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Издательство Springer, 2008, -249 pp.The rapid development of spline theory in the last five decades—and its widespread applications in many diverse areas—has not only made the subject rich and diverse, but also made it immensely popular within different research communities. It is well established that splines are a powerful tool and have tremendous problem-solving capability. Of the large number of splines discovered so far, a few have established permanent homes in computer graphics, image processing, and machine vision. In computer graphics, their significant role is well documented. Unfortunately, this is not really the case in machine vision, even though a great deal of spline-based research has already been done in this area. The situation is somewhat better for image processing. One, therefore, feels the need for something in the form of a report or book that clearly spells out the importance of spline functions while teaching a course on machine vision. It is unfortunate that despite considerable searching, not even a single book in this area was found in the market. This singular fact provides the motivation for writing this book on splines, with special attention to applications in image processing and machine vision.

The philosophy behind writing this book lies in the fact that splines are effective, efficient, easy to implement, and have a strong and elegant mathematical background as well. Its problem-solving capability is, therefore, unquestionable. The remarkable spline era in computer science started when P. E. Bґezier first published his work on UNISURF. The subject immediately caught the attention of many researchers. The same situation was repeated with the discovery of Ingrid Daubechi’s wavelets. Different wavelet splines are now well known and extensively found in the literature. As splines are rich in properties, they provide advantages in designing new algorithms and hence they have wide-scale applications in many important areas. Bґezier and wavelet splines, can, therefore, be regarded as two different landmarks in spline theory with wide application in image processing and machine vision, and this justifies the title of the book.

In writing this book, therefore, we introduce the Bernstein polynomial at the very beginning, since its importance and dominance in Bґezier spline models for curve and surface design and drawing are difficult to ignore. We omitted the design problems of curves and surfaces because they are dealt with in almost all books on computer graphics. Some applications in different image processing areas, based on the Bґezier-Bernstein model, are discussed in depth in Chapters 1, 2, 3, and 4, so that researchers and students can get a fairly good idea about them and can apply them independently. Chapter 1 provides a background for Bґezier-Bernstein (B-B) polynomial and how binary images can be viewed, approximated, and regenerated through Bґezier-Bernstein arcs. Chapter 2 explains the underlying concept of graylevel image segmentation and provides some implementation details, which can be successfully used for image compression. In Chapter 3 of this book, we will show how one can use one dimensional B-B function to segment as well as compress image data points. Chapter 4 depicts image compression in a different way, using two dimensional B-B function.

B-splines, discussed in Chapter 5, are useful to researchers and students of many different streams including computer science and information technology, physics, and mathematics. We tried to provide a reasonably comprehensive coverage. Attention has been devoted to writing this chapter so that students can independently design algorithms that are sometimes needed for their class work, projects, and research. We have also included applications of B-splines in machine vision because we believe it also has strong potential in research. The beta splines discussed in Chapter 6 are relatively new and much work remains to be done in this area. However, we tried to discuss them as much as possible and indicated possible directions of further work. In Chapter 7, discrete splines are discussed, along with the feasibility of their use in machine vision. The application is appropriate and informative. It shows how the problem of recovering surface orientations can be solved through a system of nonlinear equations. Splines in vision is an open area and much attention needs to be paid for further research work. Wavelet splines are relatively new, so we took special care to write the theory in a clear, straightforward way in Chapter 8 . To aid in understanding, we used examples whenever necessary.

Snakes and active contours are explained in Chapter 9, and we discuss their intimate relationship with mathematical splines. Minimizing snake energy using both the original calculus of variations method and the dynamic programming approach are discussed. This chapter also includes problems and pitfalls drawn from several applications to provide a better understanding of the subject. Chapter 10, on the other hand, discusses powerful globally optimal energy minimization techniques, keeping in mind the need of students and researchers in this new and promising area of image processing and machine vision.Early Background

Bernstein Polynomial and Brezier-Bernstein Spline

Image Segmentation

1d B-B Spline Polynomial and Hilbert Scan for Graylevel Image Coding

Image Compression

Intermediate Steps

B-Splines and Its Applications

Beta-Splines: A Flexible Model

Advanced Methodologies

Discrete Splines and Vision

Spline Wavelets: Construction, Implication, and Uses

Snakes and Active Contours

Globally Optimal Energy Minimization Techniques

The philosophy behind writing this book lies in the fact that splines are effective, efficient, easy to implement, and have a strong and elegant mathematical background as well. Its problem-solving capability is, therefore, unquestionable. The remarkable spline era in computer science started when P. E. Bґezier first published his work on UNISURF. The subject immediately caught the attention of many researchers. The same situation was repeated with the discovery of Ingrid Daubechi’s wavelets. Different wavelet splines are now well known and extensively found in the literature. As splines are rich in properties, they provide advantages in designing new algorithms and hence they have wide-scale applications in many important areas. Bґezier and wavelet splines, can, therefore, be regarded as two different landmarks in spline theory with wide application in image processing and machine vision, and this justifies the title of the book.

In writing this book, therefore, we introduce the Bernstein polynomial at the very beginning, since its importance and dominance in Bґezier spline models for curve and surface design and drawing are difficult to ignore. We omitted the design problems of curves and surfaces because they are dealt with in almost all books on computer graphics. Some applications in different image processing areas, based on the Bґezier-Bernstein model, are discussed in depth in Chapters 1, 2, 3, and 4, so that researchers and students can get a fairly good idea about them and can apply them independently. Chapter 1 provides a background for Bґezier-Bernstein (B-B) polynomial and how binary images can be viewed, approximated, and regenerated through Bґezier-Bernstein arcs. Chapter 2 explains the underlying concept of graylevel image segmentation and provides some implementation details, which can be successfully used for image compression. In Chapter 3 of this book, we will show how one can use one dimensional B-B function to segment as well as compress image data points. Chapter 4 depicts image compression in a different way, using two dimensional B-B function.

B-splines, discussed in Chapter 5, are useful to researchers and students of many different streams including computer science and information technology, physics, and mathematics. We tried to provide a reasonably comprehensive coverage. Attention has been devoted to writing this chapter so that students can independently design algorithms that are sometimes needed for their class work, projects, and research. We have also included applications of B-splines in machine vision because we believe it also has strong potential in research. The beta splines discussed in Chapter 6 are relatively new and much work remains to be done in this area. However, we tried to discuss them as much as possible and indicated possible directions of further work. In Chapter 7, discrete splines are discussed, along with the feasibility of their use in machine vision. The application is appropriate and informative. It shows how the problem of recovering surface orientations can be solved through a system of nonlinear equations. Splines in vision is an open area and much attention needs to be paid for further research work. Wavelet splines are relatively new, so we took special care to write the theory in a clear, straightforward way in Chapter 8 . To aid in understanding, we used examples whenever necessary.

Snakes and active contours are explained in Chapter 9, and we discuss their intimate relationship with mathematical splines. Minimizing snake energy using both the original calculus of variations method and the dynamic programming approach are discussed. This chapter also includes problems and pitfalls drawn from several applications to provide a better understanding of the subject. Chapter 10, on the other hand, discusses powerful globally optimal energy minimization techniques, keeping in mind the need of students and researchers in this new and promising area of image processing and machine vision.Early Background

Bernstein Polynomial and Brezier-Bernstein Spline

Image Segmentation

1d B-B Spline Polynomial and Hilbert Scan for Graylevel Image Coding

Image Compression

Intermediate Steps

B-Splines and Its Applications

Beta-Splines: A Flexible Model

Advanced Methodologies

Discrete Splines and Vision

Spline Wavelets: Construction, Implication, and Uses

Snakes and Active Contours

Globally Optimal Energy Minimization Techniques

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