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Vectors and Operators | |

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Linear Vector Spaces | |

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Vector addition and scalar multiplication | |

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Metric spaces and norms | |

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Sequences of vectors and complete metric spaces | |

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Scalar products and Hilbert space | |

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Basis vectors | |

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Continuous bases | |

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Types of Operators | |

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Functions and functionals | |

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Integral transforms | |

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Matrix operators | |

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Continuous-to-discrete mappings | |

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Differential operators | |

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Hilbert-Space Operators | |

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Range and domain | |

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Linearity, boundedness and continuity | |

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Compactness | |

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Inverse operators | |

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Adjoint operators | |

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Projection operators | |

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Outer products | |

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Eigenanalysis | |

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Eigenvectors and eigenvalue spectra | |

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Similarity transformations | |

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Eigenanalysis in finite-dimensional spaces | |

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Eigenanalysis of Hermitian operators | |

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Diagonalization of a Hermitian operator | |

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Simultaneous diagonalization of Hermitian matrices | |

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Singular-Value Decomposition | |

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Definition and properties | |

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Subspaces | |

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SVD representations of vectors and operators | |

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Moore-Penrose Pseudoinverse | |

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Penrose equations | |

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Pseudoinverses and SVD | |

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Properties of the pseudoinverse | |

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Pseudoinverses and projection operators | |

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Pseudoinverses and Linear Equations | |

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Nature of solutions of linear equations | |

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Existence and uniqueness of exact solutions | |

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Explicit solutions for consistent data | |

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Least-squares solutions | |

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Minimum-norm solutions | |

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Iterative calculation of pseudoinverse solutions | |

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Reproducing-Kernel Hilbert Space | |

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Positive-definite Hermitian operators | |

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Nonnegative-definite Hermitian operators | |

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The Dirac Delta and Other Generalized Functions | |

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Theory of Distributions | |

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Basic concepts | |

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Well-behaved functions | |

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Approximation of other functions | |

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Formal definition of distributions | |

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Properties of distributions | |

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Tempered distributions | |

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One-Dimensional Delta Function | |

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Intuitive definition and elementary properties | |

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Limiting representations | |

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Distributional approach | |

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Derivatives of delta functions | |

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A synthesis | |

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Delta functions as basis vectors | |

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Other Generalized Functions in 1D | |

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Generalized functions as limits | |

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Generalized functions related to the delta function | |

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Other point singularities | |

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Multidimensional Delta Functions | |

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Multidimensional distributions | |

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Multidimensional delta functions | |

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Delta functions in polar coordinates | |

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Line masses and plane masses | |

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Multidimensional derivatives of delta functions | |

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Other point singularities | |

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Angular delta functions | |

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Fourier Analysis | |

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Sines, Cosines and Complex Exponentials | |

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Orthogonality on a finite interval | |

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Complex exponentials | |

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Orthogonality on the infinite interval | |

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Discrete orthogonality | |

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View from the complex plane | |

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Fourier Series | |

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Basic concepts | |

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Convergence of the Fourier series | |

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Properties of the Fourier coefficients | |

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1D Fourier Transform | |

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Basic concepts | |

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Convergence issues | |

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Unitarity of the Fourier operator | |

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Fourier transforms of generalized functions | |

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Properties of the 1D Fourier transform | |

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Convolution and correlation | |

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Fourier transforms of some special functions | |

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Relation between Fourier series and Fourier transforms | |

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Analyticity of Fourier transforms | |

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Related transforms | |

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Multidimensional Fourier Transforms | |

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Basis functions | |

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Definitions and elementary properties | |

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Multidimensional convolution and correlation | |

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Rotationally symmetric functions | |

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Some special functions and their transforms | |

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Multidimensional periodicity | |

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Sampling Theory | |

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Bandlimited functions | |

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Reconstruction of a bandlimited function from uniform samples | |

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Aliasing | |

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Sampling in frequency space | |

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Multidimensional sampling | |

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Sampling with a finite aperture | |

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Discrete Fourier Transform | |

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Motivation and definitions | |

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Basic properties of the DFT | |

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Relation between discrete and continuous Fourier transforms | |

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Discrete-Space Fourier Transform | |

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Fast Fourier Transform | |

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Multidimensional DFTs | |

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Series Expansions and Integral Transforms | |

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Expansions in Orthogonal Functions | |

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Basic concepts | |

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Orthogonal polynomials | |

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Sturm-Liouville theory | |

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Classical orthogonal polynomials and related functions | |

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Prolate spheroidal wavefunctions | |

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Classical Integral Transforms | |

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Laplace transform | |

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Mellin transform | |

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z transform | |

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Hilbert transform | |

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Higher-order Hankel transforms | |

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Fresnel Integrals and Transforms | |

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Fresnel integrals | |

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Fresnel transforms | |

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Chirps and Fourier transforms | |

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Radon Transform | |

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2D Radon transform and its adjoint | |

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Central-slice theorem | |

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Filtered backprojection | |

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Unfiltered backprojection | |

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Radon transform in higher dimensions | |

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Radon transform in signal processing | |

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Mixed Representations | |

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Local Spectral Analysis | |

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Local Fourier transforms | |

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Uncertainty | |

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Local frequency | |

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Gabor's signal expansion | |

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Bilinear Transforms | |

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Wigner distribution function | |

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Ambiguity functions | |

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Fractional Fourier transforms | |

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Wavelets | |

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Mother wavelets and scaling functions | |

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Continuous wavelet transform | |

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Discrete wavelet transform | |

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Multiresolution analysis | |

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Group Theory | |

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Basic Concepts | |

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Definition of a group | |

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Group multiplication tables | |

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Isomorphism and homomorphism | |

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Subgroups and Classes | |

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Definitions | |

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Examples | |

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Group Representations | |

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Matrices that obey the multiplication table | |

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Irreducible representations | |

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Characters | |

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Unitary irreducible representations and orthogonality properties | |

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Some Finite Groups | |

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Cyclic groups | |

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Dihedral groups | |

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Continuous Groups | |

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Basic properties | |

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Linear, orthogonal and unitary groups | |

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Abelian and non-Abelian Lie groups | |

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Groups of Operators on a Hilbert Space | |

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Geometrical transformations of functions | |

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Invariant subspaces | |

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Irreducible subspaces | |

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Orthogonality of basis functions | |

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Quantum Mechanics and Image Science | |

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Smattering of quantum mechanics | |

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Connection with image science | |

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Symmetry group of the Hamiltonian | |

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Symmetry and degeneracy | |

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Reducibility and accidental degeneracy | |

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Parity | |

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Rotational symmetry in three dimensions | |

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Functions and Transforms on Groups | |

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Functions on a finite group | |

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Extension to infinite groups | |

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Convolutions on groups | |

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Fourier transforms on groups | |

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Wavelets revisited | |

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Deterministic Descriptions of Imaging Systems | |

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Objects and Images | |

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Objects and images as functions | |

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Objects and images as infinite-dimensional vectors | |

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Objects and images as finite-dimensional vectors | |

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Representation accuracy | |

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Uniform translates | |

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Other representations | |

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Linear Continuous-To-Continuous Systems | |

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General shift-variant systems | |

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Adjoint operators and SVD | |

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Shift-invariant systems | |

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Eigenanalysis of LSIV systems | |

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Singular-value decomposition of LSIV systems | |

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Transfer functions | |

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Magnifiers | |

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Approximately shift-invariant systems | |

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Rotationally symmetric systems | |

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Axial systems | |

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Linear Continuous-to-Discrete Systems | |

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System operator | |

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Adjoint operator and singular-value decomposition | |

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Fourier description | |

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Sampled LSIV systems | |

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Mixed CC-CD systems | |

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Discrete-to-continuous systems | |

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Linear Discrete-to-Discrete Systems | |

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System matrix | |

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Adjoint operator and singular-value decomposition | |

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Image errors | |

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Discrete representations of shift-invariant systems | |

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Nonlinear Systems | |

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Point nonlinearities | |

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Nonlocal nonlinearities | |

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Object-dependent system operators | |

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Postdetection nonlinear operations | |

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Stochastic Descriptions of Objects and Images | |

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Random Vectors | |

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Basic concepts | |

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Expectations | |

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Covariance and correlation matrices | |

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| |

Characteristic functions | |

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| |

Transformations of random vectors | |

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Eigenanalysis of covariance matrices | |

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| |

Random Processes | |

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| |

Definitions and basic concepts | |

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Averages of random processes | |

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Characteristic functionals | |

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Correlation analysis | |

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Spectral analysis | |

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Linear filtering of random processes | |

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Eigenanalysis of the autocorrelation operator | |

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Discrete random processes | |

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Normal Random Vectors and Processes | |

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Probability density functions | |

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Characteristic function | |

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Marginal densities and linear transformations | |

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Central-limit theorem | |

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Normal random processes | |

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Complex Gaussian random fields | |

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Stochastic Models for Objects | |

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Probability density functions in Hilbert space | |

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Multipoint densities | |

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Normal models | |

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Texture models | |

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Signals and backgrounds | |

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Stochastic Models for Images | |

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Linear systems | |

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Conditional statistics for a single object | |

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Effects of object randomness | |

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Signals and backgrounds in image space | |

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| |

Diffraction Theory and Imaging | |

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Wave Equations | |

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Maxwell's equations | |

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Maxwell's equations in the Fourier domain | |

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Material media | |

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Time-dependent wave equations | |

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Time-independent wave equations | |

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Plane Waves and Spherical Waves | |

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Plane waves | |

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Spherical waves | |

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Green's Functions | |

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Differential equations for the Green's functions | |

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Time-dependent Green's function | |

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Green's functions for the Helmholtz and Poisson equations | |

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Defined-source problems | |

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Boundary-value problems | |

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Diffraction by a Planar Aperture | |

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Surface at infinity | |

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Kirchhoff boundary conditions | |

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| |

Application of Green's theorem | |

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| |

Diffraction as a 2D linear filter | |

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Some useful approximations | |

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Fresnel diffraction | |

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Fraunhofer diffraction | |

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Diffraction in the Frequency Domain | |

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Angular spectrum | |

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Fresnel and Fraunhofer approximations | |

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Beams | |

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Reflection and refraction of light | |

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| |

Imaging of Point Objects | |

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Ideal thin lens | |

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| |

Imaging a monochromatic point source | |

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Transmittance of an aberrated lens | |

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Rotationally symmetric lenses | |

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Field curvature and distortion | |

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Probing the pupil | |

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Interpretation of the other Seidel aberrations | |

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| |

Imaging of Extended Planar Objects | |

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Monochromatic objects and a simple lens | |

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4f imaging system | |

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| |

More complicated lens systems | |

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| |

Random fields and coherence | |

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| |

Quasimonochromatic imaging | |

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Spatially incoherent, quasimonochromatic imaging | |

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| |

Polychromatic, incoherent imaging | |

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| |

Partially coherent imaging | |

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| |

| |

Volume Diffraction and 3D Imaging | |

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| |

Born approximation | |

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| |

Rytov approximation | |

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| |

Fraunhofer diffraction from volume objects | |

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| |

Coherent 3D imaging | |

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| |

| |

Energy Transport and Photons | |

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| |

Electromagnetic Energy Flow and Detection | |

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| |

Energy flow in classical electrodynamics | |

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| |

| |

Plane waves | |

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| |

Photons | |

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| |

Physics of photodetection | |

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| |

What do real detectors detect? | |

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| |

Radiometric Quantities and Units | |

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| |

Self-luminous surface objects | |

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Self-luminous volume objects | |

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| |

Surface reflection and scattering | |

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| |

Transmissive objects | |

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| |

Cross sections | |

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| |

Distribution function | |

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| |

Radiance in physical optics and quantum optics | |

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| |

| |

Boltzmann Transport Equation | |

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| |

Derivation of the Boltzmann equation | |

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| |

Steady-state solutions in non-absorbing media | |

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Steady-state solutions in absorbing media | |

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| |

Scattering effects | |

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| |

Spherical harmonics | |

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| |

Elastic scattering and diffusion | |

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| |

Inelastic (Compton) scattering | |

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| |

Transport Theory and Imaging | |

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| |

General imaging equation | |

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| |

| |

Pinhole imaging | |

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| |

| |

Optical imaging of a planar source | |

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| |

| |

Adjoint methods | |

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| |

Monte Carlo methods | |

| |

| |

| |

Poisson Statistics and Photon Counting | |

| |

| |

| |

Poisson Random Variables | |

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| |

| |

Poisson and independence | |

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| |

Poisson and rarity | |

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| |

Binomial selection of a Poisson | |

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| |

Doubly stochastic Poisson random variables | |

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| |

Poisson Random Vectors | |

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| |

Multivariate Poisson statistics | |

| |

| |

| |

Doubly stochastic multivariate statistics | |

| |

| |

| |

Random Point Processes | |

| |

| |

| |

Temporal point processes | |

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| |

Spatial point processes | |

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| |

| |

Mean and autocorrelation of point processes | |

| |

| |

| |

Relation between Poisson random vectors and processes | |

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| |

Karhunen-Loeve analysis of Poisson processes | |

| |

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| |

Doubly stochastic spatial Poisson random processes | |

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| |

Doubly stochastic temporal Poisson random processes | |

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| |

| |

Point processes in other domains | |

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| |

Filtered point processes | |

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| |

Characteristic functionals of filtered point processes | |

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| |

| |

Spectral properties of point processes | |

| |

| |

| |

Random Amplification | |

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| |

Random amplification in single-element detectors | |

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| |

Random amplification and generating functions | |

| |

| |

| |

Random amplification of point processes | |

| |

| |

| |

Spectral analysis | |

| |

| |

| |

Random amplification in arrays | |

| |

| |

| |

Quantum Mechanics of Photon Counting | |

| |

| |

| |

Coherent states | |

| |

| |

| |

Density operators | |

| |

| |

| |

Counting statistics | |

| |

| |

| |

Noise in Detectors | |

| |

| |

| |

Photon Noise and Shot Noise in Photodiodes | |

| |

| |

| |

Vacuum photodiodes | |

| |

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| |

Basics of semiconductor detectors | |

| |

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| |

Shot noise in semiconductor photodiodes | |

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| |

Other Noise Mechanisms | |

| |

| |

| |

Thermal noise | |

| |

| |

| |

Generation-recombination noise | |

| |

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| |

1 / f noise | |

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| |

Noise in gated integrators | |

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Arrays of noisy photodetectors | |

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X-ray and Gamma-Ray Detectors | |

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Interaction mechanisms | |

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Photon-counting semiconductor detectors | |

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Semiconductor detector arrays | |

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Position and energy estimation with semiconductor detectors | |

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Scintillation cameras | |

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Position and energy estimation with scintillation cameras | |

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Imaging characteristics of photon-counting detectors | |

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Integrating detectors | |

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K x rays and Compton scattering | |

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Statistical Decision Theory | |

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Basic Concepts | |

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Kinds of decisions | |

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Inputs to the process | |

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Classification Tasks | |

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Partitioning the data space | |

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Binary decision outcomes | |

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The ROC curve | |

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Performance measures for binary tasks | |

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Computation of AUC | |

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The likelihood ratio and the ideal observer | |

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Statistical properties of the likelihood ratio | |

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Ideal observer with Gaussian statistics | |

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Ideal observer with non-Gaussian data | |

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Signal variability and the ideal observer | |

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Background variability and the ideal observer | |

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The optimal linear discriminant | |

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Detectability in continuous data | |

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Estimation Theory | |

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Basic concepts | |

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MSE in digital imaging | |

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Bayesian estimation | |

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Maximum-likelihood estimation | |

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Likelihood and Fisher information | |

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Properties of ML estimators | |

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Other classical estimators | |

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Nuisance parameters | |

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Hybrid detection / estimation tasks | |

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Image Quality | |

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Survey of Approaches | |

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Subjective assessment | |

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Fidelity measures | |

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JND models | |

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Information-theoretic assessment | |

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Objective assessment of image quality | |

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Human Observers and Classification Tasks | |

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Methods for investigating the visual system | |

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Modified ideal-observer models | |

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Psychophysical methods for image evaluation | |

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Estimation of figures of merit | |

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Model Observers | |

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General considerations | |

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Linear observers | |

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Ideal observers | |

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Estimation tasks | |

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Sources of Images | |

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Deterministic simulation of objects | |

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Stochastic simulation of objects | |

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Deterministic simulation of image formation | |

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Stochastic simulation of image formation | |

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Gold standards | |

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| |

Inverse Problems | |

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Basic Concepts | |

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Classifications of inverse problems | |

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Discretization dilemma | |

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Estimability | |

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Positivity | |

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Choosing the best algorithm | |

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Linear Reconstruction Operators | |

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Matrix operators for estimation of expansion coefficients | |

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Reconstruction of functions from discrete data | |

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Reconstruction from Fourier samples | |

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Discretization of analytic inverses | |

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More on analytic inverses | |

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Noise with linear reconstruction operators | |

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Implicit Estimates | |

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Functional minimization | |

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Data-agreement functionals | |

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Regularizing functionals | |

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Effects of positivity | |

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Reconstruction without discretization | |

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Resolution and noise in implicit estimates | |

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Iterative Algorithms | |

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Linear iterative algorithms | |

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Noise propagation in linear algorithms | |

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Search algorithms for functional minimization | |

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Nonlinear constraints and fixed-point iterations | |

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Projections onto convex sets | |

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MLEM algorithm | |

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Noise propagation in nonlinear algorithms | |

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Stochastic algorithms | |

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| |

Planar Imaging With X Rays and Gamma Rays | |

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| |

Digital Radiography | |

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| |

| |

The source and the object | |

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| |

X-ray detection | |

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| |

Scattered radiation | |

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| |

Deterministic properties of shadow images | |

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| |

Stochastic properties | |

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| |

Image quality: Detection tasks | |

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| |

| |

Image quality: Estimation tasks | |

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| |

| |

Planar Imaging in Nuclear Medicine | |

| |

| |

| |

Basic issues | |

| |

| |

| |

Image formation | |

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| |

The detector | |

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| |

Stochastic properties | |

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| |

| |

Image quality: Classification tasks | |

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| |

| |

Image quality: Estimation tasks | |

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| |

| |

Emission Computed Tomography | |

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| |

| |

Forward Problems | |

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| |

CD formulations for parallel-beam SPECT | |

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| |

Equally spaced angles | |

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| |

Fourier analysis in the CD formulation | |

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| |

2D Radon transform and parallel-beam SPECT | |

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| |

3D transforms and cone-beam SPECT | |

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| |

| |

Attenuation | |

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| |

Inverse Problems | |

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| |

SVD of the 2D Radon transform | |

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| |

Inverses and pseudoinverses in 2D | |

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| |

| |

Inversion of the 3D x-ray transform | |

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| |

Inversion of attenuated transforms | |

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| |

| |

Discretization of analytic reconstruction algorithms | |

| |

| |

| |

Matrices for iterative methods | |

| |

| |

| |

Noise and Image Quality | |

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| |

| |

Noise in the data | |

| |

| |

| |

Noise in reconstructed images | |

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| |

| |

Artifacts | |

| |

| |

| |

Image quality | |

| |

| |

| |

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 | |

| |

| |

Epilogue: Frontiers in Image Science | |