Digital Functions and Data Reconstruction

Transcription

1 Digital Functions and Data Reconstruction

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3 Li M. Chen Digital Functions and Data Reconstruction Digital-Discrete Methods 123

4 Li M. Chen University of the District of Columbia Washington, DC, USA ISBN ISBN (ebook) DOI / Springer New York Heidelberg Dordrecht London Library of Congress Control Number: c Springer Science+Business Media, LLC 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 To my mentors and colleagues who encouraged me to continue the research in this specific area.

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7 Preface Before Newton-Leibnitzs time, mathematics was basically discrete. Since then, continuous mathematics has dominated the literature. But discrete mathematics has found new life with the appearance and widespread use of the digital computer. However, we still perfer to use the thinking involved in continuous mathematics. For example, if we had discrete information on some samples, we would assume a continuous model to do the calculation. Sometimes, we need a discrete output from the continuous solution, and it is not hard to re-digitize the continuous results. For some problems, going from discrete to continuous back to discrete may not always be necessary. In such instances, we can directly employ a methodology to go from discrete input to discrete output. The tractability and practice of the methodology using such a philosophy is certainly valid. Lets consider an example. In seismic data processing, the seismic data sets consist of synchronous records of reflected seismic signals registered by a large number of geophones (seismic sensors) placed along a straight line or in the nodes of a rectangular lattice on the earths surface. A series of explosions serve as the source of the initial seismic pulse, responses to which are averaged in a special manner. The vertical time axis forming the resulting two- or three-dimensional picture is identified with depth, so that the peculiarities of the reflected signal under the respective sensor carries information on the local properties of the rock mass at the respective point of the underground medium. In contrast to the above-lying sedimentary cover, the absence of pronounced reflecting surfaces in a crystalline body makes it difficult to infer the geological information from the basement interval of the seismic picture. We can see that layer description (or modeling) becomes a central problem. If we know a target layer in each horizontal or vertical (line) profile, we can get the entire layer in the 3D stratum. It can be transferred into a surface fitting problem where we can use a Coones surface, Bezier polynomial, or B-spline to fit the surfaces. Based on the boundary values to fill the interior, the most suitable technique is a Coones surface. However, for a layer, one must make two surfaces, one for the top of the layer and one for the bottom. The Coons surfaces have no property of preserving a fitted surface in the convex of a guiding point set. That is to say, the upper surface may intersect with the lower surface. That is not a desired solution. Since there vii

8 viii Preface are many sampled points on measured lines, the Bezier polynomial is also not a good choice. One cannot make the order of the polynomial very high. B-spline is a very good choice for the problem, but we need to do a pre-partition and coordinate transformations. In fact, for the problem, we have no special requirements for the smoothness, and we just need two reasonable surfaces to cover the layer. Another example is from computer vision. In observing an image, if you extract an object from the image, a representation of the object can sometimes be described by its boundary curve. If all values on the boundary are the same, then we can just fill the region. If the values on the boundary are not the same, and if we assume that the values are continuous on the boundary, then one needs a fitting algorithm to find a surface. How do we fill it? Its solution will directly relates to a famous mathematical problem called the Dirchet s Problem and have direct application to in data compression. If the boundary is irregular, the 2D B-spline needs to partition the boundary into four segments to form a XY-plane vs UV-plane translation. The different partitions may yield different results. Practically, the procedure of a computation is a set of discrete actions. The input of a curve is also discrete, and the output is discrete. We can, therefore, make the following arguments. Do we always need a continuous technique for surface fitting? Is it possible to have a discrete fitting algorithm to get a reasonable surface for the above problems? In 1989, L. Chen developed an algorithm to do such discrete surface fitting in 2D. The algorithm is called gradually varied fitting. Gradually varied fitting was based on so called gradually varied functions that is a type of digital functions in general sense. In 1986, A. Rosenfeld invented a basic type of the digital continuous function for the purpose of image segmentation where one is to find a continuous-looking part in a digital image, a digital space. This is book is written to different interest groups of readers. Chapters 1 3 are foundations for the entire book; Chaps. 4 and 5 are for senior students, graduate students, or researchers who are interested in digital geometry and topology. Chapter 6 is a knowledge foundation for data reconstruction. Chapters 7 and 8 is for senior students in scientific computing. Chapter 9 will not be difficult for graduate students in computer science or senior students in mathematics with computer graphics background. Chapter 10 is for senior students in mathematics. Chapters 11 and 12 deal with future topics. For the Chapters marked with * may need some advanced knowledge. Chapters 1 3, 6 and 7 are basic chapters. Chapters 4 and 5 are in discrete mathematics especially discrete geometry and topology. Chapters 8 12 are application related topics in scientific computing. Acknowledgments Many Thanks to my friends Liang Chen and Xiaoxi Tang, they helped and observed the publishing of my very first article related. That time was a difficult time to me in changing my work from a mathematical department back to computer science department. Many thanks to my daughter Boxi Cera Chen who helped me to check English for the whole book. After her college, her professional editing tone has changed

9 Preface ix a bit to literal. That gives me some hard time to change back to simple sentences since most of our math and cs readers love simple sentences! We might already have a hard time to understand tough mathematics in this book. Nevertheless, she is the best I can found; she has both music and statistics degrees. Many thanks to my wife, she has always supported my research, especially in science and math. Ten years also I was finishing my first English book Discrete Surfaces and Manifolds. I have put special thanks to my boy Kyle Landon Chen, he was just born at that time. So I found some excuses to stay home a few months no come to UDC, then I could finish bit more of that book. Now today, when I ask Kyle to help dad for some sentences, he always rejected. Why me? Ask Cera! One day I suddenly understanded! I bosted my understanding level of philosophy on human. If there is a way to escape, they will rather do it. If there is an old way of doing, people is reluctant to use a new method. That give me an idea to promote the theory of digital functions. We need to find more important real world problems in which digital techniques are the best or unique! Meanwhile, we also need to make things available when people is looking for it. That is a purpose of this book! Washington, DC, USA Li M. Chen

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11 Acknowledgements The author would like to thank many colleagues and researchers for their support. Special thanks to my friend Professor David Mount at University of Maryland, he has been supported my research for more than 15 years. He also reviewed some of the main results in the related research, which will be presented in Chaps. 3 and 4.As a Professor A. Rosenfeld s close associates, he knows the best of the methodology of the founder of digital functions. xi

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13 Contents Part I Digital Functions 1 Introduction Overview Why Digital Continuity and Gradual Variation Basic Concepts of Digital Functions Gradually Varied Functions and Extensions Digital Curve Deformation Smooth Digital Functions and Applications Algorithm Design General Digital-Discrete Data Reconstruction and Applications The Digital-Discrete Method and Harmonic Functions Gradually Varied Functions and Other Computational Methods References Functions and Relations Definitions of Functions and Relations Functions on Sets Concept of Relations Continuous Functions in Euclidean Space Differentiation and Integration of Functions Finite Differences and Approximations of Derivatives Graphs and Simple Graphs Space, Discrete Space, Digital Space Grid Space and Digital Space Triangulation Polygons and Lattice Functions on Topological Spaces References xiii

14 xiv Contents 3 Functions in Digital and Discrete Space What is a Digital Function Digital Continuous Functions Gradually Varied Functions The Algorithms for Gradually Varied Functions A Simple Algorithm The Divide-and-Conquer Algorithm Case Study: An Example of Gradually Varied Fitting Properties of Gradual Variation Mathematical Foundation of Gradually Varied Functions Remark References Gradually Varied Extensions Basic Gradually Varied Mapping and Immersion The Main Theorem of Gradually Varied Extension Gradually Varied Extensions for Indirect Connectivity of Σ m Some Counter Examples for Gradually Varied Extensions Need of Gradually Varied Mapping and Extension Remark on General Discrete Surfaces and Extensions References Digital and Discrete Deformation Deformation and Discrete Deformation Curves and Digital Curves in Space Mathematical Definitions of Discrete Deformation Geometric Deformation of Digital Curves New Consideration of Discrete Deformation and Homotopy Basic Concepts of Fundamental Groups and Algebraic Computing Fundamental Groups Homotopy Groups Homology Groups and Genus Computing Case Study: Rosenfeld s Problem of Digital Contraction The Difference Between Discrete Deformation and Continuous Deformation Remarks References Part II Digital-Discrete Data Reconstruction 6 Basic Numerical and Computational Methods Linear and Piecewise Linear Fitting Methods Piecewise Linear Method Case Study 1: Least Squares for Linear Regression Case Study 2: Algorithm for Delaunay Triangulation... 71

15 Contents xv 6.2 Smooth Curve Fitting: Interpolation and Approximation Lagrange Interpolations Bezier Polynomials Natural Splines and B-Spline Numerical Surface Fitting Coons Surfaces Bezier and Bspline Surfaces with Tensor Products Remarks: Finite Sample Points, Continuous Functions and Their Approximation References Digital-Discrete Approaches for Smooth Functions Real World Needs: Looking for an Appropriate Fitting Method Necessity of a New Method Formal Description of the Reconstruction Problem Smooth Gradually Varied Functions Derivatives of Gradually Varied Functions Definition of Gradually Varied Derivatives Recalculation of the Function Using Taylor Expansion Algorithm Design and Digital-Discrete Methods The Main Algorithm The Algorithm for the Best Fit The Algorithm for the Gradually Varied Uniform Approximation The Iterated Method for Calculating Derivatives The Multi-scaling Method Case Study: Reconstruction Experiments Derivative Calculation Using Global Methods Relationships to Functional Analysis Methods: Lipschitz Extensions McShane-Whitney Extension Theorem Comparison Between Gradually Varied Functions and Lipschitz Extension Remark References Digital-Discrete Methods for Data Reconstruction Real World Problems General Process for Digital-Discrete Methods Spatial Data Structures for Data Reconstruction Linked-Lists and Linked Graph Data Structures Data Structures for Geometric Cells Linked Cells Data Structures Implementation of Algorithms α-set (Shape) Calculation and Computing Order...105

16 xvi Contents 8.5 Manifolds and Their Discrete and Digital Forms Meshes and Discretization of Continuous Manifolds Digital Manifolds and Its Data Forms Two Discretization Methods of 2D Function Reconstruction on Manifolds Data Reconstruction for Triangle Meshes Data Reconstruction for Digitally Represented Objects Volume Data Reconstruction Practical Problems and Solving Methodology Weakness of Piecewise-Linear Methods The Digital-Discrete Method is a Nonlinear Method Digital-Discrete Method as a Universal Method Remarks References Harmonic Functions for Data Reconstruction on 3D Manifolds Overview: Real Data on Manifolds Harmonic Functions and Discrete Harmonic Functions Harmonic Functions Discrete Harmonic Functions Principle of Finite Difference Based on Discretization and Extension Piecewise Harmonic Functions and Reconstruction Procedures for One Boundary Curve Procedures for Randomly Sampled Points Case Study: The Algorithm Analysis Open Problems Implementation and Examples Further Discussions References Part III Advanced Topics 10 Gradual Variations and Partial Differential Equations Numerical Solutions for Partial Differential Equations Three Types of Partial Differential Equations The Finite Difference Method for PDE Digital-Discrete Methods for Partial Differential Equations Digital-Discrete Algorithm for Poisson Equations Digital-Discrete Algorithm for the Heat Equation Digital-Discrete Algorithm for the Wave Equation Case Study: Groundwater Flow Equations Background and Related Research Parabolic Equations and Digital-Discrete Solutions Experiments and Real Data Processing Real Data Reconstruction for a Large Region...156

17 Contents xvii 10.4 Future Remarks References Gradually Varied Functions for Advanced Computational Methods Similarities Between Gradually Varied Functions and Harmonic Functions Review of Concepts Harmonic Functions with Gradual Variation Gradually Varied Semi-preserving Functions Gradually Varied Functions for Advanced Numerical Methods The Relationship Between Gradually Varied Functions and the Finite Elements Method The Relationship Between Gradually Varied Functions and B-Splines Are There Least Square Smooth Gradually Varied Functions? Computational Meanings of Smoothness and Smooth Extensions Overview of Continuity and Discontinuity Micro- and Macro-smoothness: From Discrete Functions to Continuous Functions Natural Smoothness of Continuous Functions: From Continuous Functions to Discrete Functions Natural Smooth Functions for Discrete Sets and Gradually Varied Smooth Reconstruction Discrete Smoothness, Differentiability and Lipschitz Continuity Future Remarks References Digital-Discrete Method and Its Relations to Graphics and AI Methods Subdivision Surfaces Versus Gradually Varied Surfaces The Principle of the Subdivision Method Gradually Varied Surfaces and Subdivision Surface Combinations Mesh-Free Methods Versus Domain Decomposition The Moving Least Square Method The Methods Based on Domain Decomposition Local Smooth Gradually Varied Jets? Expanding Gradually Varied Functions to Image Segmentation λ -Connectedness and Classification λ -Connected Segmentation Algorithm λ -Connected Segmentation for Quadtree Represented Images λ -Band-Connected Search Method...193

18 xviii Contents 12.4 Expanding Gradually Varied Functions to Intelligent Data Reconstruction Three Mathematical Issues of λ -Connected Fitting λ -Connected Fitting with Gradients Case 1: With Complete First-Order Gradients Case 2: With Incomplete First-Order Gradients Case 3: With High Order Gradients Proofof the Theoremforλ Connected Fitting References Glossary Index...205

19 Acronyms N I R G =(V,E) D J GVS GVF Σ m The natural number set The integer number set The real number set AgraphG with the vertex set V and the edge set E A simply connected domain A subset of D, J usually indicates the sample points or the guiding points Gradually varied surfaces Gradually varied functions m dimensional digital space xix

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