David J. Rossi

Reconstruction from projections based on detection and estimation of objects

This paper considers the problem of observing a 2D function via its 1D projections (Radon transform); it presents a framework for detecting, locating and describing objects contained within a 2D cross-section by using noisy measurements of the Radon transform directly, rather than post-processing a reconstructed image. This framework offers the potential for significant improvements in applications where (1) attempts to perform an initial inversion with insufficient measurement data result in severely degraded reconstructions, and (2) the ultimate goal of the process is to obtain several specific pieces of information about the cross-section. To illustrate this perspective, we focus our attention on the problem of obtaining maximum-likelihood (ML) estimates of the parameters characterizing a single random object situated within a deterministic background medium, and we investigate the performance, robustness, and computational structure of the ML estimation procedure.

Object shape estimation from tomographic measurements--a performance analysis

Abstract The problem considered is that of determining the shape of an object embedded within a medium from noisy tomographic projection measurements. In particular, the issue is addressed of how accurately coarse features of object geometry—size, elongation and orientation—can be characterized from noisy projection data. A maximum likelihood parameter estimation formulation is used and estimation performance is analyzed by evaluation of the Cramer-Rao lower bound on the error variances of the estimates. It is demonstrated that for measurements available at all projection angles and at a given noise level (1) object size and orientation are more accurately determined than is the degree of object elongation, and (2) reliable orientation estimation requiresa minimum degree of object elongation, and the required degree of elongation is inversely related to the measurement signal-to-noise ratio (SNR). Based on these observations an iterative algorithm is proposed for estimation of object geometry and results illustrating algorithm performance are presented.

The reconstruction of subsurface property maps using projection onto convex sets

Obtaining a map of the spatial distribution of some physical property of the Earth subsurface from sparse data is a fundamental problem in aquifer and petroleum reservoir characterization. This problem can be stated in terms of an inverse problem as the search for a map that satisfies certain constraints, i.e., measurements and information on the spatial statistics of the property. We present the results of numerical experiments applying the method of projection onto convex sets, or POCS, that functions both as an estimation procedure, where the objective is to obtain a unique guess of the unknown spatial distribution given some constraints, and as a simulation procedure, where the objective is to obtain a set of nonunique reconstructions that are consistent with some measurements and have given spatial statistics. Numerical experiments that use point measurements at wellbores and volume integral measurements show that incorporating data from the region of the reservoir between wellbores allows better localization of large-scale heterogeneities in the reservoir volume.<<ETX>>

Maximum likelihood estimation of object size and orientation from projection data

The problem of detecting, locating and characterizing objects in a 2D cross-section from noisy projection data has been considered recently [1-3], in which objects are characterized by a finite number of parameters, which are estimated directly from noisy projection measurements. In this paper, the problem of maximum likelihood (ML) estimation of those parameters characterizing the geometry of an object (e.g. size and orientation) is considered, and estimation performance is investigated.