Questions of Uniqueness and Resolution in Reconstruction from Projections

Reconstruction from projections has revolutionized radiology and has now become one of the most important tools of medical diagnosis The E. M. I. Scanner is one example. In this text, some fundamental theoretical and practical questions are resolved. Despite recent research activity in the area, the crucial subject of the uniqueness of the reconstruction and the effect of noise in the data posed some unsettled fundamental questions. In particular, Kennan Smith proved that if we describe an object by a C^inf_o function, i.e., infinitely differentiable with compact support, then there are other objects with the same shape, i.e., support, which can differ almost arbitrarily and still have the same projections in finitely many directions. On the other hand, he proved that objects in finite dimensional function spaces are uniquely determined by a single projection for almost all angles, i.e., except on a set of measure zero. Along these lines, Herman and Rowland in "Three Methods for reconstructing objects from x-rays: a comparative study" (1973)  showed that reconstructions obtained from the commonly used algorithms can grossly misrepresent the object and that the algorithm which produced the best reconstruction when using noiseless data gave unsatisfactory results with noisy data. Equally important are reports in Science, and personal communications by radiologists indicating that in medical practice failure rates of reconstruction vary from four to twenty percent. within this work, the mathematical dilemma posed by Kennan Smith's result is discussed and clarified. 

Table of Contents:
I Description of the General Physical Problem.- The EMI Scanner — An Example of the Present State of the Art.- Reconstruction from Projections Models Many Physical Problems and Presents a Variety of Theoretical Questions.- The Difficulties Associated with the Theory of Reconstruction from Projections.- II Basic Indeterminacy of Reconstruction.- Theoretical Background.- First Theoretical Result with Practical Significance.- The Significance of the Nullspace.- Does There Exist a Restriction on the Domain of Pr? Which Makes N = (0)?.- Conclusions to Chapter II.- Proofs of Results Stated in Chapter II.- III A Reconstruction Space which does not Contain the Objective function.- A Reconstruction Space Based on the Fourier Transform.- Description of Our Choice of the Reconstruction Space.- Resolution of a Reconstruction ? Picture Resolution.- IV A Matrix Representation of the Problem.- Proofs of Propositions Stated in Chapter IV.- V Resolution in the Projection Data.- Projection Angles Affectthe Required Resolution.- Farey Series and Projection Angles.- Significance of the Farey Projection Angles.- Proofs of Results Stated in Chapter V.- VI Results Establishing the Uniqueness of a Reconstruction.- Interpretation of the Two Uniqueness Results: Proposition VI. 2 and Theorem 2.- There Is in Practice a Limitation on the Resolution in P?f.- Explanation of Theorem 2.- Uniquely Determined Picture Resolution.- Proofs of Results Stated in Chapter VI.- VII Dealing Effectively with Noisy Data.- Physical Justification of Importance and Sources of Noise.- The Effect of Noisy Data on the Uniqueness of a Reconstruction.- The Effect of Noise on the Consistency of the Data.- The Use of Least Squares — Advantages and Difficulties.- Statistical Considerations Relevant to the Use of Least Squares.- Optimizing the Stability of the Estimate of the Unknown Reconstruction.- Choosing the Best Projection Angles.- Conclusions to Chapter VII.- Appendix to Chapter VII — Statistical Reference Material.- VIII How a Reconstruction Approximates a Real Life Object.- Assumptions with their Justifications.- Consequences of the Assumptions.- Estimating ?h ? f? L2, i. e., How close is the obtained reconstruction to the unknown objective function?.- Significance and Applications of the Estimate of ?h ? f? L2.- Conclusions.- Proofs of Propositions stated in Chapter VIII.- IX A Special Case: Improving the EMI Head Scanner.- The Use of Purposefully Displaced Reconstructions.- Theorem 2 Applied to Four Sets of Purposefully Displaced Projection Data.- Estimating the Accuracy of a 74 × 74 Reconstruction, h74.- Obtaining a Uniquely Determined Reconstruction with 1 mm Resolution from 1 mm Resolution Projection Data.- Conclusions.- X A General Theory of Reconstruction from Projections and other Mathematical Considerations Related to this Problem.- A General Theory of Reconstruction from Projections.- Other Mathematical Considerations Related to Reconstruction from Projections.- Appendix — Medical Context of Reconstruction From Projections.- The Interaction of X-rays with Matter.- The Meaning of a Projection.- Thickness of the Slice.- Types of Detectors.- Parallel and Fan-beam Techniques.- Resolution of the Data.- X-Ray Exposure.- Miscellaneous Aspects of Data Collection.- The EMI Example.- Algorithms.- Representation of a Reconstruction.- The Diagnosis Problem.- References.