The Method of Approximate Inverse: Theory and Applications

ISBN-13: 9783540712268 / Angielski / Miękka / 2007 / 202 str.

Thomas Schuster

Opis

This book is concerned with the method of approximate inverse which is a regularization technique for stably solving inverse problems in various settings. It demonstrates the performance and functionality of the method on several examples from medical imaging and non-destructive testing, such as computerized tomography, Doppler tomography, SONAR, X-ray diffractometry and thermoacoustic computerized tomography.

Recenzje

From the reviews:

"The powerful method of the approximate inverse is a good bunch of regularization techniques, and this monograph presents a comprehensive outline of this method. Application to 3D Doppler tomography and the spherical mean operator is then studied in details, and further results on X-ray diffractometry, thermoacoustic computerized tomography and reconstruction kernels in 3D are attached. The book is naturally recommended for computer tomographers and graduate students heading toward computer tomography, but it contains many beneficial results for researchers of Radon transforms too." (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

"The book under review which deals with a particular class of regularization methods, the so called method of approximate inverse, is the result of continuous study of the author for more than a decade, by himself for his habilitation thesis and also in collaborations with many experts in the field, including A. K. Louis (his own teacher), A. Rieder and many others. ... No doubt, the book is a good addition to the literature on regularization of ill-posed inverse problems." (M. Thamban Nair, Zentralblatt MATH, Vol. 1171, 2009)

Spis treści

Inverse and Semi-discrete Problems.- Ill-posed problems and regularization methods.- Approximate inverse in L 2-spaces.- Approximate inverse in Hilbert spaces.- Approximate inverse in distribution spaces.- Conclusion and perspectives.- Application to 3D Doppler Tomography.- A semi-discrete setup for Doppler tomography.- Solving the semi-discrete problem.- Convergence and stability.- Approaches for defect correction.- Conclusion and perspectives.- Application to the spherical mean operator.- The spherical mean operator.- Design of a mollifier.- Computation of reconstruction kernels.- Numerical experiments.- Conclusion and perspectives.- Further Applications.- Approximate inverse and X-ray diffractometry.- A filtered backprojection algorithm.- Computation of reconstruction kernels in 3D computerized tomography.- Conclusion and perspectives.

O autorze

1990 – 1995 Study of Mathematics at Saarland University Saarbrücken (Germany)

1996 – 2004 Scientific assistant at Saarland University Saarbrücken (Germany)

1999 PhD at Saarland University Saarbrücken (Germany)

2002 – 2003 Research stay at Tufts University Medford, MA (USA)

2004 Habilitation at Saarland University Saarbrücken (Germany)

2004 – 2006 Assistant Professor at Saarland University Saarbrücken (Germany)

2007 – today Associate Professor at the Helmut Schmidt University Hamburg (Germany)

Blurb

Inverse problems arise whenever one tries to calculate a required quantity from given measurements of a second quantity that is associated to the first one. Besides medical imaging and non-destructive testing, inverse problems also play an increasing role in other disciplines such as industrial and financial mathematics. Hence, there is a need for stable and efficient solvers. The book is concerned with the method of approximate inverse which is a regularization technique for stably solving inverse problems in various settings such as L²-spaces, Hilbert spaces or spaces of distributions.  The performance and functionality of the method is demonstrated on several examples from medical imaging and non-destructive testing such as computerized tomography, Doppler tomography, SONAR, X-ray diffractometry and thermoacoustic computerized tomography. The book addresses graduate students and researchers interested in the numerical analysis of inverse problems and regularization techniques or in efficient solvers for the applications mentioned above.