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
Contents Preface to the second edition Preface How to Use This Book Notational Conventions xvii xix xxv xxxi 1 Measurements and Modeling 1 1.1 Mathematical Modeling 3 1.1.1 Finitely Many Degrees of Freedom 5 1.1.2 Infinitely Many Degrees of Freedom 16 1.2 A Simple Model Problem for Image Reconstruction 19 1.2.1 The Space of Lines in the Plane* 20 1.2.2 Reconstructing an Object from Its Shadows 23 1.2.3 Approximate Reconstructions 27 1.2.4 Can an Object Be Reconstructed from Its Width? 28 1.3 Conclusion 30 2 Linear Models and Linear Equations 31 2.1 Linear Equations and Linear Maps 32 2.1.1 Solving Linear Equations 34 2.1.2 Stability of Solutions 38 2.2 Infinite-dimensional Linear Algebra 42 2.3 Complex Numbers and Vector Spaces* 47 2.3.1 Complex Numbers 47 2.3.2 Complex Vector Spaces 50 2.4 Conclusion 52 3 A Basic Model for Tomography 53 3.1 Tomography 53 3.1.1 Beer's Law and X-ray Tomography 56 3.2 Analysis of a Point Source Device 62 3.3 Some Physical Considerations 66 vn
viii Contents 3.4 The Definition of the Radon Transform 69 3.4.1 Appendix: Proof of Lemma 3.4.1* 76 3.4.2 The Back-Projection Formula 76 3.4.3 Continuity of the Radon Transform* 77 3.5 The Radon Transform on Radial Functions 80 3.5.1 The Range of the Radial Radon Transform* 81 3.5.2 The Abel Transform* 83 3.5.3 Volterra Equations of the First Kind* 86 3.6 Conclusion 89 4 Introduction to the Fourier Transform 91 4.1 The Complex Exponential Function 91 4.2 Functions of a Single Variable 93 4.2.1 Absolutely Integrable Functions 94 4.2.2 The Fourier Transform for Integrable Functions* 94 4.2.3 Appendix: The Fourier Transform of a Gaussian* 98 4.2.4 Regularity and Decay* 101 4.2.5 Fourier Transform on L 2 (U) Ill 4.2.6 A General Principle in Functional Analysis* 116 4.3 Functions With Weak Derivatives 117 4.3.1 Functions With L 2 -Derivatives* 120 4.3.2 Fractional Derivatives and L 2 -Derivatives* 121 4.4 Some Refined Properties of the Fourier Transform 123 4.4.1 Localization Principle 123 4.4.2 The Heisenberg Uncertainty Principle* 125 4.4.3 The Paley-Wiener Theorem* 129 4.4.4 The Fourier Transform of Generalized Functions* 130 4.5 Functions of Several Variables 136 4.5.1 L'-case 136 4.5.2 Regularity and Decay 140 4.5.3 L 2 -Theory 144 4.5.4 The Fourier Transform on Radial Functions 146 4.5.5 The Failure of Localization in Higher Dimensions 148 4.6 Conclusion 149 5 Convolution 151 5.1 Convolution 154 5.1.1 Basic Properties of the Convolution Product 155 5.1.2 Shift Invariant Filters* 160 5.1.3 Convolution Equations 161 5.2 Convolution and Regularity 163 5.2.1 Approximation by Smooth Functions 164 5.2.2 Some Convergence Results* 166 5.2.3 Approximating Derivatives and Regularized Derivatives 169
Contents ix 5.2.4 The Support of /* g 170 5.3 The ^-Function* 172 5.3.1 Approximating the (5-Function in One-Dimension 173 5.3.2 Resolution and the Full-Width Half-Maximum 175 5.4 Conclusion 177 6 The Radon Transform 179 6.1 The Radon Transform 179 6.2 Inversion of the Radon Transform 184 6.2.1 The Central Slice Theorem* 184 6.2.2 The Radon Inversion Formula* 188 6.2.3 Filtered Back-Projection* 190 6.2.4 Inverting the Radon Transform, Two Examples 195 6.2.5 Back-Projection* 197 6.3 The Hilbert Transform 201 6.3.1 The Hilbert Transform as a Convolution 202 6.3.2 Mapping Properties of the Hilbert Transform* 206 6.4 Approximate Inverses for the Radon Transform 207 6.4.1 Addendum* 209 6.5 Functions with Bounded Support 210 6.6 Continuity of the Radon Transform and Its Inverse* 213 6.6.1 Data With Bounded Support 213 6.6.2 Estimates for the Inverse Transform 216 6.7 The Higher-Dimensional Radon Transform* 219 6.8 The Hilbert Transform and Complex Analysis* 223 6.9 Conclusion 225 7 Introduction to Fourier Series 227 7.1 Fourier Series in One Dimension* 227 7.2 Decay of Fourier Coefficients 233 7.2.1 Periodic Extension 234 7.2.2 Fourier Coefficients of Differentiable Functions 237 7.3 L 2 -Theory 239 7.3.1 Geometry in L 2 ([0, 1]) 239 7.3.2 The L 2 -Inversion formula 244 7.3.3 Bessel's Inequality 246 7.3.4 L 2 -Derivatives* 248 7.4 General Periodic Functions 251 7.4.1 Convolution and Partial Sums* 252 7.4.2 Dirichlet Kernel 254 7.5 The Gibbs Phenomenon 256 7.5.1 An Example of the Gibbs Phenomenon 257 7.5.2 The General Gibbs Phenomenon* 260 7.5.3 Fejer Means 262
Contents 7.5.4 Resolution 265 7.6 The Localization Principle 267 7.7 Higher-Dimensional Fourier Series 269 7.7.1 L 2 -Theory 272 7.8 Conclusion 274 Sampling 277 8.1 Sampling and Nyquist's Theorem* 278 8.1.1 Bandlimited Functions and Nyquist's Theorem 278 8.1.2 Shannon-Whittaker Interpolation 281 8.2 The Poisson Summation Formula 284 8.2.1 The Poisson Summation Formula 285 8.2.2 Undersampling and Aliasing* 288 8.2.3 Subsampling 294 8.3 The Finite Fourier Transform* 294 8.4 Quantization Errors 298 8.5 Higher-Dimensional Sampling 300 8.6 Conclusion 303 Filters 305 9.1 Basic Definitions 306 9.1.1 Examples of Filters 306 9.1.2 Linear filters 309 9.1.3 Shift Invariant Filters and the Impulse Response 310 9.1.4 Harmonic Components 314 9.1.5 The Transfer Function 319 9.1.6 Cascades of Filters 323 9.1.7 Causal Filters 325 9.1.8 Bandpass Filters 326 9.1.9 The Inverse Filter 328 9.1.10 Resolution 332 9.1.11 The Resolution of a Cascade of Filters 335 9.2 Filtering Periodic Inputs 337 9.2.1 Resolution of Periodic Filters 340 9.2.2 The Comb Filter and Poisson Summation* 341 9.3 Higher-Dimensional Filters 343 9.3.1 Isotropic Filters 347 9.3.2 Resolution 349 9.4 Some Applications of Filtering Theory 350 9.4.1 Image Processing 350 9.4.2 Linear Filter Analysis of Imaging Hardware 365 9.5 Conclusion 376
xi 10 Implementing Shift Invariant Filters 377 10.1 Sampled Data 378 10.2 Implementing Periodic Convolutions 381 10.2.1 Further Properties of the Finite Fourier Transform 381 10.2.2 The Approximation of Fourier Coefficients 382 10.2.3 Approximating Periodic Convolutions 384 10.3 Implementing Filters on Finitely Sampled Data 386 10.3.1 Zero Padding Reconsidered 390 10.4 Higher-Dimensional Filters 392 10.4.1 Riemann Sum Approximations 393 10.4.2 The Finite Fourier Transform in n Dimensions 394 10.4.3 The Fourier Representation for Shift Invariant Filters 395 10.5 Appendix: The Fast Fourier Transform 397 10.6 Conclusion 399 11 Reconstruction in X-Ray Tomography 401 11.1 Basic Setup in X-Ray Tomography 402 11.2 The Reconstruction Problem 405 11.3 Scanner Geometries 408 11.4 Algorithms for a Parallel Beam Machine 413 11.4.1 Direct Fourier Inversion 413 11.4.2 Filtered Back-Projection 415 11.4.3 Linearly Interpolated Filters 419 11.4.4 The Shepp-Logan Analysis of the Ram-Lak Filters 421 11.4.5 Sample Spacing in a Parallel Beam Machine 426 11.5 Filtered Back-Projection in the Fan Beam Case 428 11.5.1 Fan Beam Geometry 428 11.5.2 Fan Beam Filtered Back-Projection 431 11.5.3 Implementing the Fan Beam Algorithm 433 11.5.4 Data Collection for a Fan Beam Scanner 435 11.5.5 Rebinning 437 11.6 Some Mathematical Remarks* 437 11.7 Spiral Scan CT 439 11.7.1 Interpolation methods 439 11.7.2 3d-Reconstruction Formulae 442 11.8 TheGridding Method* 443 11.9 Conclusion 447 12 Imaging Artifacts in X-Ray Tomography 451 12.1 The Effect of a Finite Width X-Ray Beam 451 12.1.1 A Linear Model for Finite Beam Width 452 12.1.2 A Nonlinear Model for Finite Beam Width 454 12.1.3 The Partial Volume Effect 456 12.2 The PSF 458
xii Contents 12.2.1 Point Sources 459 12.2.2 The PSF without Sampling 459 12.3 The PSF with Sampling 466 12.3.1 Ray Sampling 466 12.3.2 View Sampling 472 12.4 The Effects of Measurement Errors 481 12.4.1 A Single Bad Ray 483 12.4.2 A Bad Ray in Each View 484 12.4.3 A Bad View 486 12.5 Beam Hardening 488 12.6 Conclusion 490 13 Algebraic Reconstruction Techniques 493 13.1 Algebraic Reconstruction 493 13.2 Kaczmarz's Method 498 13.3 A Bayesian Estimate 503 13.4 Variants of the Kaczmarz Method 505 13.4.1 Relaxation Parameters 505 13.4.2 Other Related Algorithms 506 13.5 Conclusion 507 14 Magnetic Resonance Imaging 509 14.1 Introduction 509 14.2 Nuclear Magnetic Resonance 510 14.3 The Bloch Phenomological Equation 511 14.4 The Rotating Reference Frame 513 14.5 A Basic Imaging Experiment 514 14.6 Selective Excitation 517 14.7 Spin-warp Imaging 519 14.8 Contrast and Resolution 521 14.9 Conclusion 523 15 Probability and Random Variables 525 15.1 Measure Theory* 526 15.1.1 Allowable Events 526 15.1.2 Measures and Probability 529 15.1.3 Integration 533 15.1.4 Independent Events 540 15.1.5 Conditional Probability 541 15.2 Random Variables* 543 15.2.1 Cumulative Distribution Function 546 15.2.2 The Variance 549 15.2.3 The Characteristic Function 550 15.2.4 A Pair of Random Variables 552
xiii 15.2.5 Several Random Variables 558 15.3 Some Important Random Variables 560 15.3.1 Bernoulli Random Variables 560 15.3.2 Poisson Random Variables 561 15.3.3 Gaussian Random Variables 562 15.4 Limits of Random Variables 565 15.4.1 The Central Limit Theorem 565 15.4.2 Other Examples of Limiting Distributions 568 15.5 Statistics and Measurements 571 15.6 Conclusion 574 16 Applications of Probability 577 16.1 Applications to X-Ray Imaging 577 16.1.1 Modeling a Source-Detector Pair 577 16.1.2 Beer's Law 578 16.2 Noise in the Filtered Back-Projection Algorithm 581 16.2.1 Sampled Data 582 16.2.2 A Computation of the Variance in the Measurements 585 16.2.3 The Variance of the Radon Transform 586 16.2.4 The Variance in the Reconstructed Image 588 16.2.5 Signal-to-Noise Ratio, Dosage and Contrast 589 16.3 Signal-to-Noise in Magnetic Resonance Imaging 591 16.4 Image Reconstruction in PET 593 16.4.1 Positron Emission Physics 594 16.4.2 A Probabilistic Model for PET 597 16.4.3 The Maximum Likelihood Algorithm 599 16.4.4 Determining the Transition Matrix 601 16.5 Conclusion 603 17 Random Processes 605 17.1 Random Processes in Measurements 605 17.2 Basic Definitions 607 17.2.1 Statistical Properties of Random Processes 609 17.2.2 Stationary Random Processes 610 17.2.3 Spectral Analysis of Stationary Processes* 611 17.2.4 Independent and Stationary Increments 615 17.3 Examples of Random Processes 615 17.3.1 Gaussian Random Process 615 17.3.2 The Poisson Counting Process 616 17.3.3 Poisson Arrival Process 619 17.3.4 Fourier Coefficients for Periodic Processes* 622 17.3.5 White Noise* 625 17.4 Random Inputs to Linear Systems 627 17.4.1 The Autocorrelation of the Output 629
xiv Contents 17.4.2 Thermal or Johnson Noise 631 17.4.3 Optimal Filters 633 17.5 Noise in Filtered Back-Projection 635 17.6 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
Contents xv В.8 Fubini's Theorem and Differentiation of Integrals* 739 Bibliography 743 Index 753