Image Processing and Analysis with Graphs: Theory and Practice

Summary

Covering the theoretical aspects of image processing and analysis through the use of graphs in the representation and analysis of objects, Image Processing and Analysis with Graphs: Theory and Practice also demonstrates how these concepts are indispensible for the design of cutting-edge solutions for real-world applications.

Explores new applications in computational photography, image and video processing, computer graphics, recognition, medical and biomedical imaging

With the explosive growth in image production, in everything from digital photographs to medical scans, there has been a drastic increase in the number of applications based on digital images. This book explores how graphs—which are suitable to represent any discrete data by modeling neighborhood relationships—have emerged as the perfect unified tool to represent, process, and analyze images. It also explains why graphs are ideal for defining graph-theoretical algorithms that enable the processing of functions, making it possible to draw on the rich literature of combinatorial optimization to produce highly efficient solutions.

Some key subjects covered in the book include:

• Definition of graph-theoretical algorithms that enable denoising and image enhancement
• Energy minimization and modeling of pixel-labeling problems with graph cuts and Markov Random Fields
• Image processing with graphs: targeted segmentation, partial differential equations, mathematical morphology, and wavelets
• Analysis of the similarity between objects with graph matching
• Adaptation and use of graph-theoretical algorithms for specific imaging applications in computational photography, computer vision, and medical and biomedical imaging

Use of graphs has become very influential in computer science and has led to many applications in denoising, enhancement, restoration, and object extraction. Accounting for the wide variety of problems being solved with graphs in image processing and computer vision, this book is a contributed volume of chapters written by renowned experts who address specific techniques or applications. This state-of-the-art overview provides application examples that illustrate practical application of theoretical algorithms. Useful as a support for graduate courses in image processing and computer vision, it is also perfect as a reference for practicing engineers working on development and implementation of image processing and analysis algorithms.

Table of Contents

Graph Theory Concepts and Definitions Used in Image Processing and Analysis, O. Lezoray and L. Grady

Introduction

Basic Graph Theory

Graph Representation

Paths, Trees, and Connectivity

Graph Models in Image Processing and Analysis

Graph Cuts—Combinatorial Optimization in Vision, H. Ishikawa

Introduction

Markov Random Field

Basic Graph Cuts: Binary Labels

Multi-Label Minimization

Examples

Higher-Order Models in Computer Vision, P. Kohli and C. Rother

Introduction

Higher-Order Random Fields

Patch and Region-Based Potentials

Relating Appearance Models and Region-Based Potentials

Global Potentials

Maximum a Posteriori Inference

A Parametric Maximum Flow Approach for Discrete Total Variation Regularization, A. Chambolle and J. Darbon

Introduction

Idea of the approach

Numerical Computations

Applications

Targeted Image Segmentation Using Graph Methods, L. Grady

The Regularization of Targeted Image Segmentation

Target Specification

Conclusion

A Short Tour of Mathematical Morphology on Edge and Vertex Weighted Graphs, L. Najman and F. Meyer

Introduction

Graphs and lattices

Neighborhood Operations on Graphs

Filters

Connected Operators and Filtering with the Component Tree

Watershed Cuts

MSF Cut Hierarchy and Saliency Maps

Optimization and the Power Watershed

Partial Difference Equations on Graphs for Local and Nonlocal Image Processing, A. Elmoataz, O. Lezoray, V.-T. Ta, and S. Bougleux

Introduction

Difference Operators on Weighted Graphs

Construction of Weighted Graphs

p-Laplacian Regularization on Graphs

Examples

Image Denoising with Nonlocal Spectral Graph Wavelets, D.K. Hammond, L. Jacques, and P. Vandergheynst

Introduction

Spectral Graph Wavelet Transform

Nonlocal Image Graph

Hybrid Local/Nonlocal Image Graph

Scaled Laplacian Model

Applications to Image Denoising

Conclusions

Acknowledgments

Image and Video Matting, J. Wang

Introduction

Graph Construction for Image Matting

Solving Image Matting Graphs

Data Set

Video Matting

Optimal Simultaneous Multisurface and Multiobject Image Segmentation, X. Wu, M.K. Garvin, and M. Sonka

Introduction

Motivation and Problem Description

Methods for Graph-Based Image Segmentation

Case Studies

Conclusion

Acknowledgments

Hierarchical Graph Encodings, L. Brun and W. Kropatsch

Introduction

Regular Pyramids

Irregular Pyramids Parallel construction schemes

Irregular Pyramids and Image properties

Graph-Based Dimensionality Reduction, J.A. Lee and M. Verleysen

Summary

Introduction

Classical methods

Nonlinearity through Graphs

Graph-Based Distances

Graph-Based Similarities

Graph embedding

Examples and comparisons

Graph Edit Distance—Theory, Algorithms, and Applications, M. Ferrer and H. Bunke

Introduction

Definitions and Graph Matching

Theoretical Aspects of GED

GED Computation

Applications of GED

The Role of Graphs in Matching Shapes and in Categorization, B. Kimia

Introduction

Using Shock Graphs for Shape Matching

Using Proximity Graphs for Categorization

Conclusion

Acknowledgment

3D Shape Registration Using Spectral Graph Embedding and Probabilistic Matching, A. Sharma, R. Horaud, and D. Mateus

Introduction

Graph Matrices

Spectral Graph Isomorphism

Graph Embedding and Dimensionality Reduction

Spectral Shape Matching

Experiments and Results

Discussion

Appendix: Permutation and Doubly- stochastic Matrices

Appendix: The Frobenius Norm

Appendix: Spectral Properties of the Normalized Laplacian

Modeling Images with Undirected Graphical Models, M.F. Tappen

Introduction

Background

Graphical Models for Modeling Image Patches

Pixel-Based Graphical Models

Inference in Graphical Models

Learning in Undirected Graphical Models

Tree-Walk Kernels for Computer Vision, Z. Harchaoui and F. Bach

Introduction

Tree-Walk Kernels as Graph Kernels

The Region Adjacency Graph Kernel as a Tree-Walk Kernel

The Point Cloud Kernel as a Tree-Walk Kernel

Experimental Results

Conclusion

Acknowledgments

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