Introduction to the Mathematics of Medical Imaging

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1 Introduction to the Mathematics of Medical Imaging Second Edition Charles L. Epstein University of Pennsylvania Philadelphia, Pennsylvania EiaJTL Society for Industrial and Applied Mathematics Philadelphia

2 Contents Preface to the second edition Preface How to Use This Book Notational Conventions xvii xix xxv xxxi 1 Measurements and Modeling Mathematical Modeling Finitely Many Degrees of Freedom Infinitely Many Degrees of Freedom A Simple Model Problem for Image Reconstruction The Space of Lines in the Plane* Reconstructing an Object from Its Shadows Approximate Reconstructions Can an Object Be Reconstructed from Its Width? Conclusion 30 2 Linear Models and Linear Equations Linear Equations and Linear Maps Solving Linear Equations Stability of Solutions Infinite-dimensional Linear Algebra Complex Numbers and Vector Spaces* Complex Numbers Complex Vector Spaces Conclusion 52 3 A Basic Model for Tomography Tomography Beer's Law and X-ray Tomography Analysis of a Point Source Device Some Physical Considerations 66 vn

3 viii Contents 3.4 The Definition of the Radon Transform Appendix: Proof of Lemma 3.4.1* The Back-Projection Formula Continuity of the Radon Transform* The Radon Transform on Radial Functions The Range of the Radial Radon Transform* The Abel Transform* Volterra Equations of the First Kind* Conclusion 89 4 Introduction to the Fourier Transform The Complex Exponential Function Functions of a Single Variable Absolutely Integrable Functions The Fourier Transform for Integrable Functions* Appendix: The Fourier Transform of a Gaussian* Regularity and Decay* Fourier Transform on L 2 (U) Ill A General Principle in Functional Analysis* Functions With Weak Derivatives Functions With L 2 -Derivatives* Fractional Derivatives and L 2 -Derivatives* Some Refined Properties of the Fourier Transform Localization Principle The Heisenberg Uncertainty Principle* The Paley-Wiener Theorem* The Fourier Transform of Generalized Functions* Functions of Several Variables L'-case Regularity and Decay L 2 -Theory The Fourier Transform on Radial Functions The Failure of Localization in Higher Dimensions Conclusion Convolution Convolution Basic Properties of the Convolution Product Shift Invariant Filters* Convolution Equations Convolution and Regularity Approximation by Smooth Functions Some Convergence Results* Approximating Derivatives and Regularized Derivatives 169

4 Contents ix The Support of /* g The ^-Function* Approximating the (5-Function in One-Dimension Resolution and the Full-Width Half-Maximum Conclusion The Radon Transform The Radon Transform Inversion of the Radon Transform The Central Slice Theorem* The Radon Inversion Formula* Filtered Back-Projection* Inverting the Radon Transform, Two Examples Back-Projection* The Hilbert Transform The Hilbert Transform as a Convolution Mapping Properties of the Hilbert Transform* Approximate Inverses for the Radon Transform Addendum* Functions with Bounded Support Continuity of the Radon Transform and Its Inverse* Data With Bounded Support Estimates for the Inverse Transform The Higher-Dimensional Radon Transform* The Hilbert Transform and Complex Analysis* Conclusion Introduction to Fourier Series Fourier Series in One Dimension* Decay of Fourier Coefficients Periodic Extension Fourier Coefficients of Differentiable Functions L 2 -Theory Geometry in L 2 ([0, 1]) The L 2 -Inversion formula Bessel's Inequality L 2 -Derivatives* General Periodic Functions Convolution and Partial Sums* Dirichlet Kernel The Gibbs Phenomenon An Example of the Gibbs Phenomenon The General Gibbs Phenomenon* Fejer Means 262

5 Contents Resolution The Localization Principle Higher-Dimensional Fourier Series L 2 -Theory Conclusion 274 Sampling Sampling and Nyquist's Theorem* Bandlimited Functions and Nyquist's Theorem Shannon-Whittaker Interpolation The Poisson Summation Formula The Poisson Summation Formula Undersampling and Aliasing* Subsampling The Finite Fourier Transform* Quantization Errors Higher-Dimensional Sampling Conclusion 303 Filters Basic Definitions Examples of Filters Linear filters Shift Invariant Filters and the Impulse Response Harmonic Components The Transfer Function Cascades of Filters Causal Filters Bandpass Filters The Inverse Filter Resolution The Resolution of a Cascade of Filters Filtering Periodic Inputs Resolution of Periodic Filters The Comb Filter and Poisson Summation* Higher-Dimensional Filters Isotropic Filters Resolution Some Applications of Filtering Theory Image Processing Linear Filter Analysis of Imaging Hardware Conclusion 376

6 xi 10 Implementing Shift Invariant Filters Sampled Data Implementing Periodic Convolutions Further Properties of the Finite Fourier Transform The Approximation of Fourier Coefficients Approximating Periodic Convolutions Implementing Filters on Finitely Sampled Data Zero Padding Reconsidered Higher-Dimensional Filters Riemann Sum Approximations The Finite Fourier Transform in n Dimensions The Fourier Representation for Shift Invariant Filters Appendix: The Fast Fourier Transform Conclusion Reconstruction in X-Ray Tomography Basic Setup in X-Ray Tomography The Reconstruction Problem Scanner Geometries Algorithms for a Parallel Beam Machine Direct Fourier Inversion Filtered Back-Projection Linearly Interpolated Filters The Shepp-Logan Analysis of the Ram-Lak Filters Sample Spacing in a Parallel Beam Machine Filtered Back-Projection in the Fan Beam Case Fan Beam Geometry Fan Beam Filtered Back-Projection Implementing the Fan Beam Algorithm Data Collection for a Fan Beam Scanner Rebinning Some Mathematical Remarks* Spiral Scan CT Interpolation methods d-Reconstruction Formulae TheGridding Method* Conclusion Imaging Artifacts in X-Ray Tomography The Effect of a Finite Width X-Ray Beam A Linear Model for Finite Beam Width A Nonlinear Model for Finite Beam Width The Partial Volume Effect The PSF 458

7 xii Contents Point Sources The PSF without Sampling The PSF with Sampling Ray Sampling View Sampling The Effects of Measurement Errors A Single Bad Ray A Bad Ray in Each View A Bad View Beam Hardening Conclusion Algebraic Reconstruction Techniques Algebraic Reconstruction Kaczmarz's Method A Bayesian Estimate Variants of the Kaczmarz Method Relaxation Parameters Other Related Algorithms Conclusion Magnetic Resonance Imaging Introduction Nuclear Magnetic Resonance The Bloch Phenomological Equation The Rotating Reference Frame A Basic Imaging Experiment Selective Excitation Spin-warp Imaging Contrast and Resolution Conclusion Probability and Random Variables Measure Theory* Allowable Events Measures and Probability Integration Independent Events Conditional Probability Random Variables* Cumulative Distribution Function The Variance The Characteristic Function A Pair of Random Variables 552

8 xiii Several Random Variables Some Important Random Variables Bernoulli Random Variables Poisson Random Variables Gaussian Random Variables Limits of Random Variables The Central Limit Theorem Other Examples of Limiting Distributions Statistics and Measurements Conclusion Applications of Probability Applications to X-Ray Imaging Modeling a Source-Detector Pair Beer's Law Noise in the Filtered Back-Projection Algorithm Sampled Data A Computation of the Variance in the Measurements The Variance of the Radon Transform The Variance in the Reconstructed Image Signal-to-Noise Ratio, Dosage and Contrast Signal-to-Noise in Magnetic Resonance Imaging Image Reconstruction in PET Positron Emission Physics A Probabilistic Model for PET The Maximum Likelihood Algorithm Determining the Transition Matrix Conclusion Random Processes Random Processes in Measurements Basic Definitions Statistical Properties of Random Processes Stationary Random Processes Spectral Analysis of Stationary Processes* Independent and Stationary Increments Examples of Random Processes Gaussian Random Process The Poisson Counting Process Poisson Arrival Process Fourier Coefficients for Periodic Processes* White Noise* Random Inputs to Linear Systems The Autocorrelation of the Output 629

9 xiv Contents Thermal or Johnson Noise Optimal Filters Noise in Filtered Back-Projection Conclusion 638 A Background Material 639 A.l Numbers 639 A.1.1 Integers 639 A.1.2 Rational Numbers 641 A.1.3 Real Numbers 645 A. 1.4 Cauchy Sequences 648 A.2 Vector Spaces 649 A.2.1 Euclidean «Space 650 A.2.2 General Vector Spaces 653 A.2.3 Linear Transformations and Matrices 656 A.2.4 Norms and Metrics 661 A.2.5 Inner Product Structure 665 A.2.6 Linear Transformations and Linear Equations 671 A.3 Functions, Theory, and Practice 673 A.3.1 Power Series 675 A.3.2 The Binomial Formula 679 A.3.3 Some Higher Transcendental Functions 681 A.4 Spaces of Functions* 686 A.4.1 Examples of Function Spaces 686 A.4.2 Completeness 690 A.4.3 Linear Functionals 692 A.4.4 Measurement, Linear Functionals, and Weak Convergence 694 A.4.5 Generalized Functions on IR 696 A.4.6 Generalized Functions on IR" 702 A.5 Functions in the Real World 705 A.5.1 Approximation 705 A.5.2 Sampling and Interpolation 711 A.6 Numerical Differentiation and Integration 714 A.6.1 Numerical Integration 716 A.6.2 Numerical Differentiation 719 В Basic Analysis 723 B.l Sequences 723 B.2 Series 725 B.3 Limits of Functions and Continuity 729 B.4 Differentiability 730 B.5 Higher-Order Derivatives and Taylor's Theorem 732 B.6 Integration 732 B.7 Improper Integrals 735

10 Contents xv В.8 Fubini's Theorem and Differentiation of Integrals* 739 Bibliography 743 Index 753

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