Machine Vision and Applications (1988) 1:115-126 Machine Vision and Applications 1988Spfinger-Verlag New York Inc. Model-Based Estimation Techniques for 3-D Reconstruction from Projections Yoram Bresler Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61821 USA Jeffrey A. Fessler* and Albert Macovski Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA Abstract: A parametric estimation approach to recon- struction from projections with incomplete and very noisy data is described. Embedding prior knowledge about "objects" in the probed domain and about the data acquisition process into stochastic dynamic models, we transform the reconstruction problem into a computation- ally ,challenging nonlinear state-estimation problem, where the objects' parametrized descriptions are to be directly estimated from the projection data. This paper is a review in a common framework and a comparative study of two distinct algorithms which were developed recently for the solution of this problem. The first, is an approximate, globally optimal minimum-mean- square-error recursive algorithm. The second is a hierar- chical suboptimal Bayesian algorithm. Simulation exam- ples demonstrate accurate reconstructions with as few as four views in a 135 ~ sector, at an average signal to noise ratio of 0.6. Key Words: 3D tomography, incomplete projections, Bayesian estimation, stochastic modeling I. Introduction Tomography, or the reconstruction of a multidi- mensional function from its line-integral projec- tions, is a well-studied problem, typically arising in the context of determining the internal structure of an object from the results of external probing by electromagnetic or sound waves, or by subatomic particles. The problem is usually posed and solved This work was supported by the National Institute of Health contract NO1-38045, by the National Science Foundation con- tract ECS-8213959, and General Electric contract 22-84. * Supported by a National Science Foundation Graduate Fel- lowship. in two dimensions, where a cross section, or a thin slice, is reconstructed from its projections. Most often, 3-D reconstruction is simply obtained by stacking thin reconstructed slices. The applications of reconstruction from projections are seen in di- verse disciplines, ranging from medicine and non- destructive testing to geophysical exploration, and from astronomy to electron microscopy (Deans 1984, Herman 1980). Owing to various temporal, physical, geometri- cal, or economic constraints in the data acquisition (Bresler and Macovski 1987, Rossi and Willsky 1984) it is often impossible to acquire projection data at all angles (views), and the number of views and/or rays within a view is severely restricted. This is almost always the case with 3-D reconstruc- tion, where a complete data-set is exceedingly large. An attempt to reconstruct the original distri- bution in this so-called incomplete data case results in images that suffer from artifacts such as streaking and geometric distortion, poor resolution, and high noise level, because the problem is ill-posed, and in extreme cases, because of the inherent nonunique- ness of the solution (Louis and Natterer 1983). Con- sequently, although 3-D reconstruction would be an ideal tool in a variety of medical (Bloch and Udupa 1983) and other applications, it is rarely attempted in practice with limited data. In this paper we consider the incomplete data case in the extreme situation when the data are re- stricted and heavily corrupted by noise to the point where all current limited data reconstruction meth- ods (Gordon and Herman 1974; Kak 1979; Ran- gayan et al. 1984; Stark 1987) produce unacceptable results. (For example, we are unaware of a current method producing diagnostically useful reconstruc- tions from four views at a signal-to-noise ratio