# Multitaper estimation on arbitrary domains

## TL;DR

This paper introduces a new approach for multitaper spectral density estimation on arbitrary domains, providing theoretical performance bounds and demonstrating improved accuracy in cryo-electron microscopy applications.

## Contribution

It shows that multitaper estimators depend only on the span of tapers, allowing the use of proxy tapers computed with standard algorithms, and establishes performance bounds for arbitrary domains.

## Key findings

- Reduces mean squared error by over 50% in cryo-EM data
- Validates use of proxy tapers for arbitrary domains
- Provides theoretical bounds for estimator performance

## Abstract

Multitaper estimators have enjoyed significant success in estimating spectral densities from finite samples using as tapers Slepian functions defined on the acquisition domain. Unfortunately, the numerical calculation of these Slepian tapers is only tractable for certain symmetric domains, such as rectangles or disks. In addition, no performance bounds are currently available for the mean squared error of the spectral density estimate. This situation is inadequate for applications such as cryo-electron microscopy, where noise models must be estimated from irregular domains with small sample sizes. We show that the multitaper estimator only depends on the linear space spanned by the tapers. As a result, Slepian tapers may be replaced by proxy tapers spanning the same subspace (validating the common practice of using partially converged solutions to the Slepian eigenproblem as tapers). These proxies may consequently be calculated using standard numerical algorithms for block diagonalization. We also prove a set of performance bounds for multitaper estimators on arbitrary domains. The method is demonstrated on synthetic and experimental datasets from cryo-electron microscopy, where it reduces mean squared error by a factor of two or more compared to traditional methods.