Discrete Calculus

Applied Analysis on Graphs for Computational Science

By: Jonathan R. Polimeni, Leo J. Grady

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Hardcover | 2 August 2010 | Edition Number 2010

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Hardcover
384 Pages

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23.5 x 15.5 x 2.54

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The goal of this book is to present the topic of discrete calculus to scientists and - gineers and to show how the theory can be applied to solving a wide variety of re- world problems. We feel that discrete calculus allows us to unify many approaches to data analysis and content extraction while being accessible enough to be widely applied in many ?elds and disciplines. This project initially began as a tutorial on discrete calculus and its applications, and we hope that this work can provide an introduction to discrete calculus and demonstrate its effectiveness in many problem domains. This book bene?ted enormously from the friends and colleagues who provided software, data, and their time in reading the manuscript. In particular, for software we would like to thank S©bastien Bougleux for help with total variation code for an arbitrary graph and Camille Couprie for powerwatershed code. For use of their data, we wish to thank Jason Bohland, James Fowler, David Gleich, and Robert S- ner. For comments on the manuscript we would like to thank Christopher Alvino, Mukund Balasubramanian, Jason Bohland, S©bastien Bougleux, Ga«lle Desbordes, Mathieu Desbrun, Noha El-Zehiry, Gareth Funka-Lea, David Gleich, Oliver Hinds, Anil Hirani, Hiroshi Ishikawa, Robert Kotiuga, Elliot Saltzman, Dheeraj Singaraju, Ganesh Sundaramoorthi and Enzo Tonti. We thank Eric Schwartz for all his help and support over the years. We would like to thank Wayne Wheeler, Simon Rees, and Catherine Brett at Springer for their enthusiasm and for initially proposing this project.

Industry Reviews

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"The book introduces some best practices and advanced techniques in the field of discrete calculus. The articles are written by a team of experts in the field and are aimed at scientists and engineers. The book shows how the theory can be applied to solving a wide variety of real-world problems. ... All the examples are well described and illustrated with source code and screen shots. I recommend this book to anyone interested in discrete calculus and its application." (Todor Jordanov Todorov, Mathematical Reviews, Issue 2012 d)

"This book illustrates the concepts of discrete calculus with examples in circuit theory, chosen because of its natural connection to algebraic topology and maturity. ... this is a unique book that brings together research in complex networks and algorithm content extraction. I have no doubt that the book will be a valuable addition to the libraries of many researchers and graduate students working in the areas of complex networks, graphics, and algorithm design." (Alexander Tzanov, ACM Computing Reviews, July, 2011)

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Calculus Computing & I.T.Computer Programming & Software Development Algorithms & Data Structures Discrete Mathematics Computer Science Image Processing Numerical Analysis Graphical & Digital Media Applications

Mathematical Theory of Computation Maths for Computer Scientists Science Physics Mathematical Physics Artificial Intelligence Computer Vision

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Paperback

Published: 9th October 2014

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Discrete Calculus: History and Future	p. 1
Discrete Calculus	p. 1
Origins of Vector Calculus	p. 2
Origins of Discrete Calculus	p. 3
Discrete vs. Discretized	p. 4
Complex Networks	p. 6
Content Extraction	p. 7
Organization of the Book	p. 8
Intended Audience	p. 8
A Brief Review of Discrete Calculus
Introduction to Discrete Calculus	p. 13
Topology and the Fundamental Theorem of Calculus	p. 14
Differential Forms	p. 16
Exterior Algebra and Antisymmetric Tensors	p. 17
Differentiation and Integration of Forms	p. 26
The Hodge Star Operator	p. 32
Differential Forms and Linear Pairings	p. 37
Discrete Calculus	p. 38
Discrete Domains	p. 39
Discrete Forms and the Coboundary Operator	p. 48
Primal and Dual Complexes	p. 50
The Role of a Metric: the Metric Tensor, the Discrete Hodge Star Operator, and Weighted Complexes	p. 54
The Dual Coboundary Operator	p. 59
The Discrete Laplace-de Rham Operator	p. 60
Structure of Discrete Physical Laws	p. 63
Examples of Discrete Calculus	p. 63
Fundamental Theorem of Calculus and the Generalized Stokes' Theorem	p. 65
The Helmholtz Decomposition	p. 73
Matrix Representation of Discrete Calculus Identities	p. 77
Elliptic Equations	p. 79
Diffusion	p. 82
Advection	p. 86
Concluding Remarks	p. 88
Circuit Theory and Other Discrete Physical Models	p. 91
Circuit Laws	p. 93
Steady-State Solutions	p. 94
Dependent Sources	p. 97
Energy Minimization	p. 99
AC Circuits	p. 101
Connections Between Circuit Theory and Other Discrete Domains	p. 104
Spring Networks	p. 104
Random Walks	p. 106
Gaussian Markov Random Fields	p. 110
Tree Counting	p. 112
Linear Algebra Applied to Circuit Analysis	p. 117
Conclusion	p. 121
Applications of Discrete Calculus
Building a Weighted Complex from Data	p. 125
Determining Edges and Cycles	p. 126
Defining an Edge Set	p. 126
Defining a Cycle Set	p. 129
Deriving Edge Weights	p. 135
Edge Weights to Reflect Geometry	p. 135
Edge Weights to Penalize Data Outliers	p. 136
Edge Weights to Cause Repulsion	p. 143
Edge Weights to Represent Joint Statistics	p. 144
Deducing Edge Weights from Observations	p. 144
Obtaining Higher-Order Weights to Penalize Outliers	p. 147
Weights Beyond Flows	p. 149
Metrics Defined on a Complex	p. 150
Conclusion	p. 153
Filtering on Graphs	p. 155
Fourier and Spectral Filtering on a Graph	p. 156
Graphs that Are Not Shift-Invariant	p. 159
The Origins of High Frequency Noise	p. 162
Energy Minimization Methods for Filtering	p. 163
The Basic Energy Minimization Model	p. 163
Extended Basic Energy Model	p. 167
The Total Variation Model	p. 168
Filtering with Implicit Discontinuities	p. 170
Filtering with Explicit, but Unknown, Discontinuities	p. 172
Filtering by Gradient Manipulation	p. 174
Nonlocal Filtering	p. 174
Filtering Vectors and Flows	p. 175
Translating Scalar Filtering to Flow Filtering	p. 176
Filtering Higher-Order Cochains	p. 179
Applications	p. 180
Image Processing	p. 180
Three-Dimensional Mesh Filtering	p. 185
Filtering Data on a Surface	p. 187
Geospatial Data	p. 192
Filtering Flow Data-Brain Connectivity	p. 193
Conclusion	p. 197
Clustering and Segmentation	p. 199
Targeted Clustering	p. 200
Primal Targeted Clustering	p. 201
Dual Targeted Clustering	p. 210
Untargeted Clustering	p. 215
Primal Untargeted Clustering	p. 216
Dual Untargeted Clustering	p. 220
Semi-targeted Clustering	p. 221
The k-Means Model	p. 221
Clustering Higher-Order Cells	p. 227
Clustering Edges	p. 227
Applications	p. 229
Image Segmentation	p. 229
Social Networks	p. 235
Machine Learning and Classification	p. 236
Gene Expression	p. 240
Conclusion	p. 242
Manifold Learning and Ranking	p. 243
Manifold Learning	p. 244
Multidimensional Scaling and Isomap	p. 245
Laplacian Eigenmaps and Spectral Coordinates	p. 247
Locality Preserving Projections	p. 249
Relationship to Clustering	p. 251
Manifold Learning on Edge Data	p. 251
Ranking	p. 253
PageRank	p. 253
HITS	p. 256
Applications	p. 257
Shape Characterization	p. 257
Point Correspondence	p. 260
Web Search	p. 262
Judicial Citation	p. 264
Conclusion	p. 266
Measuring Networks	p. 267
Measures of Graph Connectedness	p. 268
Graph Distance	p. 268
Node Centrality	p. 269
Distance-Based Properties of a Graph	p. 270
Measures of Graph Separability	p. 274
Clustering Measures	p. 274
Small-World Graphs	p. 277
Topological Measures	p. 279
Geometric Measures	p. 281
Discrete Gaussian Curvature	p. 282
Discrete Mean Curvature	p. 283
Applications	p. 285
Social Networks	p. 285
Chemical Graph Theory	p. 286
Conclusion	p. 288
Representation and Storage of a Graph and Complex	p. 291
General Representations for Complexes	p. 291
Cells List Representation	p. 291
Operator Representation	p. 292
Representation of 1-Complexes	p. 293
Neighbor List Representation	p. 293
Optimization	p. 295
Real-Valued Optimization	p. 296
Unconstrained Direct Solutions	p. 297
Constrained Direct Solutions	p. 298
Descent Methods	p. 308
Nonconvex Energy Optimization over Real Variables	p. 312
Integer-Valued Optimization	p. 312
Linear Objective Functions	p. 312
Quadratic Objective Functions	p. 314
General Integer Programming Problems	p. 318
The Hodge Theorem: A Generalization of the Helmholtz Decomposition	p. 319
The Helmholtz Theorem	p. 319
The Hodge Decomposition	p. 326
Summary of Notation	p. 331
References	p. 335
Index	p. 353
Color Plates	p. 361
Table of Contents provided by Ingram. All Rights Reserved.

Discrete Calculus

Applied Analysis on Graphs for Computational Science

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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