Reconstruction of Missing Data using Iterative Harmonic Expansion

TL;DR

This paper analyzes an iterative harmonic expansion method for reconstructing missing data in cosmic maps, showing its effectiveness depends on the prior, mask size, and fluctuation properties, with improvements over naive methods.

Contribution

The paper formulates an asymptotic expansion for the harmonic reconstruction method and compares its performance with naive approaches, highlighting the impact of priors and mask characteristics.

Findings

Reconstruction accuracy improves with smaller masks and more iterations.

Gaussian priors with Harrison--Zel'dovich spectrum yield better results than naive methods.

Anisotropic non-Gaussian priors can enhance reconstruction accuracy.

Abstract

In the cosmic microwave background or galaxy density maps, missing fluctuations in masked regions can be reconstructed from fluctuations in the surrounding unmasked regions if the original fluctuations are sufficiently smooth. One reconstruction method involves applying a harmonic expansion iteratively to fluctuations in the unmasked region. In this paper, we discuss how well this reconstruction method can recover the original fluctuations depending on the prior of fluctuations and property of the masked region. The reconstruction method is formulated with an asymptotic expansion in terms of the size of mask for a fixed iteration number. The reconstruction accuracy depends on the mask size, the spectrum of the underlying density fluctuations, the scales of the fluctuations to be reconstructed and the number of iterations. For Gaussian fluctuations with the Harrison--Zel'dovich spectrum, the reconstruction method provides more accurate restoration than naive methods based on brute--forth matrix inversion or the singular value decomposition. We also demonstrate that an isotropic non-Gaussian prior does not change the results but an anisotropic non-Gaussian prior can yield a higher reconstruction accuracy compared to the Gaussian prior case.