Topographical Tools for Filtering and Segmentation, Watersheds on Node- or Edge-weighted Graphs by Meyer, Fernand -- Read -- Imperial Library of Trantor

Index

Cover Table of Contents Notations Introduction PART 1: Getting Started

1 A Primer to Flooding, Razing and Watersheds

1.1. Topographic reliefs and topographic features 1.2. Flooding, razing and morphological filters 1.3. Catchment zones of flooded surfaces 1.4. The waterfall hierarchy 1.5. Size-driven hierarchies 1.6. Separating overlapping particles in n dimensions 1.7. Catchment zones and lakes of region neighborhood graphs 1.8. Conclusion

2 Watersheds and Flooding: a Segmentation Golden Braid

2.1. Watersheds, offsprings and parallel branches 2.2. Flooding and connected operators 2.3. Connected operators and hierarchies 2.4. Hierarchical segmentation: extinction values

3 Mathematical Notions

3.1. Summary of the chapter 3.2. Complete lattices 3.3. Operators between complete lattices 3.4. The adjunction: a cornerstone of mathematical morphology 3.5. Openings and closings 3.6. Complete lattices of functions

PART 2: The Topography of Weighted Graphs

4 Weighted Graphs

4.1. Summary of the chapter 4.2. Reminders on graphs 4.3. Weight distributions on the nodes or edges of a graph 4.4. Exploring the topography of graphs by following a drop of water 4.5. Node-weighted graphs 4.6. Edge-weighted graphs 4.7. Comparing the topography of node-weighted graphs and edge-weighted graphs

5 Flowing Graphs

5.1. Summary of the chapter 5.2. Towards a convergence between node- and edge-weighted graphs 5.3. The flowing adjunction 5.4. Flowing edges under closer scrutiny 5.5. Illustration as a hydrographic model

6 The Topography of Digraphs

6.1. Summary of the chapter 6.2. Status report 6.3. The topography of unweighted digraphs 6.4. The topography of gravitational digraphs

PART 3: Reducing the Overlapping of Catchment Zones

7 Measuring the Steepness of Flowing Paths

7.1. Summary of the chapter 7.2. Why do the catchment zones overlap? 7.3. The lexicographic pre-order relation of length k

8 Pruning a Flow Digraph

8.1. Summary of the chapter 8.2. The pruning operator 8.3. Evolution of the catchment zones with pruning

9 Constructing an ∞-steep Digraph by Flooding

9.1. Summary of the chapter 9.2. Characterization of ∞ - steep graphs 9.3. The core-expanding flooding algorithm

10 Creating Steep Watershed Partitions

11 An Historical Intermezzo

PART 4: Segmenting with Dead Leaves Partitions

12 Intermezzo: Encoding the Digraph Associated with an Image

12.1. Summary of the theoretical developments seen so far 12.2. Summary of the chapter 12.3. Representing a node-weighted digraph as two images 12.4. Defining labels

13 Two Paradigms for Creating a Partition or a Partial Partition on a Graph

14 Dead Leaves Segmentation

14.1. Summary of the chapter 14.2. Segmenting with a watershed 14.3. The evolution of a dead leaves tessellation with pruning 14.4. Local correction of overlapping zones 14.5. Local correction of the overlapping zones on a DEM 14.6. Segmentation of some marked regions

15 Propagating Segmentations

15.1. Summary of the chapter 15.2. Step-by-step segmentation 15.3. Marker-based segmentation

Appendix: Mathematical Recap and Notations References Index End User License Agreement