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Flow Computations on

Imprecise Terrains

Anne Driemel,Herman Haverkort,Maarten Lofflerand Rodrigo Silveira p

q

EuroCG 2011, Morschach

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Why study water flow on terrains?

Analysis of flash floods

Streamflow forecasting Erosion prediction for river beds

..

Interesting geometric problems

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p

Assumption: Water flows downwards inthe direction of steepest descent.

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q

p

The watershed W(q):the area that drains to q.

Assumption: Water flows downwards inthe direction of steepest descent.

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q

p

q

p flows to q

The watershed W(q):the area that drains to q.

Assumption: Water flows downwards inthe direction of steepest descent.

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q

p

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q

p

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qq

p

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q

p

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p

q

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q

p

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p

q

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p?

q

Can we computefuzzy watersheds?

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p?

q

Can we computefuzzy watersheds?In this model:NP-hard

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In a grid elevation model..

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In a grid elevation model..

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In a grid elevation model..

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Each grid cell drainsto one of its eightneighbors.

In a grid elevation model..

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Each grid cell drainsto one of its eightneighbors.

In a grid elevation model..

q

p

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Flow network

q

p

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Flow network

Watersheds arediscrete point sets

q

p

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Definitions

An imprecise terrain is a graph in which each node

v has the form v= (x,y, [z, z+]).

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Definitions

An imprecise terrain is a graph in which each node

v has the form v= (x,y, [z, z+]).

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Definitions

An imprecise terrain is a graph in which each node

v has the form v= (x,y, [z, z+]).

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Definitions

An imprecise terrain is a graph in which each node

v has the form v= (x,y, [z, z+]).

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Definitions

A realization R is the graph with v= (x , y , z),

such that z[z, z+].

R

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Definitions

In a fixed realization water flows from a node to

its steepest descent neighbor.

p

R

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Definitions

q

The potential watershed :

W(q) :=R

{p:p flows to q in R}

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Definitions

q

The core watershed :

W(q) :=R

{p:p flows to q in R}

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Results

The potential watershed :

The core watershed :

W(Q) :=R

{p:p flows to q in R, qQ}

W(Q) :=

R

{p:p flows to q in R, qQ}

q

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Results

The potential watershed :

The core watershed :

W(Q) :=R

{p:p flows to q in R, qQ}

W(Q) :=

R

{p:p flows to q in R, qQ}

q

We can compute bothin O(n log n) time;on grid terrains: O(n)

C

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Core watersheds do not give a good definition of

persistent water flow..

Understanding Core Watersheds

W(q) =

RRT

{p: p flows to q in R}

U d d C W h d

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Core watersheds do not give a good definition of

persistent water flow..

Understanding Core Watersheds

W(q) =

RRT

{p: p flows to q in R}

q

upper terrain

lower terrain

U d d C W h d

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Core watersheds do not give a good definition of

persistent water flow..

W(q)

Understanding Core Watersheds

W(q) =

RRT

{p: p flows to q in R}

q

upper terrain

lower terrain

realization

U d di C W h d

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Core watersheds do not give a good definition of

persistent water flow..

W(q) ={q} W(q)

Understanding Core Watersheds

.. only contains q!

W(q) =

RRT

{p: p flows to q in R}

q

upper terrain

lower terrain

realization

U d di C W h d

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Core watersheds do not give a good definition of

persistent water flow..

W(q) ={q}

Understanding Core Watersheds

p

.. only contains q!potential localminimum

W(q) =

RRT

{p: p flows to q in R}

q

upper terrain

lower terrain

realization

U d di C W h d

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Understanding Core Watersheds

W(q) =

RRT

{p: p flows to q in R}

Core Watersheds are the complement of the set of

nodes that have alternative destinations

U d di C W h d

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Understanding Core Watersheds

= RRT

{p: p does not flow to q in R}c

W(q) =

RRT

{p: p flows to q in R}

Core Watersheds are the complement of the set of

nodes that have alternative destinations

U d di C W h d

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Understanding Core Watersheds

= RRT

{p: p does not flow to q in R}c

W(q) =

RRT

{p: p flows to q in R}

(ii) nodes outsideW(q)

(i) potential local minima

Core Watersheds are the complement of the set of

nodes that have alternative destinations

Contained in this set:

(iii) nodes with flow paths to (i) or (ii)

U d di C W h d

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Understanding Core Watersheds

= RRT

{p: p does not flow to q in R}c

W(q) =

RRT

{p: p flows to q in R}

(ii) nodes outsideW(q)

(i) potential local minima

Core Watersheds are the complement of the set of

nodes that have alternative destinations

Contained in this set:

(iii) nodes with flow paths to (i) or (ii)

Vmin

U d di C W h d

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Understanding Core Watersheds

= RRT

{p: p does not flow to q in R}c

W(q) =

RRT

{p: p flows to q in R}

(ii) nodes outsideW(q)

(i) potential local minima

Core Watersheds are the complement of the set of

nodes that have alternative destinations

Contained in this set:

(iii) nodes with flow paths to (i) or (ii)

Vmin (W(q))c

U d t di C W t h d

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Understanding Core Watersheds

= RRT

{p: p does not flow to q in R}c

W(q) =

RRT

{p: p flows to q in R}

(ii) nodes outsideW(q)

(i) potential local minima

Core Watersheds are the complement of the set of

nodes that have alternative destinations

Contained in this set:

(iii) nodes with flow paths to (i) or (ii)

W ( Vmin (W(q))c )

U d t di C W t h d

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Understanding Core Watersheds

= RRT

{p: p does not flow to q in R}c

W(q) =

RRT

{p: p flows to q in R}

(ii) nodes outsideW(q)

(i) potential local minima

Core Watersheds are the complement of the set of

nodes that have alternative destinations

Contained in this set:

(iii) nodes with flow paths to (i) or (ii)

W ( Vmin (W(q))c )

Caution! Avoid theflow paths through q.

U d t di C W t h d

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Understanding Core Watersheds

= RRT

{p: p does not flow to q in R}c

W(q) =

RRT

{p: p flows to q in R}

(ii) nodes outsideW(q)

(i) potential local minima

Core Watersheds are the complement of the set of

nodes that have alternative destinations

Contained in this set:

(iii) nodes with flow paths to (i) or (ii)

W\q ( Vmin (W(q))

c )

Caution! Avoid theflow paths through q.

U d t di C W t h d

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Understanding Core Watersheds

= RRT

{p: p does not flow to q in R}c

W(q) =

RRT

{p: p flows to q in R}

(ii) nodes outsideW(q)

(i) potential local minima

=

W\q ( Vmin (W(q))

c )

c

Core Watersheds are the complement of the set of

nodes that have alternative destinations

Contained in this set:

(iii) nodes with flow paths to (i) or (ii)

U d t di C W t h d

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Understanding Core Watersheds

= RRT

{p: p does not flow to q in R}c

W(q) =

RRT

{p: p flows to q in R}

(ii) nodes outsideW(q)

(i) potential local minima

=

W\q ( Vmin (W(q))

c )

c

Core Watersheds are the complement of the set of

nodes that have alternative destinations

Contained in this set:

(iii) nodes with flow paths to (i) or (ii)

Alternative Definition:

W(q) = W\q ( ( W(q) )

c)

c

U d t di C W t h d

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Understanding Core Watersheds

= RRT

{p: p does not flow to q in R}c

W(q) =

RRT

{p: p flows to q in R}

(ii) nodes outsideW(q)

(i) potential local minima

=

W\q ( Vmin (W(q))

c )

c

Core Watersheds are the complement of the set of

nodes that have alternative destinations

Contained in this set:

(iii) nodes with flow paths to (i) or (ii)

Alternative Definition:

W(q) = W\q ( ( W(q) )

c)

c

Persistent Watersheds

P i t t W t h d P ti

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Persistent Watersheds Properties

In general, persistent watersheds and potential

watersheds are not nested!

Persistent Watersheds Properties

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Persistent Watersheds Properties

In general, persistent watersheds and potential

watersheds are not nested!

On regular terrains, we can prove:

q

p

Let p W(q)

after removing

avoidable local minima

Persistent Watersheds Properties

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Persistent Watersheds Properties

In general, persistent watersheds and potential

watersheds are not nested!

On regular terrains, we can prove:

q

p

Let p W(q)

(i) W(p)W(q)

after removing

avoidable local minima

Persistent Watersheds Properties

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Persistent Watersheds Properties

In general, persistent watersheds and potential

watersheds are not nested!

On regular terrains, we can prove:

q

p

Let p W(q)

(i) W(p)W(q)

(ii)W(p)W(q)

after removing

avoidable local minima

Fuzzy watersheds

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Fuzzy watersheds

Fuzzy watersheds

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Fuzzy watersheds

Persistent

Minimum

Fuzzy watersheds

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Fuzzy watersheds

PotentialWatersheds

Persistent

Minimum

Fuzzy watersheds

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Fuzzy watersheds

PotentialWatersheds

Persistent

Watershed

Persistent

Minimum

Fuzzy watersheds

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Fuzzy watersheds

PotentialWatersheds

Fuzzy ridge!

Persistent

Watershed

Persistent

Minimum

Fuzzy watersheds

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Fuzzy watersheds

PotentialWatersheds

Fuzzy ridge!

Persistent

Watershed

Thank you!

Persistent

Minimum