Multi-Scale Dual Morse Multi-Scale Dual Morse ComplexesComplexes
for for Representing Terrain Representing Terrain
MorphologyMorphologyE. Danovaro Free University of
Bolzano, Italy
L. De Floriani University of Genova, Italy and
University of Maryland, MD, USA
M. Vitali, P. Magillo University of Genova, Italy
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OverviewOverview
Introduction: motivations and goals Terrain morphology: model and
representation Multi-scale terrain morphology Results Conclusions
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Modeling Terrain GeometryModeling Terrain Geometry
Terrain Terrain (scalar field)(scalar field)– function function z = f(x,y)z = f(x,y)– known at a set of pointsknown at a set of points
Represented as a Represented as a Triangulated Irregular NetworkTriangulated Irregular Network (TIN)(TIN)
– 2D triangle mesh2D triangle mesh– Piecewise linear interpolationPiecewise linear interpolation
TIN represents terrain TIN represents terrain geometrygeometry
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From Geometry to MorphologyFrom Geometry to Morphology
Geometric models (as TINs) give accurate representations
But too verbose, especially for large data sets
basin
peak
passvalley
Morphological models can give a more effective overview of terrain shape
Important for terrain analysis and exploration
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From Geometry to MorphologyFrom Geometry to Morphology
Start from geometry Understand and represent
morphological structure
basins
mountains
Face complexity by usinga multi-scale model
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Multi-Scale ModelsMulti-Scale Models
Implicitly encode a range of representations at variable scale
multi-scale
model
Allow extracting representations according to user requirements
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OverviewOverview
Introduction: motivations and goals Terrain morphology: model and
representation Multi-scale terrain morphology Results Conclusions
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Morse and Morse-Smale TheoryMorse and Morse-Smale Theory
Mathematical theory for modeling morphology of continuous and differentiable scalar fields
Version for discrete case (TINs) [ICIAP’07]
Morse function iff its critical points are non-degenerate (=isolated): no horizontal lines or areas
Critical point iff null gradient (=horizontal tangent plane): maximum, minimum, saddle
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Morse and Morse-Smale TheoryMorse and Morse-Smale Theory
Local neighborhood of a critical point How many terrain sectors lie above/below the tangent
plane?– Totally above = minimum– Totally below = maximum– Mixed = simple saddle or multiple saddle
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Morse and Morse-Smale TheoryMorse and Morse-Smale Theory
Integral line: maximal path which is everywhere tangent to the gradient– Minimum = only
incoming integral lines– Maximum = only outgoing integral lines– Saddle = both
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Stable and Unstable Morse Stable and Unstable Morse ComplexesComplexes
Union of incoming integral lines of a minimum =
stable cell (basin)
Union of outgoing integral lines of a maximum =
unstable cell (mountain)
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Morse-Smale FunctionsMorse-Smale Functions
Morse-Smale function iff boundaries of stable and unstable Morse complexes intersect only at saddle points
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Morse-Smale FunctionsMorse-Smale Functions
Morse-Smale complex = overlay of stable and unstable Morse complexes
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Morse-Smale ComplexMorse-Smale Complex
Each saddle is connected with two maxima and two minima
Terrain is decomposed into quadrangles bounded by saddle-minimum-saddle-maximum
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Stable and Unstable Complexes are Stable and Unstable Complexes are DualDual
Stable region (basin around minimum) = unstable vertex (minimum)
Stable edge (ridge through saddle) = unstable edge (valley through saddle)
Stable vertex (maximum) = unstable region (basin around minimum)
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Morphological StructureMorphological Structure
In [ICIAP’07] we proposed a multi-scale topological model for TINs that assumes:– Morse-Smale function– Simple saddles
Here, we extend to:– Morse, but not Morse-Smale, function– Multiple saddles
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Morse (not Smale) ComplexesMorse (not Smale) Complexes
Stable and unstable Morse complexes may intersect in edges, which connect saddle points
k=2
Macro-saddle = maximal connected set of edges where both endpoints are saddles
A macro-saddle contains k saddles, k>1
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Morse (not Smale) ComplexesMorse (not Smale) Complexes
We can “shrink a macro-saddle to a point”
Obtain the same properties as a normal saddle
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Encoding Morphological StructureEncoding Morphological Structure
Two-level representation of the overlay of the stable and unstable Morse complexes:
Top-level = description of “Morse-Smale complex” by considering macro-saddles instead of individual saddles
Lower-level = description of each macro-saddle
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Top-level data structureTop-level data structure
Minimum radially sorted list of adjacent macro-saddles
Maximum radially sorted list of adjacent macro-saddles
Macro-saddle radially sorted list of adjacent maxima and minima
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Lower-level data structure for a macro-Lower-level data structure for a macro-saddlesaddle
Graph describing relations of individual saddles and minima within macro-saddle
Graph describing relations of individual saddles and maxima within macro-saddle
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OverviewOverview
Introduction: motivations and goals Terrain morphology: model and
representation Multi-scale terrain morphology Results Conclusions
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Multi-Scale (Geometric) ModelMulti-Scale (Geometric) Model
Built off-lineBuilt off-line through a simplification process
Encodencode all simplification steps
multi-scale
model
simplification
query
query
Queried on-line to extract variable-resolution representations
User gives resolution requirements
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Multi-Scale (Geometric) ModelMulti-Scale (Geometric) Model The most simplified TINThe most simplified TIN
A partially ordered set of A partially ordered set of refinement stepsrefinement steps which reverse the simplification process and which reverse the simplification process and progressively refine it into the full-scaleprogressively refine it into the full-scale TINTIN
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Extraction of a TIN at variable scale:Extraction of a TIN at variable scale: Perform a subset of refinements which is Perform a subset of refinements which is
consistent with the partial orderconsistent with the partial order Driven by user requirementsDriven by user requirements
Multi-Scale (Geometric) ModelMulti-Scale (Geometric) Model
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Extract TINs at variable resolution with Extract TINs at variable resolution with consistent morphologyconsistent morphology
Preserve the structure of the dual Morse Preserve the structure of the dual Morse complexcomplex
Simplification of geometry onlySimplification of geometry only
Morphology-preserving geometric Morphology-preserving geometric simplification [ACM-GIS’03]simplification [ACM-GIS’03]
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Morphology SimplificationMorphology Simplification
Morphology is often too complex, especially for Morphology is often too complex, especially for large terrainslarge terrains
Many meaningless features due to noiseMany meaningless features due to noise
Simplification to remove undesired featuresSimplification to remove undesired features Simplify the dual Morse complexesSimplify the dual Morse complexes
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Morphology simplification Morphology simplification for Morse-Smale complex for Morse-Smale complex [ICIAP’07][ICIAP’07]
Simplification operators: minimum-saddle-minimum (collapse to 1 minimum)
maximum-saddle-maximum (collapse to 1 maximum)
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Morphology simplification Morphology simplification for dual Morse Complexesfor dual Morse Complexes
Each operator becomes three: Minimum-saddle-minimum
– for an isolated (= non-macro) simple saddle– (unchanged) remove saddle
Minimum-saddleh-minimum– for an isolated (= non-macro) multiple saddle– decrease multiplicity of saddle to h-1
Minimum-k-saddle-minimum– for an macro-saddle containing k simple saddles – decrease cardinality of macro-saddle to k-1
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Morphology simplification for dual Morphology simplification for dual Morse ComplexesMorse Complexes
Minimum-saddleh-minimum– for an isolated (= non-macro) multiple saddle – collapse 2 consecutive minima and reduce multiplicity of
saddle
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Morphology simplification for dual Morphology simplification for dual Morse ComplexesMorse Complexes
Minimum-k-saddle-minimum– for an macro-saddle containing k simple saddles – collapse 2 minima and the simple saddle
connecting them to 1 minimum, and reduce cardinality of macro-saddle
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Multi-Scale Morphological ModelMulti-Scale Morphological Model
Each simplification step affects a limited number of entities (vertices, edges, regions) in the dual Morse complexes
(Detailed description of such entities is in the paper)
It can be described as one set of entities replaced by another set of entities: u=(u+, u-)
Each simplification step can be reversed into a refinement step: rev(u) = (u-,u+)
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Multi-Scale Morphological ModelMulti-Scale Morphological Model
The multi-scale models is composed of: Coarse model (final result of simplification) Refinement steps that allow reconstructing the
initial full-scale model (inverse of performed simplification steps)
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Multi-Scale Morphological ModelMulti-Scale Morphological Model
Partial order based on dependency relation– A refinement step u2 = (u2-,u2+) depends on u1 =
(u1-,u1+) if u2 removes some element created by u1, I.e., u1+ and u2- have non-empty intersection
Represented as a directed Acyclic Graph (DAG)
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Multi-Scale Morphological ModelMulti-Scale Morphological Model
Each modification has a persistence value (approximation induced by morphology simplification)
User gives a persistence threshold over the whole domain, or in a region of interest
Extract the coarsest representation fitting the threshold
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OverviewOverview
Introduction: motivations and goals Terrain morphology: model and
representation Multi-scale terrain morphology Results Conclusions
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Experimental ResultsExperimental Results
Multi-scale TIN (geometry) Multi-scale dual Morse complexes
(morphology)
Synchronization simultaneous extraction of matching geometry and morphology for user criteria
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Elba island at uniform resolutionElba island at uniform resolution
Full resolution: 785 minima 509 maxima
(75k triangles)
Persistence = 3m: 157 minima 199 maxima
(13k triangles)
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Elba island with focus on a regionElba island with focus on a region
Full resolution:158 minima114 maxima
(9.9k triangles)
Persistence = 3m:52 minima69 maxima
(2.7k triangles)
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OverviewOverview
Introduction: motivations and goals Terrain morphology: model and
representation Multi-scale terrain morphology Results Conclusions
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ConclusionsConclusions
Model and data structure to represent terrain morphology of Morse (but not Morse-Smale) functions
Multi-Scale version of such model and construction through morphological simplification
Extraction of variable-scale morphology representations
Matching extracted morphology and geometry (not in the paper)