We consider the problem of reconstructing remotely obtained images from image-plane detector arrays. Although the individual detectors may be larger than the blur spot of the imaging optics, high-resolution reconstructions can be obtained by scanning or rotating the image with respect to the detector. As an alternative to matrix inversion or least-squares estimation [Appl. Opt. 26, 3615 (1987)], the method of convex projections is proposed. We show that readily obtained prior knowledge can be used to obtain good-quality imagery with reduced data. The effect of noise on the reconstruction process is considered.

High-resolution image recovery from image-plane arrays, using convex projections.

Developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined. Particular attention is paid to common estimation structures and to practical problems not properly solved yet. The problem of image reconstruction and restoration is first formulated. Some of the current regularization approaches used to solve the problem are then described. The concepts of a priori information and compound criterion are introduced. A Bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated. The practical aspects of computing the solution, first when the hyperparameters are known and second when they must be estimated, are then considered. Conclusions are drawn, and points that still need to be investigated are outlined. >

Image reconstruction and restoration: overview of common estimation structures and problems

Explains set theoretic estimation, which is governed by the notion of feasibility and produces solutions whose sole property is to be consistent with all information arising from the observed data and a priori knowledge. Each piece of information is associated with a set in the solution space, and the intersection of these sets, the feasibility set, represents the acceptable solutions. The practical use of the set theoretic framework stems from the existence of efficient techniques for finding these solutions. Many scattered problems in systems science and signal processing have been approached in set theoretic terms over the past three decades. The author synthesizes a single, general framework from these various approaches, examines its fundamental philosophy, goals, and analytical techniques, and relates it to conventional methods. >

The foundations of set theoretic estimation

Publisher Summary Image recovery is a broad discipline that encompasses the large body of inverse problems, in which an image h is to be inferred from the observation of data x consisting of signals physically or mathematically related to it. Image restoration and image reconstruction are the two main sub-branches of image recovery. The term “image restoration” usually applies to the problem of estimating the original form h of a degraded image x . The following four basic elements are required to solve an image recovery problem: (1) a data formation model, (2) a priori information, (3) a recovery criterion, and (4) a solution method. The recovery criterion defines the class of images that are acceptable as solutions to the problem. It is chosen by the user on grounds that may include experience, compatibility with the available a priori knowledge, personal convictions on the best way to solve the problem, and ease of implementation. The traditional approach has been to use a criterion of optimality, which usually leads to a single best solution. An alternative approach is to use a criterion of feasibility, in which consistency with all prior information and the data defines a set of equally acceptable solutions. The solution method is a numerical algorithm that will produce a solution to the recovery problem—that is, an image that satisfies the recovery criterion. Modification can be made in two directions: in the conventional image recovery framework, one seeks to preserve the notion of an optimal solution, whereas in the set theoretic framework the emphasis is placed on feasibility.

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The Convex Feasibility Problem in Image Recovery

An algorithm for lost signal restoration in block-based still image and video sequence coding is presented. Problems arising from imperfect transmission of block-coded images result in lost blocks. The resulting image is flawed by the absence of square pixel regions that are notably perceived by human vision, even in real-time video sequences. Error concealment is aimed at masking the effect of missing blocks by use of temporal or spatial interpolation to create a subjectively acceptable approximation to the true error-free image. This paper presents a spatial interpolation algorithm that addresses concealment of lost image blocks using only intra-frame information. It attempts to utilize spatially correlated edge information from a large local neighborhood of surrounding pixels to restore missing blocks. The algorithm is a Gerchberg (1974) type spatial domain/spectral domain constraint-satisfying iterative process, and may be viewed as an alternating projections onto convex sets method. >

Concealment of damaged block transform coded images using projections onto convex sets

We consider the problem of reconstructing CAT imagery by the direct Fourier method (DFM) when not all view data are available. To restore the missing information we use the method of projections onto convex sets (POCS). POCS is a recursive image restoration technique that finds a solution consistent with the measured data and a priori known constraints in both the space and Fourier domain. Because DFM reconstruction is a frequency-domain technique it is ideally matched to POCS restoration when, for one reason or another, we are forced to generate an image from a less than complete set of view data. We design and apply an algorithm (PRDF) which interpolates/extrapolates the missing Fourier domain information by POCS and reconstructs an image by DFM. A simulated human thorax cross section is restored and reconstructed. The restorations using POCS are compared with the Gerchberg-Papoulis extrapolation method and shown to be superior. Applications of PRDF to other types of medical imaging modalities are discussed.

M. Ibrahim Sezan

Papers

Tomographic Image Reconstruction from Incomplete View Data by Convex Projections and Direct Fourier Inversion