Introduction to the mathematics of medical imaging

   Book for Physicists  »  Introduction to the mathematics of medical imaging

Table of Contents

• 1 Introduction (Introduction to: Mathematics of Medical Imaging)
• 2 Integral Geometry and Integral Transforms
  • 2.1 Introduction ( Introduction to: Integral geometry))
  • 2.2 Represenation of a Line and other linear geometrical elements
  • 2.3 The 2D Radon transform
  • 2.4 The sinogram
  • 2.5 The properties of the Radon transform
  • 2.6 The Hilbert-transform
  • 2.7 The Digital Radon Transform
  • 2.8 The Radon Transform in multiple dimensions
• 3 Analytical Reconstruction techniques
  • 3.1 Introduction (Analyitical Reconstruction)
  • 3.2 The Central Slice Theorem
  • 3.3 The Filtered Backprojection
  • 3.4 Relaization of the filtered backprojection
  • 3.5 Multidimensional Central Slice Theorem and the Fourier Inverson Formula
  • 3.6 Interpretation of the inverse Radon transform
  • 3.7 Inverse Radon transfrom with Riesz potentials
  • 3.8 Filter Design for the Filtered Backprojection
  • 3.9 3D reconstruction
• 4 Algebraic Image Reconstruction
  • 4.1 Introduction (Introduction to: Algebraic Image Reconstruction)
  • 4.2 Descrete Base for the reconstruction
  • 4.3 Non-statistical iterative reconstructions
  • 4.4 Statistical image reconstruction strategies
    • 4.4.1 Example to a Maximum Likelihood estimation
  • 4.5 The ML-EM algorithm
  • 4.6 The ML-EM algorithm for emission tomography
  • 4.7 ML-EM variations: MAP-EM,OSEM
• 5 The DICOM standard
  • 5.1 Abstract
  • 5.2 Introduction (DICOM)
  • 5.3 A simplified DICOM for beginners
    • 5.3.1 Introduction to the digital representation of alphanumeric data
    • 5.3.2 Introduction to the a simplified toy-DICOM file format I (Problems for the reader)
    • 5.3.3 Introduction to the a simplified toy-DICOM file format II (Solutions of the problems)
    • 5.3.4 Introduction to the a simplified toy-DICOM file format III
    • 5.3.5 Introduction to the a simplified 'toy-DICOM' file format IV (Further development: the Value Representation)
  • 5.4 A few words about the real DICOM format
    • 5.4.1 Value representation in DICOM
    • 5.4.2 The DICOM Identifiers and the Concept of the DICOM Study
    • 5.4.3 The structure of a DICOM file, the DICOM tags
    • 5.4.4 DICOM Data Coding Image Related Information
    • 5.4.5 A few words about the DICOM services
  • 5.5 Appendix
• 6 Mathematical Methods of Linear Model Based Image Processing Procedures
  • 6.1 Introduction.
  • 6.2 Linear Operators
  • 6.3 Characteristic Input Functions
  • 6.4 General Input Functions - Fourier Transformation
  • 6.5 Laplace-Transformation
    • 6.5.1 Analysis of linear systems in extended frequency space
    • 6.5.2 Properties and rules of Laplace transformation
    • 6.5.3 Laplace transformation of characteristic and any other typical functions
    • 6.5.4 Mathematical description of the system properties functions
    • 6.5.5 Inverse transformation method for linear invariant systems
    • 6.5.6 Inverse transformation of the proper rational function
    • 6.5.7 Transfer Function
    • 6.5.8 Transfer characteristic function (Modulation Transfer Function MTF)
    • 6.5.9 Linear shift invariant system description by the step response function
    • 6.5.10 Relation between the step response function and weighting function of linear shift invariant system
  • 6.6 Problems (Linear Systems)
  • 6.7 Theory and basic laws of sampling
  • 6.8 Planar imaging as a linear system
  • 6.9 Appendix.
    • 6.9.1 Theorems, Detailed explanations
      • 6.9.1.1 Deriving of Fourier Theory, Fourier Series
      • 6.9.1.2 Description of Fourier series by complex expression
      • 6.9.1.3 Parzeval theorem
      • 6.9.1.4 Response function in general case
      • 6.9.1.5 Deriving of Duhamel Theorem
    • 6.9.2 Solution of problems
      • 6.9.2.1 S1.
      • 6.9.2.2 S2.
      • 6.9.2.3 S3.
      • 6.9.2.4 S4.
      • 6.9.2.5 S5.
      • 6.9.2.6 S6.
      • 6.9.2.7 S7.
      • 6.9.2.8 S8.
      • 6.9.2.9 S9.
      • 6.9.2.10 S10.
      • 6.9.2.11 S11.
• 7 Monte Carlo Methods (English)
  • 7.1 Introduction (introduction to: Monte Carlo Methods)
  • 7.2 Sampling
  • 7.3 Sampling the free flight distance
  • 7.4 Sampling an interaction
  • 7.5 Detection
• 8 References