This book publishes a collection of original scientific research articles that address the state-of-art in using partial differential equations for image and signal processing. Coverage includes: level set methods for image segmentation and construction, denoising techniques, digital image inpainting, image dejittering, image registration, and fast numerical algorithms for solving these problems.

Partial differential equations and variational methods were introduced into image processing about 15 years ago, and intensive research has been carried out since then. The main goal of this work is to present the variety of image analysis applications and the precise mathematics involved. It is intended for two audiences. The first is the mathematical community, to show the contribution of mathematics to this domain and to highlight some unresolved theoretical questions. The second is the computer vision community, to present a clear, self-contained, and global overview of the mathematics involved in image processing problems. The book is divided into five main parts. Chapter 1 is a detailed overview. Chapter 2 describes and illustrates most of the mathematical notions found throughout the work. Chapters 3 and 4 examine how PDEs and variational methods can be successfully applied in image restoration and segmentation processes. Chapter 5, which is more applied, describes some challenging computer vision problems, such as sequence analysis or classification. This book will be useful to researchers and graduate students in mathematics and computer vision.

This book provides an introduction to the use of geometric partial differential equations in image processing and computer vision. This research area 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 in this book. Applications covered include image segmentation, shape analysis, image enhancement, and tracking. This book will be a useful resource for researchers and practitioners. It is intended to provide information for people investigating new solutions to image processing problems as well as for people searching for existent advanced solutions.

In image processing and computer vision applications such as medical or scientific image data analysis, as well as in industrial scenarios, images are used as input measurement data. It is good scientific practice that proper measurements must be equipped with error and uncertainty estimates. For many applications, not only the measured values but also their errors and uncertainties, should be—and more and more frequently are—taken into account for further processing. This error and uncertainty propagation must be done for every processing step such that the final result comes with a reliable precision estimate. The goal of this book is to introduce the reader to the recent advances from the field of uncertainty quantification and error propagation for computer vision, image processing, and image analysis that are based on partial differential equations (PDEs). It presents a concept with which error propagation and sensitivity analysis can be formulated with a set of basic operations. The approach discussed in this book has the potential for application in all areas of quantitative computer vision, image processing, and image analysis. In particular, it might help medical imaging finally become a scientific discipline that is characterized by the classical paradigms of observation, measurement, and error awareness. This book is comprised of eight chapters. After an introduction to the goals of the book (Chapter 1), we present a brief review of PDEs and their numerical treatment (Chapter 2), PDE-based image processing (Chapter 3), and the numerics of stochastic PDEs (Chapter 4). We then proceed to define the concept of stochastic images (Chapter 5), describe how to accomplish image processing and computer vision with stochastic images (Chapter 6), and demonstrate the use of these principles for accomplishing sensitivity analysis (Chapter 7). Chapter 8 concludes the book and highlights new research topics for the future.

This book addresses the mathematical aspects of modern image processing methods, with a special emphasis on the underlying ideas and concepts. It discusses a range of modern mathematical methods used to accomplish basic imaging tasks such as denoising, deblurring, enhancing, edge detection and inpainting. In addition to elementary methods like point operations, linear and morphological methods, and methods based on multiscale representations, the book also covers more recent methods based on partial differential equations and variational methods. Review of the German Edition: The overwhelming impression of the book is that of a very professional presentation of an appropriately developed and motivated textbook for a course like an introduction to fundamentals and modern theory of mathematical image processing. Additionally, it belongs to the bookcase of any office where someone is doing research/application in image processing. It has the virtues of a good and handy reference manual. (zbMATH, reviewer: Carl H. Rohwer, Stellenbosch)

This book introduces the mathematical concept of partial differential equations (PDE) for virtual image restoration. It provides insight in mathematical modelling, partial differential equations, functional analysis, variational calculus, optimisation and numerical analysis. It is addressed towards generally informed mathematicians and graduate students in mathematics with an interest in image processing and mathematical analysis.

This dissertation, "Partial Differential Equation Based Methods in Medical Image Processing" by Kwok-wing, Anthony, Sum, 岑國榮, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled Partial Di(R)erential Equation Based Methods in Medical Image Processing Submitted by Anthony Kwok Wing SUM for the degree of Doctor of Philosophy at The University of Hong Kong in August 2007 Medical image analysis is essential for clinical diagnosis and surgical planning. To cope with the rapid development of modern imaging technologies, there is a continuingneedforadvancedimageprocessingtechniquestoimproveimagequality and automate the analytical processes. The two most important and fundamental image processing techniques required for fully utilizing and e(R)ectively interpreting the acquired images are image segmentation and image ltering. They play an indispensableroleintheentiremedicalimageanalysisprocess. Inthisthesis, image segmentation and ltering methods using partial di(R)erential equation (PDE) are studied and explored. iiIn daily clinical practice, physicians are required to identify anatomical struc- tures from a large number of medical images. This identication process can be aidedbyimagesegmentationtechniques. Inthisthesis, newdevelopmentsinactive contour models are introduced for image segmentation. First, parametric active contoursaredesirabletoextractobjectswithaconnedboundary. Arobustpara- metric active contour model with a novel external force, namely boundary vector eld (BVF), is proposed. This new model is shown to be more ecient than other existing parametric active contour models in terms of ease of initialization, extrac- tion capability and speed. Second, geometric active contour models are found to be well suited for extracting topologically complex objects such as vessels in an- giograms. However, angiograms and other medical images commonly su(R)er from a nonuniform illumination artifact. This artifact induces serious problem in object extraction during image segmentation. Thus, a novel segmentation scheme is pro- posed based on level set methods and incorporating local contrast information in the formulation. This scheme improves the extraction outcomes even if the image su(R)ers from nonuniform illuminations artifacts. Di(R)erent imaging modalities and imaging environments may generate di(R)erent levels of noise during the data acquisition phase. Image ltering is therefore an essential technique for reducing the noise level and improving the visual quality of an image. Anisotropic di(R)usion is a PDE based ltering method, which has found useful practical applications since its introduction. The kernel of an anisotropic iiidi(R)usionmodelisthedi(R)usioncoecient, whichcharacterizestheoverallbehavior of the entire model. In this study, a new class of anisotropic di(R)usion model is formulated and its outstanding performance is demonstrated with experimental results. Itisshownthatbothsignal-to-noise ratio andvisualqualityofthe ltered images using the new di(R)usion model are improved. In summary, several creative and innovative developments of low level image processing techniques are reported in the thesis. These low level techniques are a critical requirement for advanced high level image analysis procedures, and are indispensable for the automation of many medical image analysis tasks. An abstract of exactly 434 words iv DOI: 10.5353/th_b3895862 Subjects: Differential equations, Partial Diagnostic imaging - Mathematical models Image processing - Mathematics