Foundations of Image Science - Barrett, Harrison H

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Résumé :
1. VECTORS AND OPERATORS.

1.1 LINEAR VECTOR SPACES.

1.1.1 Vector addition and scalar multiplication.

1.1.2 Metric spaces and norms.

1.1.3 Sequences of vectors and complete metric spaces.

1.1.4 Scalar products and Hilbert space.

1.1.5 Basis vectors.

1.1.6 Continuous bases.

1.2 TYPES OF OPERATORS.

1.2.1 Functions and functionals.

1.2.2 Integral transforms.

1.2.3 Matrix operators.

1.2.4 Continuous-to-discrete mappings.

1.2.5 Differential operators.

1.3 HILBERT-SPACE OPERATORS.

1.3.1 Range and domain.

1.3.2 Linearity, boundedness and continuity.

1.3.3 Compactness.

1.3.4 Inverse operators.

1.3.5 Adjoint operators.

1.3.6 Projection operators.

1.3.7 Outer products.

1.4 EIGENANALYSIS.

1.4.1 Eigenvectors and eigenvalue spectra.

1.4.2 Similarity transformations.

1.4.3 Eigenanalysis infinite-dimensional spaces.

1.4.4 Eigenanalysis of Hermitian operators.

1.4.5 Diago nalization of a Hermitian operator.

1.4.6 Simultaneo us diagonalization of Hermitian matrices.

1.5 SINGULAR-VALUE DECOMPOSITION.

1.5.1 Definition and properties.

1.5.2 Subspaces.

1.5.3 SVD representation of vectors and operators.

1.6 MOORE-PENROSE PSEUDOINVERSE.

1.6.1 Penrose equations.

1.6.2 Pseudoinverses and SVD.

1.6.3 Properties of the pseudoinverse.

1.6.4 Pseudoinverses and projection operators.

1.7 PSEUDOINVERSES AND LINEAR EQUATIONS.

1.7.1 Nature of solutions of linear equations.

1.7.2 Existence and uniqueness of exact solutions.

1.7.3 Explicit solutions for consistent data.

1.7.4 Least-squares solutions.

1.7.5 Minimum-norm solutions.

1.7.6 Iterative calculation of pseudoinverse solution.

1.8 REPRODUCING-KERNEL HILBERT SPACES.

1.8.1 Positive-definite Hermitian operators.

1.8.2 Nonnegative-definite Hermitian operators.

2. THE DIRAC DELTA AND OTHER GENERALIZED FUNCTIONS.

2.1 THEORY OF DISTRIBUTIONS.

2.1.1 Basic concepts.

2.1.2 Well-behaved functions.

2.1.3 Approximation of other functions.

2.1.4 Formal definition of distributions.

2.1.5 Properties of distributions.

2.1.6 Tempered distributions.

2.2 ONE-DIMENSIONAL DELTA FUNCTION.

2.2.1 Intuitive definition and elementary properties.

2.2.2 Limiting representations.

2.2.3 Distributional approach.

2.2.4 Derivatives of delta functions.

2.2.5 A synthesis.

2.2.6 Delta functions as basis vectors.

2.3 OTHER GENERALIZED FUNCTIONS IN 1D.

2.3.1 Generalized functions as limits.

2.3.2 Generalized functions related to the delta function.

2.3.3 Other point singularities.

2.4 MULTIDIMENSIONAL DELTA FUNCTIONS.

2.4.1 Multidimensional distributions.

2.4.2 Multidimensional delta functions.

2.4.3 Delta functions in polar coordinates.

2.4.4 Line masses and plane masses.

2.4.5 Multidimensional derivatives of delta functions.

2.4.6 Other point singularities.

2.4.7 Angular delta functions.

3. FOURIER ANALYSIS.

3.1 SINES, COSINES AND COMPLEX EXPONENTIALS.

3.1.1 Orthogonality on a finite interval.

3.1.2 Complex exponentials.

3.1.3 Orthogonality on the infinite interval.

3.1.4 Discrete orthogonality.

3.1.5 The view from the complex plane.

3.2 FOURIER SERIES.

3.2.1 Basic concepts.

3.2.2 Convergence of the Fourier series.

3.2.3 Properties of the Fourier coefficients.

3.3 1D FOURIER TRANSFORM.

3.3.1 Basic concepts.

3.3.2 Convergence i...

Sommaire:
HARRISON H. BARRETT received a BS in physics from Virginia Polytechnic Institute, an MS in Physics from MIT, and a PhD in applied physics from Harvard. Dr. Barrett is a professor in the Optical Sciences Center, the Department of Radiology, and the Program in Applied Mathematics and serves as Director of the Center for Gamma-ray Imaging. The holder of twenty-three U.S. patents, he is the recipient of the IEEE Medical Imaging Scientist Award and a Humboldt Prize and the coauthor, with William Swindell, of Radiological Imaging: The Theory of Image Formation, Detection, and Processing.

KYLE J. MYERS received a BS in Mathematics and Physics from Occidental College and an MS and PhD in Optical Sciences from the University of Arizona. Dr. Myers is the Chief of the Medical Imaging and Computer Applications Branch of the Center for Devices and Radiological Health of the U.S. Food and Drug Administration. She is a member of the SPIE, the Optical Society of America, and the Medical Image Perception Society (MIPS), and recently served as cochair of the Medical Image Perception Conference sponsored by MIPS.