Per Christian Hansen – författare

Here is an overview of modern computational stabilization methods for linear inversion, with applications to a variety of problems in audio processing, medical imaging, tomography, seismology, astronomy, and other areas.Rank-deficient problems involve matrices that are either exactly or nearly rank deficient. Such problems often arise in connection with noise suppression and other problems where the goal is to suppress unwanted disturbances of the given measurements.Discrete ill-posed problems arise in connection with the numerical treatment of inverse problems, where one typically wants to compute information about some interior properties using exterior measurements. Examples of inverse problems are image restoration and tomography, where one needs to improve blurred images or reconstruct pictures from raw data.This book describes, in a common framework, new and existing numerical methods for the analysis and solution of rank-deficient and discrete ill-posed problems. The emphasis is on insight into the stabilizing properties of the algorithms and on the efficiency and reliability of the computations. The setting is that of numerical linear algebra rather than abstract functional analysis, and the theoretical development is complemented with numerical examples and figures that illustrate the features of the various algorithms.

When we use a camera, we want the recorded image to be a faithful representation of the scene that we see, but every image is more or less blurry. In image deblurring, the goal is to recover the original, sharp image by using a mathematical model of the blurring process. The key issue is that some information on the lost details is indeed present in the blurred image, but this “hidden” information can be recovered only if we know the details of the blurring process. Deblurring Images describes the deblurring algorithms and techniques collectively known as spectral filtering methods, in which the singular value decomposition - or a similar decomposition with spectral properties - is used to introduce the necessary regularization or filtering in the reconstructed image. The concise MATLAB® implementations described in the book provide a template of techniques that can be used to restore blurred images from many applications.This book’s treatment of image deblurring is unique in two ways: it includes algorithmic and implementation details; and by keeping the formulations in terms of matrices, vectors, and matrix computations, it makes the material accessible to a wide range of readers. Students and researchers in engineering will gain an understanding of the linear algebra behind filtering methods, while readers in applied mathematics, numerical analysis, and computational science will be exposed to modern techniques to solve realistic large-scale problems in image processing.With a focus on practical and efficient algorithms, Deblurring Images includes many examples, sample image data, and MATLAB codes that allow readers to experiment with the algorithms. It also incorporates introductory material, such as how to manipulate images within the MATLAB environment, making it a stand-alone text. Pointers to the literature are given for techniques not covered in the book.

Inverse problems arise when we reconstruct a sharper image from a blurred one or reconstruct the underground mass density from measurements of the gravity above the ground. When we solve an inverse problem, we compute the source that gives rise to some observed data using a mathematical model for the relation between the source and the data.This book gives an introduction to the practical treatment of inverse problems by means of numerical methods, with a focus on basic mathematical and computational aspects. To solve inverse problems, we demonstrate that insight about them goes hand in hand with algorithms.Discrete Inverse Problems includes a number of tutorial exercises that give the reader hands-on experience with the methods, difficulties, and challenges associated with the treatment of inverse problems. It also includes examples and figures that illustrate the theory and algorithms.

As one of the classical statistical regression techniques, and often the first to be taught to new students, least squares fitting can be a very effective tool in data analysis. Given measured data, we establish a relationship between independent and dependent variables so that we can use the data predictively. The main concern of "Least Squares Data Fitting with Applications" is how to do this on a computer with efficient and robust computational methods for linear and nonlinear relationships. The presentation also establishes a link between the statistical setting and the computational issues. In a number of applications, the accuracy and efficiency of the least squares fit is central, and Per Christian Hansen, Victor Pereyra, and Godela Scherer survey modern computational methods and illustrate them in fields ranging from engineering and environmental sciences to geophysics. Anyone working with problems of linear and nonlinear least squares fitting will find this book invaluable as a hands-on guide, with accessible text and carefully explained problems.Included are: an overview of computational methods together with their properties and advantages; topics from statistical regression analysis that help readers to understand and evaluate the computed solutions; and many examples that illustrate the techniques and algorithms. "Least Squares Data Fitting with Applications" can be used as a textbook for advanced undergraduate or graduate courses and professionals in the sciences and in engineering.