1. Efficient Computation and Simplification of Discrete Morse Decompositions on Triangulated Terrains
Riccardo Fellegara
University of Genova
Genova, Italy
Federico Iuricich
University of Maryland
College Park (MD), USA
Leila De Floriani
University of Genova
Genova, Italy
Kenneth Weiss
Lawrence Livermore National Laboratory, USA
22nd ACM SIGSPATIAL GIS, Dallas, Texas, USA: Nov 4-7, 2014

2. Overview The problem: Tool: discrete Morse complexes
Computing morphological decompositions on large Triangulated Irregular Networks (TINs) efficiently
Simplifying such decompositions effectively
Tool: discrete Morse complexes
based on a discrete version of Morse theory, called Forman theory

3. Contributions Compact encoding for the Forman gradient
attached only to the triangles of the TIN
Algorithms for computing and simplifying a Forman gradient on
a spatio-topological data structure (the PR-star quadtree)
a topological data structure

4. Outline Discrete Morse theory (Forman’s theory) The PR-star quadtree
Forman gradient
Discrete Morse features
Forman gradient simplification
The PR-star quadtree
Definition
Working paradigm
Forman gradient on the PR-star quadtree
Morphological feature extraction
Experimental results

5. Outline Discrete Morse theory (Forman’s theory) The PR-star quadtree
Forman gradient
Discrete Morse features
Forman gradient simplification
The PR-star quadtree
Definition
Working paradigm
Forman gradient on the PR-star quadtree
Morphological feature extraction
Experimental results

6. Morse Theory
Analysis of scalar fields requires extracting morphological features (like critical points, integral lines and surfaces)
An integral line is a maximal path everywhere tangent to the gradient of f
Each integral line connects two critical points called origin and destination
Integral line
The morphological analysis of a terrain is generally performed decomposing the tarrain based on its morphological features corresponding to critical points, integral lines and surfaces. Morse theory offers tools for perceiving such task, decomposining the terrain in regions of influences of its critical points.
Fundamental elements…
saddle
minimum
maximum

7. Morse complex
Descending cell
Descending Morse Complex
Descending cell of p: set of integral lines which origin point is p
Descending Morse complex: collection of all the descending cells.
Ascending cell
Ascending Morse Complex
Ascending cell of p: set of integral lines which destination point is p.
Ascending Morse complex: collection of all the ascending cells.
Among such cells we can find the ridge-lines, critical net or the basins

8. Discrete Morse Theory
Given a triangle mesh Σ, DMT defines a discrete gradient vector field V (Forman gradient) .
Collection C of ordered pairs of simplices in a triangle mesh Σ (i.e. vertices, edges, triangles) such that each simplex of Σ is in at most one pair in C.
Two types of pairs in 2D:
Vertex-Edge
Edge-Triangle
Simplices that are not paired are critical
Maximum – Triangle
Saddle – Edge
Minimum – Vertex
Forman gradient V: C plus the critical simplices
Working in the discrete case we need a discrete representation for both critical points and integral lines and such representation is provided by the discrete Morse theory. DMT defines a fundamental tool, called …., which is a combinatorial representation of the gradient of a function.

9. Morphological feature extraction
The regions of influence of the critical elements (minima, maxima and saddles) form the discrete Morse decomposition
They are retrieved by navigating the Forman gradient
Starting from a critical i-simplex, we will call the corresponding region of influence descending or ascending i-cell
Descending 2-cells
The Forman gradient implicitly represents the morphological features of our terrain and the way we can extract them explicitly is navigating….
Descending 1-cells

10. Morphological feature extraction (example)
Descending 2-cell extraction
Starting critical simplex – triangle (2-simplex)
Gradient visiting order – direct (descending) order
Just to give you a brief example of how these regions are extracted…..

11. Forman gradient simplification
Simplification based on 0- and 1-cancellations:
0-cancellation deletes a minimum (critical vertex) p and a saddle (critical edge) q if and only if there is a single path between them
1-cancellation deletes a saddle (critical edge) p and a maximum (critical triangle) q if and only if there is a single path between them
As an effect arrows along the path are reversed.
The most important operation we want to perform in morphological analysis in general is the simplification of the morphological representation. Such operation is obtained using a well established operator called i-cancellation. Without going in the details of its theoretical definition an i-cancellation deletes a pair of critical simplexes locally modifying the gradient arrows connecting them….
1-cancellation between a maximum p and a saddle q

12. Morphological features simplification
Simplifications on Forman gradient implicitly produces simplifications of the discrete Morse cells.
When a path between two critical simplexes is reversed:
corresponding 2-cells are merged
corresponding 1-cells are contracted
Since the Forman gradient is an implicit representation of the regions, i-cancellation implicitly modify such features also….

13. Simplification algorithms
Simplification algorithms are based on a graph representation of gradient, the Morse Incidence Graph (MIG) G=(N,A)
Based on the gradient V
Nodes in N – critical simplexes of V
Arcs in A – paths connecting critical simplexes
Based on the Morse cells
Nodes in N – discrete Morse cells
Arcs in A – incidence relations among the Morse cells
For implementing a simpification algorithm usually a support data structure is used for representing all the possible simplifications at each step (critical point connected by a sequence of arrows)

14. Outline Discrete Morse theory (Forman’s theory) The PR-star quadtree
Forman gradient
Discrete Morse features
Forman gradient simplification
The PR-star quadtree
Definition
Working paradigm
Forman gradient on the PR-star quadtree
Morphological feature extraction
Experimental results

15. The PR-star quadtree
Compact data structure for encoding simplicial meshes (triangle, tetrahedral, etc.) embedded in space
alternative to topological data structures (that explicitly encode adjacencies among simplices)
supports efficient retrieval of topological connectivity relations on demand
The PR-star tree uses a spatial index (quadtree in 2D and octree in 3D) to generate efficient local application-dependent topological data structures at run-time
In contrast to topological data structures, which explicitly encode the connectivity among mesh elements, or to spatial data structures, which index the elements for efficient spatial queries, PR-star quadtrees and octrees use the spatial index induced by a quadtree (in 2D) or an octree (in 3D) to efficiently generate local application-dependent topological data structures at runtime.
K. Weiss, R. Fellegara, L. De Floriani, and M. Velloso. The PR-star octree: A spatio-topological data structure for tetrahedral meshes. In Proceedings ACM SIGSPATIAL GIS, GIS '11. ACM, November 2011

16. The PR-star quadtree: structure
Based on the Point-Region (PR) quadtree
a spatial index on a set of points in a d-dimensional domain
Structures:
a global array V of vertices: geometry of the mesh
a global array T of triangles
an augmented PR-quadtree
each full leaf block in the PR-star quadtree:
indexes in array V of the vertices inside the block
indexes in array T of the triangles incident in vertices of V inside the block
The PR-star tree is based on the Point Region quadtree (PR quadtree), which is a spatial index on a set of points P in a d-dimensional domain.
The PR-star quadtree for a triangle mesh S consists of
an array P of S’s vertices, which encode the geometry of the mesh;
an array T of triangles (in 2D). Each element in T is encoded in terms of the indices of its three vertices within P;
an augmented PR quadtree N, whose leaf nodes index a subset of vertices from P, as well as all the elements from T that are incident in these vertices.

17. PR-star quadtree – working paradigm
Locally process the mesh in a streaming manner by iterating through the leaf blocks of the tree
For each leaf block, generate a local application-dependent data structure, that can be discarded after processing the block.
PR-star quadree requires less than 60% of IA’s storage space
The basic paradigm for performing operations on a mesh encoded as a PR-star quad tree is to locally process the mesh in a streaming manner by iterating through the leaf nodes of the tree. For each leaf node n, a local application-dependent data structure is built, which is then used to process the local geometry. After we finish processing node n, we discard the local data structure and move on to the next node.
Compared to the most compact topological data structure, the Indexed Mesh with Adjacencies data structure, it requires up to 60% of space for the base structure.
Current leaf block
Triggered leaf block
Stored leaf block

18. Outline Discrete Morse theory (Forman’s theory) The PR-star quadtree
Forman gradient
Discrete Morse features
Forman gradient simplification
The PR-star quadtree
Definition
Working paradigm
Forman gradient on the PR-star quadtree
Morphological feature extraction
Experimental results

19. Forman gradient on the PR-star quadtree
The Forman gradient is computed in a streaming manner on the PR-star quadtree
the algorithm extends the one in [Robins et al, 2011] to simplicial meshes
gradient V stored as a global structure
Gradient is efficiently encoded by attaching information only to the triangles (1 byte per triangle)
The Forman gradient is visited in a streaming manner for extraction the morphological features
We have developed algorithms and a compact encoding for performing morphological analysis with the Forman gradient on a PR-star.
With have implemented different types of algorithms for evaluating the performance of the PR-star compare to the a compact topological data structure called IA.

20. Simplification on the PR-star quadtree
Two different simplification algorithms based on a streaming approach:
local representation of the MIG
global representation of the MIG
Local strategy
a coherent MIG is built inside each leaf node
only the paths inside the leaf are considered
Global strategy
all paths are considered
Both strategies use a streaming approach for the simplification of the gradient.
simplifications are performed one leaf at time
LOCAL: the mig is encoded within a leaf node
GLOBAL: the mig is encoded as a global data structure (i.e. computed in a pre-processing phase)

21. Outline Discrete Morse theory (Forman’s theory) The PR-star quadtree
Forman gradient
Discrete Morse features
Forman gradient simplification
The PR-star quadtree
Definition
Working paradigm
Forman gradient on the PR-star quadtree
Morphological feature extraction
Experimental results

22. Experimental results – storage costMB
Experimental results – storage cost MB
Base mesh – information on the mesh encoded by both the data structures
Topological overhead – remaining storage cost
PR-star topological overhead is 85% to 90% less wrt to the base mesh. IA topological overhead is only 20% less.
At runtime PR-star requires from 20% to 35% less memory than the IA data structure MB
We started comparing the storage costs of the two data structures. We took apart the storage cost common to the two structures (Base mesh) and then we have compared this cost to the (topological overhead) that is the remaining storage cost corresponding to the two structures.
Then we have compared the maximum peak of memory obtained with the two structures and still the PR-star performaces are better respect to the IA.

23. Experimental results - timings
Sec.
Forman gradient computation
PR-star quadtree from 10% to 15% faster than IA data structure
Feature extraction PR-star quadtree vs IA data structure
ascending cells: from 2 to 4 times slower
descending cells: from 8 to 12 times slower
Morse incidence graph 1.2 times faster
Sec.
Considering the timings instead, during the Forman gradient computation the PR-star perform better. It is thery bad respect to the feature extraction algorithm but still is better or comparable when we are extracting the MIG.
In this slide we are basically presenting the different types of compuation we can perform:
- Computation extremely local, requiring a lot of navigation on the tree, requiring a lot of navigation but also massive extraction of the topological relations.

24. Experimental results - simplification
Topological simplification
Timings
PR-star quadtree: about twice slower
the timings of local and global strategies are dataset-dependent
Storage cost
local strategy requires 60% to 80% less memory on PR-star quadtree wrt IA data structure
Global strategy requires 55% to 75% less memory on PR-star quadtree wrt IA data structure
Sec. MB

25. Summary
Efficient tool for computing and simplifying terrain morphology
Compact representation for the discrete gradient field
modular approach to the simplification of the discrete gradient based on the PR-star quadtree
The tool is independent of
the mesh topology
the Forman gradient computation algorithm
the morphological feature extraction algorithm

26. Future work
Encoding the extracted Morse-Smale cells for reconstructing the topological connectivity of the complexes
Parallel/distributed implementation of the local algorithm
Dimension-independent version of the PR-star tree
homology computation
persistence-based simplification in any dimension
analysis of time-varying datasets defined on simplicial meshes or tetrahedral shapes in 4D space.