Reconstruction from projections based on detection and estimation of objects

Abstract

The problem of reconstructing a multi-dimensional field from noisy, limited projection measurements is approached using an object-based stochastic field model. Objects within a cross-section are characterized by a finite-dimensional set of parameters, which are estimated directly from limited, noisy projection measurements using maximum likelihood estimation. In Part I, the computational structure and performance of the ML estimation procedure are investigated for the problem of locating a single object in a deterministic background; simulations are also presented. In Part H, the issue of robustness to modeling errors is addressed. L INTRODUCTIONThe problem of reconstructing an n-dimensional function from its (n-1)-dimensional projections arises, typically in the context of cross-sectional imaging, in a diversity of disciplines. In the two-dimensional (2D) problem, let f(x) represent the value of the cross-sectional function (for example x-ray attenuation coefficient) at a point specified by the vector x -(xl x 2 )'. The projection of f(x) at any angle 9 is a ID function denoted as g(t,9) shown in Figure 1. For a given value of projection angle 9, the projection evaluated at the point t is the integral g(t,9) -f ff(x)8(t-x') dxIdx 2 -f(x)ds X [Rf](t,9) along the line This work was conducted at the M.I.T. Laboratory for Information and Decision Systems, with support from the National Science Foundation under Grant ECS-8012668. Figure 2. In (1), 8(t) is the Dirac delta function. Equation (1) corresponds to the Radon transformation, which maps the 2D function f: R 2 -R into the function on a half-cylinder g: Y-'R; g(t,O) is called the Radon transform of f(x) [1][2][3], and is also denoted by [Rf](t, ). The reconstruction problem, determining the function f from its projections g, is an inverse problem, since the measurements are specified by the integral equation in (1) which must be inverted to recover an estimate of the original function.Reconstruction from projections has been employed successfully in radio astronomy, electron microscopy, medical CAT scanning and other applications. Recently, it has been suggested that these techniques be similarly applied to a number of technologically demanding and novel applications, for example real-time monitoring of high production rate manufacturing processes, mesoscale oceanographic thermal mapping, quality control nondestructive evaluation, and 'stop action' imaging of very rapidly changing media [4,51. Virtually all of these applications, as well as many current applications, are characterized by a limited availability of measurement data, due to: * economic constraints that limit the total number of measurement transducers--e.g. oceanographic transducers are sophisticated, low power units that are costly to build, place and maintain [5].* time constraints that limit measurement quality --e.g. limited measurement time interval in "stop action' imaging of very rapid events, or in nondestructive testing of high production rate processes such as steel manufactu...

Object shape estimation from tomographic measurements—A performance analysis

“…The probability of obtaining an anomolous estimate may be characterized from knowledge of the ambiguity function [9,10,15], and becasue of this the ambiguity function plays a key role in assessing ML estimation performance. The ambiguity function also plays a key role in assessing local estimation performance, that is, in identifying the estimate error variance in the case when the estimate is not anomolous and occurs close to the true parameter value.…”

“…For simplicity, and in order to establish insight, it is assumed that the background fb(x) is known (and without loss of generality taken to equal zero) and that only a single object (N=1) is present at a known location cl. The effect of errors in these assumptions is considered in the robustness study in [9], where it is shown that object shape determination is quite robust to errors in the assumed object location and to the presence of additional unmodeled objects having small Radon transform energy (as defined in the next section). The single object in the cross-section is considered to have unknown size, shape and orientation (i.e., y is unknown) and these parameters are estimated directly from noisy tomographic data.…”

“…The issue of estimation robustness or sensitivity to modeling errors is not considered in this paper; see [9] for a discussion of these issues.…”

“…In Section II, notation is reviewed for both the tomographic reconstruction problem and for the object-based profile model described in [9,10]. In Section III the profile model from [9,101 is restricted to objects capturing the three features of object geometry already mentioned -size, elongation and orientation.…”

“…where the convolving kernel v(t, 0) is 0-independent with a Fourier transform with respect In the object-based model from [9,10], the 2D cross-section f(x) is represented as the superposition of a background and N objects,…”