of moderate degree ($\ell \le 100$). The recovery is highly satisfactory in the absence of noise and even in the presence of moderate noise.

Figures (8)

• Figure 1: Original Gaussian random field
• Figure 2: An axially symmetric $C^3$ mask with $a_z = \frac{\pi}{2} - \frac{10\pi}{180}$ and $b_z = \frac{\pi}{2}- \frac{20\pi}{180}$.
• Figure 3: The $C^3$ mask $v(\theta,\phi)$ with $a = \frac{10\pi}{180}$, $b = \frac{20\pi}{180}$.
• Figure 4: The masked noisy random field for $\tau = 10^{-4}$
• Figure 5: Singular values and condition numbers for the the matrices $E^{(m)}$ with $L=100$ and $K=900$ and the radially symmetric mask in Figure \ref{['fig:mask_theta']}.

Theorems & Definitions (15)

• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 4.1
• Remark 1
• proof
• Remark 2