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)

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End of talk…..

Thank you!

Questions?