[Z] Introduction to the Level Set Method

From: http://www.cnblogs.com/tabatabaye/articles/891232.html

Level Set Method Introduction:

The Level Set method was proposed by sethian and Osher in 1988 and has been widely used in the past decade. Simply put, the level set method increases some low-dimensional computing to a higher one, and regards the n-dimensional description as a level of N + 1. For example, a circle in a two-dimensional plane, such as x ^ 2 + y ^ 2 = 1, can be regarded as a binary function f (x, y) = x ^ 2 + y ^ 2 1 level. Therefore, when calculating the variation of this circle, we can first calculate the variation of f (x, y), and then obtain its 1 level set. The advantage of doing so is that, first, the topology changes in low dimensions are no longer a problem in high dimensions; second, low dimensions need to be re-parameterized from time to time, which is not required in high dimensions; third, the high-dimensional computation is more accurate and more robust. Fourth, the level set method can be easily promoted to the higher-dimensional computation. Finally, it is very important that, after rising to the high-dimensional space, many mature algorithms can be used directly, and there are very mature analysis tools in this regard, such as the theory and numerical of partial differential equations. Of course, the most criticized method is that it increases the amount of computing, but the emergence of new fast algorithms makes this not a big problem.

 

Applicability of Level Set:

This is just to list some classic fields, but not all of them. If you can find new applications in your own field, congratulations. The initial application field of level set is the motion of the hidden curve (surface, now level set has been widely used in image restoration, image enhancement, image segmentation, object tracking, Shape Detection and Recognition, surface reconstruction, minimum surface, optimization, and fluid mechanics.

 

Level Set knowledge:

To learn and apply level set, you need to master the theory of partial differential equations and their numerical methods. In addition, you should focus on the conversation law in partial differential equations, the theory of viscosity solution (viscous solution) and Hamilton-jakalequation (Hamilton-jakyton equation) and its numerical method. At the same time, when learning level set, we often encounter some content of the variational method and the measurement theory, but the requirements for these two aspects are not high. Just take a look.

 

Recommended books for Level Set:

(1) Stanley Osher and Ronald fedkiw. level Set methods and dynamic implicit surfaces. springer-Verlag (2002 ). comment: This book was written by Osher, one of the founders. This book is one of the most complete books on Level Set, focusing more on the high-precision solution of numerical, the application field involves image processing and computing physics.
(2) James. sethian. level Set methods and fast marching methods. cambridge University Press (1999 ). comments: this is the work of sethian, another founder. It focuses on and complements Osher's books. This book focuses more on fast marching methods and unstructured grids, and covers a wider range of application fields.
(3) Guillermo sapiro, geometric partial differential equations and image analysis, Cambridge University. comment: This book is also very helpful for understanding level set. It focuses more on geometric features in the image, such as curvature, and gives a detailed introduction to the geometric partial differential equations.
(4) Gilles Aubert and Pierre kornprobst,
Mathematical Problems in image processing: partial differential equations and the calculus of variation, Springer, Applied Mathematical Sciences, vol 147,200 2. This book has a strong mathematical taste, and most people are not interested in reading it. But if you really want to lay a better theoretical foundation for your method, this book will be very useful, it can guide you in what aspects. In addition, the preface and the first chapter of this book are very good and worth reading.
General evaluation: (1) and (2) are the manuals for learning level set standing case head. If you want to dive deeper, (3) and (4) should also take a look.

 

Level Set recommended articles

(1) Osher, S ., and sethian, J. A ., fronts propagating with curvature-dependent speed: Algorithms Based on Hamilton-jakformulations, Journal of Computational Physics, 79, pp. 12-49,198 8. comments: this is an article about level set. It must be read.
Http://math.berkeley.edu /~ Sethian/publications/publications.html
You can download the document here, but the document downloaded here only contains text without a graph. To view the original version, copy it in the library.

(2) Osher, S. and fedkiw, R ., "Level Set Methods: an overview and some recent results", J. comput. phys. 169,463-502 (2001 ). comments: this is an early review, with the ucia cam report 00-08.
Http://www.math.ucla.edu/%7eimagers/htmls/reports.html can be downloaded.

(3) Richard Tsai and Stanly Osher, Level Set Methods and Their Applications in Image Science, Comm. math. sci. vol. 1, No. 4, pp. 623-656 comments: the summary is richer and the results are newer. Intlpress.com/cms/issue4/levelset_imaging_chapter.pdf can be downloaded.
General comment: there are too many articles about Level Set that cannot be listed one by one. It is strongly recommended that you visit the following website, where there are the latest articles. Http://www.math.ucla.edu /~ Imagers/reports.htm

 

Level Set recommended websites:

(1) http://math.berkeley.edu /~ Sethian/level_set.html
Comments: This is the sethian website, which has many discussions about level set, which are classified and clear.
(2) http://www.math.ucla.edu /~ Imagers/
Comments: This is the research group of ucia, founded by Osher. The new progress in Level Set is almost related to them. This website is the best place to focus on the latest news of Level Set.

Level Set Toolkit:
Http://www.cs.ubc.ca /~ Mitchell/toolboxls/index.html comments: this is a tool kit developed by Michell. It has good versatility, but it is very troublesome to modify it by yourself. We recommend that you re-write these functions by yourself. You can use this toolkit to verify whether your write is correct.

More level set books please: http://shop34470562.taobao.com/

The Level Set method was proposed by sethian and Osher in 1988 and has been widely used in the past decade. Simply put, the level set method increases some low-dimensional computing to a higher one, and regards the n-dimensional description as a level of N + 1. For example, a circle in a two-dimensional plane, such as x ^ 2 + y ^ 2 = 1, can be regarded as a binary function f (x, y) = x ^ 2 + y ^ 2 1 level. Therefore, when calculating the variation of this circle, we can first calculate the variation of f (x, y), and then obtain its 1 level set. The advantage of doing so is that, first, the topology changes in low dimensions are no longer a problem in high dimensions; second, low dimensions need to be re-parameterized from time to time, which is not required in high dimensions; third, the high-dimensional computation is more accurate and more robust. Fourth, the level set method can be easily promoted to the higher-dimensional computation. Finally, it is very important that, after rising to the high-dimensional space, many mature algorithms can be used directly, and there are very mature analysis tools in this regard, such as the theory and numerical of partial differential equations. Of course, the most criticized method is that it increases the amount of computing, but the emergence of new fast algorithms makes this not a big problem.

 

Applicability of Level Set:

This is just to list some classic fields, but not all of them. If you can find new applications in your own field, congratulations. The initial application field of level set is the motion of the hidden curve (surface, now level set has been widely used in image restoration, image enhancement, image segmentation, object tracking, Shape Detection and Recognition, surface reconstruction, minimum surface, optimization, and fluid mechanics.

 

Level Set knowledge:

To learn and apply level set, you need to master the theory of partial differential equations and their numerical methods. In addition, you should focus on the conversation law in partial differential equations, the theory of viscosity solution (viscous solution) and Hamilton-jakalequation (Hamilton-jakyton equation) and its numerical method. At the same time, when learning level set, we often encounter some content of the variational method and the measurement theory, but the requirements for these two aspects are not high. Just take a look.

 

Recommended books for Level Set:

(1) Stanley Osher and Ronald fedkiw. level Set methods and dynamic implicit surfaces. springer-Verlag (2002 ). comment: This book was written by Osher, one of the founders. This book is one of the most complete books on Level Set, focusing more on the high-precision solution of numerical, the application field involves image processing and computing physics.
(2) James. sethian. level Set methods and fast marching methods. cambridge University Press (1999 ). comments: this is the work of sethian, another founder. It focuses on and complements Osher's books. This book focuses more on fast marching methods and unstructured grids, and covers a wider range of application fields.
(3) Guillermo sapiro, geometric partial differential equations and image analysis, Cambridge University. comment: This book is also very helpful for understanding level set. It focuses more on geometric features in the image, such as curvature, and gives a detailed introduction to the geometric partial differential equations.
(4) Gilles Aubert and Pierre kornprobst,
Mathematical Problems in image processing: partial differential equations and the calculus of variation, Springer, Applied Mathematical Sciences, vol 147,200 2. This book has a strong mathematical taste, and most people are not interested in reading it. But if you really want to lay a better theoretical foundation for your method, this book will be very useful, it can guide you in what aspects. In addition, the preface and the first chapter of this book are very good and worth reading.
General evaluation: (1) and (2) are the manuals for learning level set standing case head. If you want to dive deeper, (3) and (4) should also take a look.

 

Level Set recommended articles

(1) Osher, S ., and sethian, J. A ., fronts propagating with curvature-dependent speed: Algorithms Based on Hamilton-jakformulations, Journal of Computational Physics, 79, pp. 12-49,198 8. comments: this is an article about level set. It must be read.
Http://math.berkeley.edu /~ Sethian/publications/publications.html
You can download the document here, but the document downloaded here only contains text without a graph. To view the original version, copy it in the library.

(2) Osher, S. and fedkiw, R ., "Level Set Methods: an overview and some recent results", J. comput. phys. 169,463-502 (2001 ). comments: this is an early review, with the ucia cam report 00-08.
Http://www.math.ucla.edu/%7eimagers/htmls/reports.html can be downloaded.

(3) Richard Tsai and Stanly Osher, Level Set Methods and Their Applications in Image Science, Comm. math. sci. vol. 1, No. 4, pp. 623-656 comments: the summary is richer and the results are newer. Intlpress.com/cms/issue4/levelset_imaging_chapter.pdf can be downloaded.
General comment: there are too many articles about Level Set that cannot be listed one by one. It is strongly recommended that you visit the following website, where there are the latest articles. Http://www.math.ucla.edu /~ Imagers/reports.htm

 

Level Set recommended websites:

(1) http://math.berkeley.edu /~ Sethian/level_set.html
Comments: This is the sethian website, which has many discussions about level set, which are classified and clear.
(2) http://www.math.ucla.edu /~ Imagers/
Comments: This is the research group of ucia, founded by Osher. The new progress in Level Set is almost related to them. This website is the best place to focus on the latest news of Level Set.

Level Set Toolkit:
Http://www.cs.ubc.ca /~ Mitchell/toolboxls/index.html comments: this is a tool kit developed by Michell. It has good versatility, but it is very troublesome to modify it by yourself. We recommend that you re-write these functions by yourself. You can use this toolkit to verify whether your write is correct.

More level set books please: http://shop34470562.taobao.com/