Mathematical Foundations of Infinite-dimensional Statistical Models

In nonparametric and high-dimensional statistical models, the classical Gauss-Fisher-Le Cam theory of the optimality of maximum likelihood estimators and Bayesian posterior inference does not apply, and new foundations and ideas have been developed in the past several decades. This book gives a coherent account of the statistical theory in infinite-dimensional parameter spaces. The mathematical foundations include self-contained 'mini-courses' on the theory of Gaussian and empirical processes, on approximation and wavelet theory, and on the basic theory of function spaces. The theory of statistical inference in such models - hypothesis testing, estimation and confidence sets - is then presented within the minimax paradigm of decision theory. This includes the basic theory of convolution kernel and projection estimation, but also Bayesian nonparametrics and nonparametric maximum likelihood estimation. In a final chapter the theory of adaptive inference in nonparametric models is developed, including Lepski's method, wavelet thresholding, and adaptive inference for self-similar functions.

Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing

This textbook introduces sparse and redundant representations with a focus on applications in signal and image processing. The theoretical and numerical foundations are tackled before the applications are discussed. Mathematical modeling for signal sources is discussed along with how to use the proper model for tasks such as denoising, restoration, separation, interpolation and extrapolation, compression, sampling, analysis and synthesis, detection, recognition, and more. The presentation is elegant and engaging. Sparse and Redundant Representations is intended for graduate students in applied mathematics and electrical engineering, as well as applied mathematicians, engineers, and researchers who are active in the fields of signal and image processing.

Dimension Reduction：A Guided Tour

We give a tutorial overview of several geometric methods for dimension reduction. We divide the methods into projective methods and methods that model the manifold on which the data lies. For projective methods, we review projection pursuit, principal component analysis (PCA), kernel PCA, probabilistic PCA, canonical correlation analysis, oriented PCA, and several techniques for sufficient dimension reduction. For the manifold methods, we review multidimensional scaling (MDS), landmark MDS, Isomap, locally linear embedding, Laplacian eigenmaps and spectral clustering. The Nystr¨om method, which links several of the manifold algorithms, is also reviewed. The goal is to provide a self-contained overview of key concepts underlying many of these algorithms, and to give pointers for further reading.

Riemannian Geometry and Statistical Machine Learning

Statistical machine learning algorithms deal with the problem of selecting an appropriate statistical model from a model space  based on a training set {xi}N i=1 ⊂ X or {(xi, yi)}N i=1 ⊂ X × Y. In doing so they either implicitly or explicitly make assumptions on the geometries of the model space  and the data space X. Such assumptions are crucial to the success of the algorithms as different geometries are appropriate for different models and data spaces. By studying these assumptions we are able to develop new theoretical results that enhance our understanding of several popular learning algorithms. Furthermore, using geometrical reasoning we are able to adapt existing algorithms such as radial basis kernels and linear margin classifiers to non-Euclidean geometries. Such adaptation is shown to be useful when the data space does not exhibit Euclidean geometry. In particular, we focus in our experiments on the space of text documents that is naturally associated with the Fisher information metric on corresponding multinomial models.

GEOMETRIC PARTIAL DIFFERENTIAL EQUATIONS AND IMAGE ANALYSIS

This book provides an introduction to the use of geometric partial differential equations in image processing and computer vision. It brings a number of new concepts into the field, providing a very fundamental and formal approach to image processing. State-of-the-art practical results in a large number of real problems are achieved with the techniques described. Applications covered include image segmentation, shape analysis, image enhancement, and tracking. The volume provides information for people investigating new solutions to image processing problems as well as for people searching for existent advanced solutions.

Mathematica in Action: Problem Solving Through Visualization and Computation

In this third edition of Mathematica in Action, award-winning author Stan Wagon guides beginner and veteran users alike through Mathematica's powerful tools for mathematical exploration. The transition to Mathematica 7 is made smooth with plenty of examples and case studies that utilize Mathematica's newest tools, such as dynamic manipulations and adaptive three-dimensional plotting. Mathematica in Action also emphasizes the breadth of Mathematica and the impressive results of combining techniques from different areas. This material enables the reader to use Mathematica to solve a variety of complex problems of mathematics. Case studies ranging from elementary to sophisticated are provided throughout. Whenever possible, the book shows how Mathematica can be used to discover new things. Striking examples include the design of a road on which a square wheel bike can ride, the design of a drill that can drill square holes, an illustration of the Banach-Tarski Paradox via hyperbolic geometry, new and surprising formulas for p, the discovery of shadow orbits for chaotic systems, and the use of powerful new capabilities for three-dimensional graphics. Visualization is emphasized throughout, with finely crafted graphics in each chapter. All Mathematica code is included on a CD, saving the reader hours of typing. Wagon is the author of nine books on mathematics, including A Course in Computational Number Theory, named one of the ten best math books of 2000 by the American Library Association. He has written extensively on the educational applications of Mathematica, including the books VisualDSolve: Visualizing Differential Equations with Mathematica, and Animating Calculus: Mathematica Notebooks for the Laboratory.

Moments and Moment Invariants in Pattern Recognition

Moments as projections of an image’s intensity onto a proper polynomial basis can be applied to many different aspects of image processing. These include invariant pattern recognition, image normalization, image registration, focus/ defocus measurement, and watermarking. This book presents a survey of both recent and traditional image analysis and pattern recognition methods, based on image moments, and offers new concepts of invariants to linear filtering and implicit invariants. In addition to the theory, attention is paid to efficient algorithms for moment computation in a discrete domain, and to computational aspects of orthogonal moments. The authors also illustrate the theory through practical examples, demonstrating moment invariants in real applications across computer vision, remote sensing and medical imaging. Key features: Presents a systematic review of the basic definitions and properties of moments covering geometric moments and complex moments. Considers invariants to traditional transforms – translation, rotation, scaling, and affine transform - from a new point of view, which offers new possibilities of designing optimal sets of invariants. Reviews and extends a recent field of invariants with respect to convolution/blurring. Introduces implicit moment invariants as a tool for recognizing elastically deformed objects. Compares various classes of orthogonal moments (Legendre, Zernike, Fourier-Mellin, Chebyshev, among others) and demonstrates their application to image reconstruction from moments. Offers comprehensive advice on the construction of various invariants illustrated with practical examples. Includes an accompanying website providing efficient numerical algorithms for moment computation and for constructing invariants of various kinds, with about 250 slides suitable for a graduate university course. Moments and Moment Invariants in Pattern Recognition is ideal for researchers and engineers involved in pattern recognition in medical imaging, remote sensing, robotics and computer vision. Post graduate students in image processing and pattern recognition will also find the book of interest.

Nonlinear operators in image restoration

We firstly present a variational approach such that during image restoration, edges detected in the original image are being preserved, and then we compare in a second part, the mathematical foundation of this method with respect to some of the well known methods recently proposed in the literature within the class of PDE based algorithms (anisotropic diffusion, mean curvature motion, min/max flow technique,...). The performance of our approach is carefully examined and compared to the classical methods. Experimental results on synthetic and real images will illustrate the capabilities of all the studied approaches.

Wiley.The.Foundations.Of.Signal.Integrity.Nov.2009.eBook-ELOHiM

The first book to focus on the electromagnetic basis of signal integrity The Foundations of Signal Integrity is the first of its kind—a reference that examines the physical foundation of system integrity based on electromagnetic theory derived from Maxwell's Equations. Drawing upon the cutting-edge research of Professor Paul Huray's team of industrial engineers and graduate students, it develops the physical theory of wave propagation using methods of solid state and high-energy physics, mathematics, chemistry, and electrical engineering before addressing its application to modern high-speed systems. Coverage includes: All the necessary electromagnetic theory needed for a complete understanding of signal integrity Techniques for obtaining analytic solutions to Maxwell's Equations for ideal materials and boundary conditions Plane electromagnetic waves Plane waves in compound media Transmission lines and waveguides Ideal models vs. real-world systems Complex permittivity of propagating media Surface roughness Advanced signal integrity Signal integrity simulations Problem sets for each chapter With its thorough coverage of this relatively new discipline, the book serves as an ideal textbook for senior undergraduate and junior graduate students, as well as a resource for practicing engineers in this burgeoning field. At the end of each section, it typically stimulates the reader with open-ended questions that might lead to future theses or dissertation research.

Machine Learning for Human Motion Analysis Theory and Practice

"With the ubiquitous presence of video data and its increasing importance in a wide range of real-world applications, it is becoming increasingly necessary to automatically analyze and interpret object motions from large quantities of footage. Machine Learning for Human Motion Analysis: Theory and Practice highlights the development of robust and effective vision-based motion understanding systems. This advanced publication addresses a broad audience including practicing professionals working with specific vision applications such as surveillance, sport event analysis, healthcare, video conferencing, and motion video indexing and retrieval."