Download or read book Shape Curve Analysis Using Curvature written by James Miller and published by . This book was released on 2009 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Statistical shape analysis is a field for which there is growing demand. One of the major drivers for this growth is the number of practical applications which can use statistical shape analysis to provide useful insight. An example of one of these practical applications is investigating and comparing facial shapes. An ever improving suite of digital imaging technology can capture data on the three-dimensional shape of facial features from standard images. A field for which this offers a large amount of potential analytical benefit is the reconstruction of the facial surface of children born with a cleft lip or a cleft lip and palate. This thesis will present two potential methods for analysing data on the facial shape of children who were born with a cleft lip and/or palate using data from two separate studies. One form of analysis will compare the facial shape of one year old children born with a cleft lip and/or palate with the facial shape of control children. The second form of analysis will look for relationships between facial shape and psychological score for ten year old children born with a cleft lip and/or palate. While many of the techniques in this thesis could be extended to different applications much of the work is carried out with the express intention of producing meaningful analysis of the cleft children studies. Shape data can be defined as the information remaining to describe the shape of an object after removing the effects of location, rotation and scale. There are numerous techniques in the literature to remove the effects of location, rotation and scale and thereby define and compare the shapes of objects. A method which does not require the removal of the effects of location and rotation is to define the shape according to the bending of important shape curves. This method can naturally provide a technique for investigating facial shape. When considering a child s face there are a number of curves which outline the important features of the face. Describing these feature curves gives a large amount of information on the shape of the face. This thesis looks to define the shape of children s faces using functions of bending, called curvature functions, of important feature curves. These curvature functions are not only of use to define an object, they are apt for use in the comparison of two or more objects. Methods to produce curvature functions which provide an accurate description of the bending of face curves will be introduced in this thesis. Furthermore, methods to compare the facial shape of groups of children will be discussed. These methods will be used to compare the facial shape of children with a cleft lip and/or palate with control children. There is much recent literature in the area of functional regression where a scalar response can be related to a functional predictor. A novel approach for relating shape to a scalar response using functional regression, with curvature functions as predictors, is discussed and illustrated by a study into the psychological state of ten year old children who were born with a cleft lip or a cleft lip and palate. The aim of this example is to investigate whether any relationship exists between the bending of facial features and the psychological score of the children, and where relationships exist to describe their nature. The thesis consists of four parts. Chapters 1 and 2 introduce the data and give some background to the statistical techniques. Specifically, Chapter 1 briefly introduces the idea of shape and how the shape of objects can be defined using curvature. Furthermore, the two studies into facial shape are introduced which form the basis of the work in this thesis. Chapter 2 gives a broad overview of some standard shape analysis techniques, including Procrustes methods for alignment of objects, and gives further details of methods based on curvature. Functional data analysis techniques which are of use throughout the thesis are also discussed. Part 2 consists of Chapters 3 to 5 which describe methods to find curvature functions that define the shape of important curves on the face and compare these functions to investigate differences between control children and children born with a cleft lip and/or palate. Chapter 3 considers the issues with finding and further analysing the curvature functions of a plane curve whilst Chapter 4 extends the methods to space curves. A method which projects a space curve onto two perpendicular planes and then uses the techniques of Chapter 3 to calculate curvature is introduced to facilitate anatomical interpretation. Whilst the midline profile of a control child is used to illustrate the methods in Chapters 3 and 4, Chapter 5 uses curvature functions to investigate differences between control children and children born with a cleft lip and/or palate in terms of the bending of their upper lips. Part 3 consists of Chapters 6 and 7 which introduce functional regression techniques and use these to investigate potential relationships between the psychological score and facial shape, defined by curvature functions, of cleft children. Methods to both display graphically and formally analyse the regression procedure are discussed in Chapter 6 whilst Chapter 7 uses these methods to provide a systematic analysis of any relationship between psychological score and facial shape. The final part of the thesis presents conclusions discussing both the effectiveness of the methods and some brief anatomical/psychological findings. There are also suggestions of potential future work in the area.

Download or read book Designing Fair Curves and Surfaces written by Nickolas S. Sapidis and published by SIAM. This book was released on 1994-01-01 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors define fairness mathematically, demonstrate how newly developed curve and surface schemes guarantee fairness, and assist the user in identifying and removing shape aberrations in a surface model without destroying the principal shape characteristics of the model. A valuable resource for engineers working in CAD, CAM, or computer-aided engineering.

Download or read book Shape Analysis and Structuring written by Leila de Floriani and published by Springer Science & Business Media. This book was released on 2007-12-24 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: With a lot of recent developments in the field, this much-needed book has come at just the right time. It covers a variety of topics related to preserving and enhancing shape information at a geometric level. The contributors also cover subjects that are relevant to effectively capturing the structure of a shape by identifying relevant shape components and their mutual relationships.

Download or read book Modern Differential Geometry of Curves and Surfaces with Mathematica Second Edition written by mary Gray and published by CRC Press. This book was released on 1997-12-29 with total page 1094 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Second Edition combines a traditional approach with the symbolic manipulation abilities of Mathematica to explain and develop the classical theory of curves and surfaces. You will learn to reproduce and study interesting curves and surfaces - many more than are included in typical texts - using computer methods. By plotting geometric objects and studying the printed result, teachers and students can understand concepts geometrically and see the effect of changes in parameters. Modern Differential Geometry of Curves and Surfaces with Mathematica explains how to define and compute standard geometric functions, for example the curvature of curves, and presents a dialect of Mathematica for constructing new curves and surfaces from old. The book also explores how to apply techniques from analysis. Although the book makes extensive use of Mathematica, readers without access to that program can perform the calculations in the text by hand. While single- and multi-variable calculus, some linear algebra, and a few concepts of point set topology are needed to understand the theory, no computer or Mathematica skills are required to understand the concepts presented in the text. In fact, it serves as an excellent introduction to Mathematica, and includes fully documented programs written for use with Mathematica. Ideal for both classroom use and self-study, Modern Differential Geometry of Curves and Surfaces with Mathematica has been tested extensively in the classroom and used in professional short courses throughout the world.

Download or read book Surfaces with Constant Mean Curvature written by Katsuei Kenmotsu and published by American Mathematical Soc.. This book was released on 2003 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: The mean curvature of a surface is an extrinsic parameter measuring how the surface is curved in the three-dimensional space. A surface whose mean curvature is zero at each point is a minimal surface, and it is known that such surfaces are models for soap film. There is a rich and well-known theory of minimal surfaces. A surface whose mean curvature is constant but nonzero is obtained when we try to minimize the area of a closed surface without changing the volume it encloses. An easy example of a surface of constant mean curvature is the sphere. A nontrivial example is provided by the constant curvature torus, whose discovery in 1984 gave a powerful incentive for studying such surfaces. Later, many examples of constant mean curvature surfaces were discovered using various methods of analysis, differential geometry, and differential equations. It is now becoming clear that there is a rich theory of surfaces of constant mean curvature. In this book, the author presents numerous examples of constant mean curvature surfaces and techniques for studying them. Many finely rendered figures illustrate the results and allow the reader to visualize and better understand these beautiful objects. The book is suitable for advanced undergraduates, graduate students and research mathematicians interested in analysis and differential geometry.

Download or read book The Shapes of Things written by Shawn W. Walker and published by SIAM. This book was released on 2015-06-25 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many things around us have properties that depend on their shape--for example, the drag characteristics of a rigid body in a flow. This self-contained overview of differential geometry explains how to differentiate a function (in the calculus sense) with respect to a "shape variable." This approach, which is useful for understanding mathematical models containing geometric partial differential equations (PDEs), allows readers to obtain formulas for geometric quantities (such as curvature) that are clearer than those usually offered in differential geometry texts. Readers will learn how to compute sensitivities with respect to geometry by developing basic calculus tools on surfaces and combining them with the calculus of variations. Several applications that utilize shape derivatives and many illustrations that help build intuition are included.

Download or read book Statistics and Analysis of Shapes written by Hamid Krim and published by Springer Science & Business Media. This book was released on 2007-12-31 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of pattern analysis and recognition pervades many aspects of our daily lives, including user authentication in banking, object retrieval from databases in the consumer sector, and the omnipresent surveillance and security measures around sensitive areas. Shape analysis, a fundamental building block in many approaches to these applications, is also used in statistics, biomedical applications (Magnetic Resonance Imaging), and many other related disciplines. With contributions from some of the leading experts and pioneers in the field, this self-contained, unified volume is the first comprehensive treatment of theory, methods, and algorithms available in a single resource. Developments are discussed from a rapidly increasing number of research papers in diverse fields, including the mathematical and physical sciences, engineering, and medicine.

Download or read book Curvature of Space and Time with an Introduction to Geometric Analysis written by Iva Stavrov and published by American Mathematical Soc.. This book was released on 2020-11-12 with total page 243 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces advanced undergraduates to Riemannian geometry and mathematical general relativity. The overall strategy of the book is to explain the concept of curvature via the Jacobi equation which, through discussion of tidal forces, further helps motivate the Einstein field equations. After addressing concepts in geometry such as metrics, covariant differentiation, tensor calculus and curvature, the book explains the mathematical framework for both special and general relativity. Relativistic concepts discussed include (initial value formulation of) the Einstein equations, stress-energy tensor, Schwarzschild space-time, ADM mass and geodesic incompleteness. The concluding chapters of the book introduce the reader to geometric analysis: original results of the author and her undergraduate student collaborators illustrate how methods of analysis and differential equations are used in addressing questions from geometry and relativity. The book is mostly self-contained and the reader is only expected to have a solid foundation in multivariable and vector calculus and linear algebra. The material in this book was first developed for the 2013 summer program in geometric analysis at the Park City Math Institute, and was recently modified and expanded to reflect the author's experience of teaching mathematical general relativity to advanced undergraduates at Lewis & Clark College.

Download or read book Geometric Curve Evolution and Image Processing written by Frédéric Cao and published by Springer Science & Business Media. This book was released on 2003-02-27 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: In image processing, "motions by curvature" provide an efficient way to smooth curves representing the boundaries of objects. In such a motion, each point of the curve moves, at any instant, with a normal velocity equal to a function of the curvature at this point. This book is a rigorous and self-contained exposition of the techniques of "motion by curvature". The approach is axiomatic and formulated in terms of geometric invariance with respect to the position of the observer. This is translated into mathematical terms, and the author develops the approach of Olver, Sapiro and Tannenbaum, which classifies all curve evolution equations. He then draws a complete parallel with another axiomatic approach using level-set methods: this leads to generalized curvature motions. Finally, novel, and very accurate, numerical schemes are proposed allowing one to compute the solution of highly degenerate evolution equations in a completely invariant way. The convergence of this scheme is also proved.

Download or read book Shape Classification and Analysis written by Luciano da Fona Costa and published by CRC Press. This book was released on 2018-10-03 with total page 693 pages. Available in PDF, EPUB and Kindle. Book excerpt: Because the properties of objects are largely determined by their geometric features, shape analysis and classification are essential to almost every applied scientific and technological area. A detailed understanding of the geometrical features of real-world entities (e.g., molecules, organs, materials and components) can provide important clues about their origin and function. When properly and carefully applied, shape analysis offers an exceedingly rich potential to yield useful applications in diverse areas ranging from material sciences to biology and neuroscience. Get Access to the Authors’ Own Cutting-Edge Open-Source Software Projects—and Then Actually Contribute to Them Yourself! The authors of Shape Analysis and Classification: Theory and Practice, Second Edition have improved the bestselling first edition by updating the tremendous progress in the field. This exceptionally accessible book presents the most advanced imaging techniques used for analyzing general biological shapes, such as those of cells, tissues, organs, and organisms. It implements numerous corrections and improvements—many of which were suggested by readers of the first edition—to optimize understanding and create what can truly be called an interactive learning experience. New Material in This Second Edition Addresses Graph and complex networks Dimensionality reduction Structural pattern recognition Shape representation using graphs Graphically reformulated, this edition updates equations, figures, and references, as well as slides that will be useful in related courses and general discussion. Like the popular first edition, this text is applicable to many fields and certain to become a favored addition to any library. Visit http://www.vision.ime.usp.br/~cesar/shape/ for Useful Software, Databases, and Videos

Download or read book The Design and Analysis of Computer Experiments written by Thomas J. Santner and published by Springer. This book was released on 2019-01-08 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes methods for designing and analyzing experiments that are conducted using a computer code, a computer experiment, and, when possible, a physical experiment. Computer experiments continue to increase in popularity as surrogates for and adjuncts to physical experiments. Since the publication of the first edition, there have been many methodological advances and software developments to implement these new methodologies. The computer experiments literature has emphasized the construction of algorithms for various data analysis tasks (design construction, prediction, sensitivity analysis, calibration among others), and the development of web-based repositories of designs for immediate application. While it is written at a level that is accessible to readers with Masters-level training in Statistics, the book is written in sufficient detail to be useful for practitioners and researchers. New to this revised and expanded edition: • An expanded presentation of basic material on computer experiments and Gaussian processes with additional simulations and examples • A new comparison of plug-in prediction methodologies for real-valued simulator output • An enlarged discussion of space-filling designs including Latin Hypercube designs (LHDs), near-orthogonal designs, and nonrectangular regions • A chapter length description of process-based designs for optimization, to improve good overall fit, quantile estimation, and Pareto optimization • A new chapter describing graphical and numerical sensitivity analysis tools • Substantial new material on calibration-based prediction and inference for calibration parameters • Lists of software that can be used to fit models discussed in the book to aid practitioners

Download or read book Curvature Scale Space Representation Theory Applications and MPEG 7 Standardization written by F. Mokhtarian and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 444 pages. Available in PDF, EPUB and Kindle. Book excerpt: MPEG-7 is the first international standard which contains a number of key techniques from Computer Vision and Image Processing. The Curvature Scale Space technique was selected as a contour shape descriptor for MPEG-7 after substantial and comprehensive testing, which demonstrated the superior performance of the CSS-based descriptor. Curvature Scale Space Representation: Theory, Applications, and MPEG-7 Standardization is based on key publications on the CSS technique, as well as its multiple applications and generalizations. The goal was to ensure that the reader will have access to the most fundamental results concerning the CSS method in one volume. These results have been categorized into a number of chapters to reflect their focus as well as content. The book also includes a chapter on the development of the CSS technique within MPEG standardization, including details of the MPEG-7 testing and evaluation processes which led to the selection of the CSS shape descriptor for the standard. The book can be used as a supplementary textbook by any university or institution offering courses in computer and information science.

Download or read book Design and Analysis of Experiments written by Douglas C. Montgomery and published by Wiley. This book was released on 2005 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This bestselling professional reference has helped over 100,000 engineers and scientists with the success of their experiments. The new edition includes more software examples taken from the three most dominant programs in the field: Minitab, JMP, and SAS. Additional material has also been added in several chapters, including new developments in robust design and factorial designs. New examples and exercises are also presented to illustrate the use of designed experiments in service and transactional organizations. Engineers will be able to apply this information to improve the quality and efficiency of working systems.

Download or read book Collapse Analysis for Shells of General Shape written by Bo O. Almroth and published by . This book was released on 1972 with total page 94 pages. Available in PDF, EPUB and Kindle. Book excerpt: F33615-69-C-1523, AF-1467146703, AFFDLTR-71-8-Vol-1(*shells(structural forms), failure(mechanics)), structural properties, numerical analysis, buckling, plastic properties, tensor analysis, computer programming newton- raphson method, collapse, plates(structural members), finite difference theory, stags computer program, structural analysis. The report presents a theory for nonlinear collapse analysis of shells with general shape. The theory combines energy principals and finite difference methods to obtain a system of nonlinear equations; these are solved by a modified Newton-Raphson technique. For greater economy and flexibility in the analysis a capability is provided for use of variable spacing finite difference grids. Inelastic material behavior, as predicted by the White-Besseling Theory, is incorporated into the analysis. A computer code, STAGS, based on the theory has been written and used to solve a number of sample problems. Results for these problems are presented.

Download or read book Shape Interrogation for Computer Aided Design and Manufacturing written by Nicholas M. Patrikalakis and published by Springer Science & Business Media. This book was released on 2002-02-14 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: Shape interrogation is the process of extraction of information from a geometric model. It is a fundamental component of Computer Aided Design and Manufacturing (CAD/CAM) systems. The authors focus on shape interrogation of geometric models bounded by free-form surfaces. Free-form surfaces, also called sculptured surfaces, are widely used in the bodies of ships, automobiles and aircraft, which have both functionality and attractive shape requirements. Many electronic devices as well as consumer products are designed with aesthetic shapes, which involve free-form surfaces. This book provides the mathematical fundamentals as well as algorithms for various shape interrogation methods including nonlinear polynomial solvers, intersection problems, differential geometry of intersection curves, distance functions, curve and surface interrogation, umbilics and lines of curvature, geodesics, and offset curves and surfaces. This book will be of interest both to graduate students and professionals.

Download or read book The Analytical Development of Curves and Streamline Shapes written by Harry Hermann Haase and published by . This book was released on 1948 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Elastic Shape Analysis of Curves and Surfaces written by Zhe Su and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The shape analysis of curves and surfaces has caught more and more attention in recent years. For curves in Euclidean space, the SRVF framework allows us to define and efficiently compute a distance on the shape space of curves. In this dissertation we give a generalization of the SRVF to curves with values in a homogeneous space and show that, under mild conditions, there always exist optimal reparametrizations realizing the quotient distance on the space of shapes. We give concrete examples to demonstrate the efficiency of our framework. Next, we introduce a diffeomorphism-invariant Riemannian metric on the space of vector valued one-forms. The metric is motivated by applications in the field of shape analysis and by connections to the Ebin metric on the space of all Riemannian metrics. We calculate the geodesic equations and obtain explicit solutions for the corresponding initial value problem. Using this, we study the geodesic and metric incompleteness of the space of one-forms and exhibit some totally geodesic subspaces. We also calculate its sectional curvature and observe that, depending on the dimensions of the base manifold and the ambient Euclidean space, it either has a semidefinite sign or admits both signs. After that, we give two data-driven frameworks for analyzing shapes of immersed surfaces in $\RR^3$. In the first framework, we introduce a 4-parameter family of elastic metrics on the space of surfaces as a pullback of a family of metrics on the space of vector-valued one-forms. This 4-parameter family of elastic metrics is invariant under rigid motions and reparametrizations. It generalizes a previously studied 3-parameter family of elastic metrics called the general elastic metric, and in particular the Square Root Normal Field (SRNF) metric, which has proved to be successful in various applications. In the second framework, we define a new representation for surfaces in $\RR^3$ by combining the induced surface metric and the SRNF of each surface. Using the DeWitt metric on the space of metrics and the $L^2$ metric on the space of SRNFs, we obtain a 3-parameter family of elastic metrics on surfaces, which forms an open subset of the general elastic metric. The new representation avoids the degeneracy of the SRNF representation, while it still results in an explicit extrinsic distance function on the space of surfaces and thus makes the framework computationally efficient. For both of these two frameworks we provide numerical algorithms and show several examples to validate the frameworks. The second framework, although it does not give a computation of geodesics, provides fast and accurate registration of surfaces. Finally we conclude with a summary of this dissertation and a discussion of possible future work.