Questions of uniqueness and resolution in reconstruction from projection

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 Affect the 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 x 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.