Foundations of Image Science

Transcription

1 Foundations of Image Science Harrison H. Barrett Kyle J. Myers

2 2004 by John Wiley & Sons,, Hoboken,

3 1 VECTORS AND OPERATORS LINEAR VECTOR SPACES Vector addition and scalar multiplication Metric spaces and norms Sequences of vectors and complete metric spaces Scalar products and Hilbert space Basis vectors Continuous bases TYPES OF OPERATORS Functions and functionals Integral transforms Matrix operators Continuous-to-discrete mappings Differential operators HILBERT-SPACE OPERATORS Range and domain Linearity, boundedness and continuity Compactness Inverse operators Adjoint operators Projection operators Outer products 21 EIGENANALYSIS Eigenvectors and eigenvalue spectra Similarity transformations Eigenanalysis in finite-dimensional spaces Eigenanalysis of Hermitian operators Diagonalization of a Hermitian operator Simultaneous diagonalization of Hermitian matrices SINGULAR-VALUE DECOMPOSITION Definition and properties Subspaces SVD representations of vectors and operators MOORE-PENROSE PSEUDOINVERSE Penrose equations Pseudoinverses and SVD Properties of the pseudoinverse Pseudoinverses and projection operators PSEUDOINVERSES AND LINEAR EQUATIONS Nature of solutions of linear equations Existence and uniqueness of exact solutions Explicit solutions for consistent data Least-squares solutions Minimum-norm solutions Iterative calculation of pseudoinverse solutions 53

4 1.8 REPRODUCING-KERNEL HILBERT SPACE Positive-definite Hermitian operators Nonnegative-definite Hermitian operators 59 2 THE DIRAC DELTA AND OTHER GENERALIZED FUNCTIONS THEORY OF DISTRIBUTIONS Basic concepts Well-behaved functions Approximation of other functions Formal definition of distributions Properties of distributions Tempered distributions ONE-DIMENSIONAL DELTA FUNCTION Intuitive definition and elementary properties Limiting representations Distributional approach Derivatives of delta functions A synthesis Delta functions as basis vectors OTHER GENERALIZED FUNCTIONS IN 1D Generalized functions as limits Generalized functions related to the delta function Other point singularities MULTIDIMENSIONAL DELTA FUNCTIONS Multidimensional distributions Multidimensional delta functions Delta functions in polar coordinates Line masses and plane masses Multidimensional derivatives of delta functions Other point singularities Angular delta functions 94 3 FOURIER ANALYSIS SINES, COSINES AND COMPLEX EXPONENTIALS Orthogonality on a finite interval Complex exponentials Orthogonality on the infinite interval Discrete orthogonality View from the complex plane FOURIER SERIES Basic concepts Convergence of the Fourier series Properties of the Fourier coefficients D FOURIER TRANSFORM Basic concepts Convergence issues Unitarity of the Fourier operator Fourier transforms of generalized functions Properties of the 1D Fourier transform 120

5 3.3.6 Convolution and correlation Fourier transforms of some special functions Relation between Fourier series and Fourier transforms Analyticity of Fourier transforms Related transforms MULTIDIMENSIONAL FOURIER TRANSFORMS Basis functions Definitions and elementary properties Multidimensional convolution and correlation Rotationally symmetric functions 146 transforms Some special functions and their Multidimensional periodicity 149 SAMPLING THEORY Bandlimited functions Reconstruction of a bandlimited function from uniform samples Aliasing Sampling in frequency space Multidimensional sampling Sampling with a finite aperture DISCRETE FOURIER TRANSFORM Motivation and definitions Basic properties of the DFT Relation between discrete and continuous Fourier transforms Discrete-Space Fourier Transform Fast Fourier Transform Multidimensional DFTs SERIES EXPANSIONS AND INTEGRAL TRANSFORMS EXPANSIONS IN ORTHOGONAL FUNCTIONS Basic concepts Orthogonal polynomials Sturm-Liouville theory Classical orthogonal polynomials and related functions Prolate spheroidal wavefunctions CLASSICAL INTEGRAL TRANSFORMS Laplace transform Mellin transform z transform Hilbert transform Higher-order Hankel transforms FRESNEL INTEGRALS AND TRANSFORMS Fresnel integrals Fresnel transforms Chirps and Fourier transforms RADON TRANSFORM D Radon transform and its adjoint Central-slice theorem Filtered backprojection 206

6 4.4.4 Unfiltered backprojection Radon transform in higher dimensions Radon transform in signal processing MIXED REPRESENTATIONS LOCAL SPECTRAL ANALYSIS Local Fourier transforms Uncertainty Local frequency Gabor's signal expansion BILINEAR TRANSFORMS Wigner distribution function Ambiguity functions Fractional Fourier transforms WAVELETS Mother wavelets and scaling functions Continuous wavelet transform Discrete wavelet transform Multiresolution analysis GROUP THEORY BASIC CONCEPTS Definition of a group Group multiplication tables Isomorphism and homomorphism SUBGROUPS AND CLASSES Definitions Examples GROUP REPRESENTATIONS Matrices that obey the multiplication table Irreducible representations Characters Unitary irreducible representations and orthogonality properties SOME FINITE GROUPS Cyclic groups Dihedral groups CONTINUOUS GROUPS Basic properties Linear, orthogonal and unitary groups Abelian and non-abelian Lie groups GROUPS OF OPERATORS ON A HILBERT SPACE Geometrical transformations of functions Invariant subspaces Irreducible subspaces Orthogonality of basis functions 255 QUANTUM MECHANICS AND IMAGE SCIENCE Smattering of quantum mechanics Connection with image science 257

7 6.7.3 Symmetry group of the Hamiltonian Symmetry and degeneracy Reducibility and accidental degeneracy Parity Rotational symmetry in three dimensions FUNCTIONS AND TRANSFORMS ON GROUPS Functions on a finite group Extension to infinite groups Convolutions on groups Fourier transforms on groups Wavelets revisited DETERMINISTIC DESCRIPTIONS OF IMAGING SYSTEMS OBJECTS AND IMAGES Objects and images as functions Objects and images as infinite-dimensional vectors Objects and images as finite-dimensional vectors Representation accuracy Uniform translates Other representations LINEAR CONTINUOUS-TO-CONTINUOUS SYSTEMS General shift-variant systems Adjoint operators and SVD Shift-invariant systems Eigenanalysis of LSIV systems Singular-value decomposition of LSIV systems Transfer functions Magnifiers Approximately shift-invariant systems Rotationally symmetric systems Axial systems LINEAR CONTINUOUS-TO-DISCRETE SYSTEMS System operator Adjoint operator and singular-value decomposition Fourier description Sampled LSIV systems Mixed CC-CD systems Discrete-to-continuous systems LINEAR DISCRETE-TO-DISCRETE SYSTEMS System matrix Adjoint operator and singular-value decomposition Image errors Discrete representations of shift-invariant systems NONLINEAR SYSTEMS Point nonlinearities Nonlocal nonlinearities Object-dependent system operators Postdetection nonlinear operations 359

8 8 STOCHASTIC DESCRIPTIONS OF OBJECTS AND IMAGES RANDOM VECTORS Basic concepts Expectations Covariance and correlation matrices Characteristic functions Transformations of random vectors Eigenanalysis of covariance matrices RANDOM PROCESSES Definitions and basic concepts Averages of random processes Characteristic functionals Correlation analysis Spectral analysis Linear filtering of random processes Eigenanalysis of the autocorrelation operator Discrete random processes NORMAL RANDOM VECTORS AND PROCESSES Probability density functions Characteristic function Marginal densities and linear transformations Central-limit theorem Normal random processes Complex Gaussian random fields STOCHASTIC MODELS FOR OBJECTS Probability density functions in Hilbert space Multipoint densities Normal models Texture models Signals and backgrounds STOCHASTIC MODELS FOR IMAGES Linear systems Conditional statistics for a single object Effects of object randomness Signals and backgrounds in image space DIFFRACTION THEORY AND IMAGING WAVE EQUATIONS Maxwell's equations Maxwell's equations in the Fourier domain Material media Time-dependent wave equations Time-independent wave equations PLANE WAVES AND SPHERICAL WAVES Plane waves Spherical waves GREEN'S FUNCTIONS Differential equations for the Green's functions Time-dependent Green's function 468

9 Green's functions for the Helmholtz and Poisson equations Defined-source problems Boundary-value problems 473 DIFFRACTION BY A PLANAR APERTURE Surface at infinity Kirchhoff boundary conditions Application of Green ' s theorem Diffraction as a 2D linear filter Some useful approximations Fresnel diffraction Fraunhofer diffraction 483 DIFFRACTION IN THE FREQUENCY DOMAIN Angular spectrum Fresnel and Fraunhofer approximations Beams Reflection and refraction of light IMAGING OF POINT OBJECTS Ideal thin lens Imaging a monochromatic point source Transmittance of an aberrated lens Rotationally symmetric lenses Field curvature and distortion Probing the pupil Interpretation of the other Seidel aberrations IMAGING OF EXTENDED PLANAR OBJECTS Monochromatic objects and a simple lens f imaging system More complicated lens systems Random fields and coherence Quasimonochromatic imaging Spatially incoherent, quasimonochromatic imaging Polychromatic, incoherent imaging Partially coherent imaging VOLUME DIFFRACTION AND 3D IMAGING Born approximation Rytov approximation Fraunhofer diffraction from volume objects Coherent 3D imaging ENERGY TRANSPORT AND PHOTONS ELECTROMAGNETIC ENERGY FLOW AND DETECTION Energy flow in classical electrodynamics Plane waves Photons Physics of photodetection What do real detectors detect? RADIOMETRIC QUANTITIES AND UNITS Self-luminous surface objects Self-luminous volume objects 574

10 Surface reflection and scattering Transmissive objects Cross sections Distribution function Radiance in physical optics and quantum optics BOLTZMANN TRANSPORT EQUATION Derivation of the Boltzmann equation Steady-state solutions in non-absorbing media Steady-state solutions in absorbing media Scattering effects Spherical harmonics Elastic scattering and diffusion Inelastic (Compton) scattering TRANSPORT THEORY AND IMAGING General imaging equation Pinhole imaging Optical imaging of a planar source Adjoint methods Monte Carlo methods POISSON STATISTICS AND PHOTON COUNTING POISSON RANDOM VARIABLES Poisson and independence Poisson and rarity Binomial selection of a Poisson Doubly stochastic Poisson random variables POISSON RANDOM VECTORS Multivariate Poisson statistics Doubly stochastic multivariate statistics RANDOM POINT PROCESSES Temporal point processes Spatial point processes Mean and autocorrelation of point processes Relation between Poisson random vectors and processes Karhunen-Loeve analysis of Poisson processes Doubly stochastic spatial Poisson random processes Doubly stochastic temporal Poisson random processes Point processes in other domains Filtered point processes Characteristic functionals of filtered point processes Spectral properties of point processes RANDOM AMPLIFICATION Random amplification in single-element detectors Random amplification and generating functions Random amplification of point processes Spectral analysis Random amplification in arrays 683

11 11.5 QUANTUM MECHANICS OF PHOTON COUNTING Coherent states Density operators Counting statistics NOISE IN DETECTORS PHOTON NOISE AND SHOT NOISE IN PHOTODIODES Vacuum photodiodes Basics of semiconductor detectors Shot noise in semiconductor photodiodes OTHER NOISE MECHANISMS Thermal noise Generation-recombination noise /f noise Noise in gated integrators Arrays of noisy photodetectors X-RAY AND GAMMA-RAY DETECTORS Interaction mechanisms Photon-counting semiconductor detectors Semiconductor detector arrays Position and energy estimation with semiconductor detectors Scintillation cameras Position and energy estimation with scintillation cameras Imaging characteristics of photon-counting detectors Integrating detectors K x rays and Compton scattering STATISTICAL DECISION THEORY BASIC CONCEPTS Kinds of decisions Inputs to the process CLASSIFICATION TASKS Partitioning the data space Binary decision outcomes The ROC curve Performance measures for binary tasks Computation of AUC The likelihood ratio and the ideal observer Statistical properties of the likelihood ratio Ideal observer with Gaussian statistics Ideal observer with non-gaussian data Signal variability and the ideal observer Background variability and the ideal observer The optimal linear discriminant Detectability in continuous data ESTIMATION THEORY Basic concepts MSE in digital imaging Bayesian estimation 883

12 Maximum-likelihood estimation Likelihood and Fisher information Properties of ML estimators Other classical estimators Nuisance parameters Hybrid detection/estimation tasks IMAGE QUALITY SURVEY OF APPROACHES Subjective assessment Fidelity measures JND models Information-theoretic assessment Objective assessment of image quality HUMAN OBSERVERS AND CLASSIFICATION TASKS Methods for investigating the visual system Modified ideal-observer models Psychophysical methods for image evaluation Estimation of figures of merit MODEL OBSERVERS General considerations Linear observers Ideal observers Estimation tasks SOURCES OF IMAGES Deterministic simulation of objects Stochastic simulation of objects Deterministic simulation of image formation Stochastic simulation of image formation Gold standards INVERSE PROBLEMS BASIC CONCEPTS Classifications of inverse problems Discretization dilemma Estimability Positivity Choosing the best algorithm LINEAR RECONSTRUCTION OPERATORS Matrix operators for estimation of expansion coefficients Reconstruction of functions from discrete data Reconstruction from Fourier samples Discretization of analytic inverses More on analytic inverses Noise with linear reconstruction operators IMPLICIT ESTIMATES Functional minimization Data-agreement functionals Regularizing functionals 1035

13 Effects of positivity Reconstruction without discretization Resolution and noise in implicit estimates ITERATIVE ALGORITHMS Linear iterative algorithms Noise propagation in linear algorithms Search algorithms for functional minimization Nonlinear constraints and fixed-point iterations Projections onto convex sets MLEM algorithm Noise propagation in nonlinear algorithms Stochastic algorithms PLANAR IMAGING WITH X RAYS AND GAMMA RAYS DIGITAL RADIOGRAPHY The source and the object X-ray detection Scattered radiation Deterministic properties of shadow images Stochastic properties Image quality: Detection tasks Image quality: Estimation tasks PLANAR IMAGING IN NUCLEAR MEDICINE Basic issues Image formation The detector Stochastic properties Image quality: Classification tasks Image quality: Estimation tasks SINGLE-PHOTON EMISSION COMPUTED TOMOGRAPHY FORWARD PROBLEMS CD formulations for parallel-beam SPECT Equally spaced angles Fourier analysis in the CD formulation D Radon transform and parallel-beam SPECT D transforms and cone-beam SPECT Attenuation INVERSE PROBLEMS SVD of the 2D Radon transform Inverses and pseudoinverses in 2D Inversion of the 3D x-ray transform Inversion of attenuated transforms Discretization of analytic reconstruction algorithms Matrices for iterative methods 1200

14 17.3 NOISE AND IMAGE QUALITY Noise in the data Noise in reconstructed images Artifacts Image quality COHERENT IMAGING AND SPECKLE BASIC CONCEPTS Elementary statistical considerations Speckle in imaging SPECKLE IN A NONIMAGING SYSTEM Description of the ground glass Some simplifying assumptions Propagation of characteristic functionals Central-limit theorem Statistics of the irradiance SPECKLE IN AN IMAGING SYSTEM The imaging system Propagation of characteristic functionals Effect of the detector NOISE AND IMAGE QUALITY Measurement noise Random objects Task performance POINT-SCATTERING MODELS AND NON-GAUSSIAN SPECKLE Object fields and objects Image fields Univariate statistics of the image field and irradiance COHERENT RANGING System configurations Deterministic analysis Statistical analysis Task performance IMAGING IN FOURIER SPACE FOURIER MODULATORS Data acquisition Noise Reconstruction Image quality INTERFEROMETERS Young's double-slit experiment Visibility estimation Michelson stellar interferometer Interferometers with multiple telescopes 1368 Epilogue: Frontiers in Image Science 1375

15 Appendix A: Matrix Algebra 1383 Appendix B: Complex Variables 1413 Appendix C: Probability 1427 Bibliography 1473 Index 1523