Abstract
SIAM J. Imaging Sciences, vol. 6, no. 4, pp. 2047-2074, Oct. 2013 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.