Level Set Estimation from Projection Measurements: Performance Guarantees and Fast Computation

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 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.

Silhouette estimation

“…What parametric model should we use to represent the silhouette of a walker for gait recognition or for the silhouette of a satellite for pose estimation? Other candidate parametric models include level sets [23][24][25] and various shape-coding models [26,27]. Perhaps the parametric silhouette model could be built from the data itself, by adaptively inserting spline control points until the data no longer support further model complexity.…”

Robust Super-Level Set Estimation Using Gaussian Processes

“…In the literature of level set estimation, when level sets are estimated from an existing dataset, several approaches are available [8], [9], [10]. In the context of active learning, a topic of growing interest [11], [12], [13], one technique is known as the Straddle heuristic [14], where the expected value and variance given by a Gaussian process are combined to characterize the uncertainty at each candidate point, based on which the next query point is chosen.…”