Level set estimation from projection measurements: Performance guarantees and fast computation

TL;DR

This paper introduces a rapid, direct method for estimating level sets from projection measurements without reconstructing the original function, supported by theoretical performance guarantees based on random matrix theory.

Contribution

It proposes the projective level set estimator, a novel approach that bypasses intermediate reconstruction, offering faster computation and rigorous performance analysis.

Findings

Estimator achieves accurate level set estimation directly from projections.

Method significantly reduces computation time compared to traditional plug-in methods.

Theoretical analysis provides performance guarantees based on measurement geometry and function norms.

Abstract

Estimation of the level set of a function (i.e., regions where the function exceeds some value) is an important problem with applications in digital elevation mapping, medical imaging, astronomy, etc. In many applications, the function of interest is not observed directly. Rather, it is acquired through (linear) projection measurements, such as tomographic projections, interferometric measurements, coded-aperture measurements, and random projections associated with compressed sensing. This paper describes a new methodology for rapid and accurate estimation of the level set from such projection measurements. The key defining characteristic of the proposed method, called the projective level set estimator, is its ability to estimate the level set from projection measurements without an intermediate reconstruction step. This leads to significantly faster computation relative to heuristic "plug-in" methods that first estimate the function, typically with an iterative algorithm, and then threshold the result. The paper also includes a rigorous theoretical analysis of the proposed method, which utilizes the recent results from the non-asymptotic theory of random matrices results from the literature on concentration of measure and characterizes the estimator's performance in terms of geometry of the measurement operator and 1-norm of the discretized function.