Computational topology for biomedical imageand data analysis [electronic resource] : theory and applications / Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama.Material type: TextSeries: Publisher: Boca Raton, FL : CRC Press, Taylor & Francis Group, Description: 1 online resourceISBN: 9780429810992; 0429810997; 9780429443077; 0429443072; 9780429810978; 0429810970; 9780429810985; 0429810989Subject(s): Imaging systems in medicine -- Data processing | Topology -- Data processing | Image analysis -- Data processing | Homology theory | MEDICAL / Radiology & Nuclear Medicine | TECHNOLOGY / Imaging SystemsDDC classification: 616.07/54 LOC classification: R857.O6Online resources: Taylor & Francis | OCLC metadata license agreement
Cover; Half Title; Series Page; Title Page; Copyright Page; Dedication; Contents; Series Preface; Foreword; Preface; Contributors; Authors; Section I: Theoretical foundations; Chapter 1: Elements of topology and homology; 1.1 ELEMENTS OF GROUP THEORY; 1.1.1 Set; 1.1.2 Group; 1.1.3 Ring and field; 1.1.4 Homomorphism; 1.1.5 Modular arithmetic; 1.2 TOPOLOGICAL SPACES; 1.2.1 Connected topological spaces; 1.2.2 Continuity and continuous maps; 1.3 METRIC SPACES; 1.3.1 Topology in a metric space; 1.4 ELEMENTS OF AFFINE VECTOR SPACES; 1.4.1 Affine vector space A; 1.5 SIMPLICIAL COMPLEXES
1.5.1 Geometric simplicial complexes1.5.2 Oriented simplicial complexes; 1.5.3 Unoriented simplicial complexes; 1.5.4 Abstract simplicial complexes; 1.6 HOMOLOGY; 1.6.1 Homology of oriented simplicial complexes; 1.6.2 Homology of unoriented simplicial complexes; 1.6.3 Homology of abstract simplicial complexes; 1.7 CHAPTER FIGURES; Chapter 2: Persistent homology of images; 2.1 INTRODUCTION; 2.2 DIGITAL IMAGES; 2.3 A NEIGHBORHOOD SYSTEM FOR IMAGES; 2.3.1 Spatial-based neighborhood; 2.3.2 Intensity-based neighborhood; 2.4 COMPUTING HOMOLOGY FROM IMAGES; 2.4.1 The digital image function x
2.4.2 Projected umbra2.4.3 Inclusion tree of umbra projections; 2.4.4 Algorithms for computing homology; 2.4.5 Construction of abstract simplicial complexes; 2.5 PERSISTENT HOMOLOGY; 2.5.1 Filtration; 2.5.2 Persistence; 2.5.3 Computation of persistent homology; 2.5.4 Persistence diagram; 2.6 COMPUTATIONAL COST; 2.7 CHAPTER FIGURES; Section II: Case studies; Chapter 3: Recognizing noise; 3.1 SYNTHETIC IMAGES; 3.2 WORKFLOW; 3.3 RESULTS; 3.4 CHAPTER FIGURES; Chapter 4: Image segmentation; 4.1 INTRODUCTION; 4.2 MATERIALS AND IMAGE ACQUISITION; 4.3 FEATURE SPACE
4.4 IMAGE DISMANTLING AND INCLUSION TREE4.5 FILTRATION; 4.6 HOMOLOGY AND PERSISTENCE DIAGRAM; 4.7 STATISTICAL INFERENCE; 4.8 MASK SEGMENTATION; 4.9 VALIDATION WITH CASE STUDY DATA; 4.10 VALIDATION WITH ILASTIK; 4.11 RESULTS WITH CASE STUDY DATA; 4.12 RESULTS WITH ILASTIK; 4.13 CHAPTER FIGURES; Chapter 5: Point cloud characterization; 5.1 INTRODUCTION; 5.2 POINT CLOUD; 5.3 WORKFLOW; 5.4 RESULTS; 5.5 CHAPTER FIGURES; Bibliography; Index
This book provides an accessible yet rigorous introduction to topology and homology focused on the simplicial space. It presents a compact pipeline from the foundations of topology to biomedical applications. It will be of interest to medical physicists, computer scientists, and engineers, as well as undergraduate and graduate students interested in this topic. Features: Presents a practical guide to algebraic topology as well as persistence homology Contains application examples in the field of biomedicine, including the analysis of histological images and point cloud data
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