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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 | |
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| |
| Poisson Statistics and Photon Counting | |
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| |
| |
| 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 | |
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| |
| |
| Doubly stochastic multivariate statistics | |
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| |
| |
| Random Point Processes | |
| |
| |
| |
| Temporal point processes | |
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| |
| |
| Spatial point processes | |
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| |
| |
| Mean and autocorrelation of point processes | |
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| |
| |
| 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 | |
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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 | |
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| |
| |
| Random amplification of point processes | |
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| |
| |
| Spectral analysis | |
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| |
| |
| Random amplification in arrays | |
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| Quantum Mechanics of Photon Counting | |
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| Coherent states | |
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| Density operators | |
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| Counting statistics | |
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| Noise in Detectors | |
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| Photon Noise and Shot Noise in Photodiodes | |
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| 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 | |
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| Thermal noise | |
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| 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 | |
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| Basic issues | |
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| 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 | |
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| Matrices for iterative methods | |
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| Noise and Image Quality | |
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| Noise in the data | |
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| Noise in reconstructed images | |
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| Artifacts | |
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| Image quality | |
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| Speckle | |
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| Basic Concepts | |
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| Elementary statistical considerations | |
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| Speckle in imaging | |
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| Speckle in a Nonimaging System | |
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| Description of the ground glass | |
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| Some simplifying assumptions | |
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| Propagation of characteristic functionals | |
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| Central-limit theorem | |
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| Statistics of the irradiance | |
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| Speckle in an Imaging System | |
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| The imaging system | |
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| Propagation of characteristic functionals | |
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| Effect of the detector | |
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| Noise and Image Quality | |
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| Measurement noise | |
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| Random objects | |
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| Task performance | |
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| Point-Scattering Models and Non-Gaussian Speckle | |
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| Object fields and objects | |
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| Image fields | |
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| Univariate statistics of the image field and irradiance | |
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| Coherent Ranging | |
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| System configurations | |
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| Deterministic analysis | |
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| Statistical analysis | |
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| Task performance | |
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| Imaging in Fourier Space | |
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| Fourier Modulators | |
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| Data acquisition | |
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| Noise | |
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| Reconstruction | |
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| Image quality | |
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| Interferometers | |
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| Young's double-slit experiment | |
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| Visibility estimation | |
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| Michelson stellar interferometer | |
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| Interferometers with multiple telescopes | |
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| Epilogue: Frontiers in Image Science | |