Efficient Computation and
Simplification of Discrete Morse
Decompositions on Triangulated
Terrains
Riccardo Fellegara
Federico Iuricich
University of Genova
Genova, Italy
[email protected]
University of Maryland
College Park (MD), USA
[email protected]
Leila De Floriani
Kenneth Weiss
University of Genova Lawrence Livermore National
Genova, Italy
Laboratory, USA
[email protected]
[email protected]
22nd ACM SIGSPATIAL GIS, Dallas, Texas, USA: Nov 4-7, 2014
Overview
› The problem:
›
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
2
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
3
Outline
› Discrete Morse theory (Forman’s theory)
›
›
›
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
Forman gradient simplification
› Experimental results
4
Outline
› Discrete Morse theory (Forman’s theory)
›
›
›
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
Forman gradient simplification
› Experimental results
5
Morse Theory
› Analysis of scalar fields requires
extracting morphological
features (like critical points,
integral lines and surfaces)
Integral line
› 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
saddle
minimum
maximum
6
Morse complex
Descending cell
› Descending cell of p: set of integral
lines which origin point is p
Descending Morse
Complex
› Descending Morse complex:
collection of all the descending cells.
Ascending cell
› Ascending cell of p: set of integral
lines which destination point is p.
Ascending Morse
Complex
› Ascending Morse complex:
collection of all the ascending cells.
7
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
8
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
›
Descending 2-cells
Starting from a critical i-simplex, we will call the
corresponding region of influence descending or
ascending i-cell
Descending 1-cells
Morphological feature extraction (example)
› Descending 2-cell extraction
Starting critical simplex – triangle (2-simplex)
› Gradient visiting order – direct (descending) order
›
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.
1-cancellation between a maximum p and a saddle q
11
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
12
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
13
Outline
› Discrete Morse theory (Forman’s theory)
›
›
›
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
Forman gradient simplification
› Experimental results
14
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
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
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
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
Current leaf block
Triggered leaf block
Stored leaf block
Outline
› Discrete Morse theory (Forman’s theory)
›
›
›
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
Forman gradient simplification
› Experimental results
18
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
19
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
20
Outline
› Discrete Morse theory (Forman’s theory)
›
›
›
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
Forman gradient simplification
› Experimental results
21
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
300
200
IA
PR-star
100
0
MB
800
600
IA
Global
400
200
0
Maui Baia Puget
22
Experimental results - timings
› 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.
150
100
IA
PR-star
50
0
Sec.
250
200
150
100
50
0
IA
Prstar
Maui Baia Puget
23
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.
80
60
IA
Global
Local
40
20
0
Maui Baia Puget
MB
400
300
IA
Global
Local
200
100
0
Maui Baia Puget
24
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
25
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.
26