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Review of Texture-based Methods
• Employs texture synthesis and image processing techniques to
provide global, continuous, dense, and visually pleasing
representations without constructing intermediate geometry.
• LIC family is the most popular texture-based technique
• IBFV is an easy but flexible technique
• Both can be extended to (2.5D) surface flow visualization
• IBFV is more computationally efficient than LIC
• Extending to 3D volumetric data visualization is challenging
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white noise (fine sand)flow field (wind)
� A low-pass filter is applied to white noise along pixel-
centered bi-directionally symmetric streamlines to exploit
spatial correlation in the flow direction — anisotropic low-
pass filtering along flow lines (Cabral and Leedom SigGraph93)
� LIC synthesizes an image, providing a dense representation
analogous to the pattern of an area of wind-blown sand
⊕convolution
(blow)
� Line Integral Convolution (LIC)
LIC image (pattern)
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a point in the flow
field — the counterpart of
a pixel in the output LIC image
d( ρ(τ) ) / d τ = υ( ρ(τ) )
ρ(τ+∆τ) = ρ(τ) + ∫ττ +∆τ υ(ρ(τ))dτ
locate a set of
pixels hit by
the streamline
index the input
noise for the
texture values
obtain the value of
the target pixel in the LIC
image via texture convolution
� Pipelineυ( ρ(τ) )
ρ(τ + dτ)
ρ(τ)
dτ
∑ ( texture[i] × weight[i] )
∑ weight[i]
weighting is governed
by a low-pass filter
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Texture-based Methods on Surfaces
Surface LIC (Detlev Stalling, ZIB, Germany)
IBFVS ISA
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Volumetric Texture
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7
http://cs.swan.ac.uk/~csbob/te
aching/csM07-vis/
Arrows vs. Streamlines vs. Textures
Streamlines: selective
Arrows: simple
Textures: 2D-filling
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Additional reading
• Helwig Hauser, Robert S. Laramee, Helmut Doleisch, Frits H. Post, and Benjamin Vrolijk, The State of the Art in Flow Visualization: Direct, Texture-based, and Geometric Techniques, TR-VRVis-2002-046 Technical Report, VRVis Research Center, Vienna, Austria, December 2002.
• Robert S. Laramee, Helwig Hauser, Helmut Doleisch, Benjamin Vrolijk, Frits H. Post, and Daniel Weiskopf,The State of the Art in Flow Visualization: Dense and Texture-Based Techniques. in Computer Graphics Forum (CGF), Vol. 23, No. 2, 2004, pages 203-221.
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Vector Field Visualization:
Feature-based
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What features are in flows
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Vector Field Gradient
• Consider a vector field
����� = � � = � �, , � =
� ����
• Its gradient is
∇� =
�� ��
�� �
�� ��
�����
����
�����
�����
����
�����
It is also called the Jacobian matrix of the vector field.
Many feature detection for flow data relies on Jacobian
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Divergence and Curl• Divergence- measures the magnitude of outward flux through
a small volume around a point
���� = ∇ ∙ � =�� ��
+����
+�����
� =�
��
�
�
�
��
• Curl- describes the infinitesimal rotation around a point
����� = � × � =����
−�����
�� ��
−�����
�����
−�� �
� ∙ (� × �) = 0� × (∇∅) = 0
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Helmholtz decomposition
� = ∇" + � × #
Hodge decomposition
� = �" + � × # + $
Curl (or rotation) free Divergence free
Curl (or
rotation) free
Divergence
freeHarmonic
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Helmholtz decomposition
curl-free divergence-freeneither
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Vortices
Source:
http://www.onera.fr/cahierdelabo/english/aerod_ind03.htm
Post et al. STAR report 2003
Post et al. STAR report 2003
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Separation Flows
Separation flow on delta wing surface [Tricoche et al. AIAA 2004]
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2D Vector Fields
• Assume a 2D vector field
����� = � � = � �, =
� ��
=%� + & + ��� + ' + �
• Its Jacobian is
�� =
�� ��
�� �
�����
����
= % &� '
• Divergence is % + '
• Curl is b − d
Given a vector field defined on a discrete mesh, it is important
to compute the coefficients a, b, c, d, e, f for later analysis.
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(�*, *)
(�+, +)
(�,, ,)
(��*, �*)
(��+, �+)
(��,, �,)
- �, =� ��
=%� + & + ��� + ' + �
��*�*
=%�* + &* + ���* + '* + �
Assume a piecewise linear vector field
We have
……
Solve for a linear system to get the coefficients
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Vector Field Topology:
Introduction
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Examples
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• Abstract representation of flow field
• Characterization of global flow structures
• Basic idea (steady case):
• Interpret flow in terms of streamlines
• Classify them w.r.t. their limit sets
• Determine regions of homogenous behavior
• Graph depiction
• Fast computation (not always)
Motivation
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Let us focus on steady vector fields at this moment
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Vector Fields (Recall)
• A vector field
– is a continuous vector-valued function V(x) on a
manifold X– can be expressed as a system of ODE ẋ = V(x)
– introduces a flow ϕ : R× X → X
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Trajectories
• A trajectory of x∈X is ∪t∈Rϕ(t, x)
• Given an initial condition, there is a unique solution
x(t) = x0 + ∫0≤u≤t v(x(u)) duϕ(t0)= x0
• Under time-independent setting a trajectory is also called streamline
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Fixed Points and Periodic Orbits
• A point x∈X is a fixed point if ϕ(t, x) = x for all t∈R
• x is a periodic point if there exist a T >0 such that ϕ(T, x) = x. The trajectory of a periodic point is called a
periodic orbit.
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Limit Sets
• Limit sets reveal the long-term behaviors of
vector fields, correspond to flow recurrence
• The limit sets are:
α(x)=∩t<0 cl(ϕ((−∞, t), x))
ω(x)=∩t>0 cl(ϕ((t, ∞), x))
point (or curve) reached after forwardintegration by streamline seeded at x
point (or curve) reached after
backward integration by streamline
seeded at x
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Repellor and Attractor Manifolds
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Invariant Sets
• An invariant set S ⊂ X satisfies ϕ(R,S)=S– A trajectory is an invariant set
– Fixed points and periodic orbits are invariant sets
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Fixed Points
We specifically consider first-order fixed points -> Jacobian is not degenerate
det 0 = %' − &� 1 0 -> Jacobian matrix is full rank
The eigenvalues of the Jacobian matrix λ0 = λ3 are
λ = 4'+,, + �56+,,
If both 4'+,, 7 0, the fixed point repels flow locally.
If both 4'+,, 8 0, the fixed point attracts flow locally.
If 4'+4', 8 0, it does both and is a saddle
If either4'+,, 1 0, the fixed point is called hyperbolic and stable.
If both 4'+,, = 0 and 56+,, 1 0, the fixed point is non-hyperbolic and unstable.
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Fixed Point Extraction
• Assume piecewise linear vector field. We
adopt cell-wise analysis
– Solve linear / quadratic equation to determine
position of critical point in cell
– Compute Jacobian at that position
– Compute eigenvalues
– If type is saddle, compute eigenvectors
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Poincaré Index
• Poincarè index I(ΓΓΓΓ, V) of a simple closed curve ΓΓΓΓ in the plane
relative to a continuous vector field is the number of the positive
field rotations while traveling along ΓΓΓΓ in positive direction.
[Tricoche Thesis2002]
• By continuity, always an integer• The index of a closed curve around multiple fixed points
will be the sum of the indices of the fixed points
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Poincaré Index
• Poincaré index: sources
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Poincaré Index
• Poincaré index: saddles
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Important Poincaré Indices
• Consider an isolated fixed point x0, there is a neighborhood N enclosing x0 such that there are no other fixed points in N or on the boundary curve ∂N– if I(∂N, V) =1, x0 is either a source or a sink;
– if I(∂N, V) =-1, x0 is a saddle.
• The Poincarè index of a fixed point free region is 0
• There is a combinatorial theory that shows
21
heI
−+=
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Sectors & Separatrices
• In the vicinity of a fixed point, there are
various sectors or regions of different flow
type:
– hyperbolic: paths do not ever reach fixed point.
– parabolic: one end of all paths is at fixed point.
– elliptic: all paths begin & end at fixed point.
• A separatrix is the bounding curve (or surface)
which separates these regions
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Sectors & Separatrices
Source: A topology simplification method for 2D vector fields. Xavier Tricoche, Gerik
Scheuermann, & Hans Hagen
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Sectors & Separatrices
Source: A topology simplification method for 2D vector fields. Xavier Tricoche, Gerik
Scheuermann, & Hans Hagen
21
heI
−+=
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Conley Index
• Poincarè index for a periodic orbit is 0
• There is an index, called Conley index that we use to classify
invariant sets.
• For an isolating block M, its Conley index is the homotopy type
of the quotient space M/L where L is the exit set (the subset of
the boundary of M consisting of all exit points).
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Conley Index Computation
• Conley index can be represented as the three Betti numbers
(β0, β1, β2) under our 2D setting. Note that β0 and β2 can NOT
be both 1.– An attracting fixed point (e.g. sink): (1,0,0)
– A repelling fixed point (e.g. source): (0,0,1)
– A saddle: (0,1,0)
– An attracting periodic orbit: (1,1,0)
– A repelling periodic orbit: (0,1,1)
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• Curve-type (1D) limit set
• Attracting / repelling behavior
• Poincaré map:
– Defined over cross section
– Map each position to next intersection with cross
section along flow
– Discrete map
– Cycle intersects at fixed point
– Hyperbolic / non-hyperbolic
Periodic Orbits
Fixed pointin the map
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Periodic Orbit Extraction
• Poincaré-Bendixson theorem:
– If a region contains a limit set and no critical
point, it contains a closed orbit
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Periodic Orbit Extraction
• Detect closed cell cycle
• Check for flow exit along boundary
• Find exact position with Poincaré map(fixed point)
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Vector Field Topology
• Vector field topology provides qualitative (structural) information of the underlying dynamics
• It usually consists of certain critical features and their connectivity, which can be expressed as a graph, e.g. vector field skeleton [Helman and
Hesselink 1989]
– Fixed points
– Periodic orbits
– Separatrices
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Topological Graph
• Three layers based on the Conley index
– If β0 =1, (A)ttractors: sinks, attracting periodic orbits
– If β2 =1, (R)epellers: sources, repelling periodic orbits
– Otherwise, (S)addles
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Vector Field Equivalence
• We can call two VFs equivalent by showing a diffeomorphism which maps integral curves from the first to the second and preserves orientation
• A VF is structural stable if any perturbation to that VF results in one which is structurally equivalent
• In particular, non-hyperbolic fixed points (such as centers) mean a VF is unstable because any arbitrarily small perturbation can change the fixed point to a hyperbolic one.
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Vector Field Visualization:
Computing Topology
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Vector Field Topology (Recall)
• Vector field topology provides qualitative (structural) information of the underlying dynamics
• It usually consists of certain critical features and their connectivity, which can be expressed as a graph, e.g. Entity Connection Graph (ECG) [Chen et al. 2007]
– Fixed points
– Periodic orbits
– Separatrices
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2D Vector Field Topology
• Differential topology
– Topological skeleton [Helman and Hesselink 1989; CGA91]
– Entity connection graph [Chen et al. TVCG07]
• Discrete topology
– Morse decomposition [Conley 78] [Chen et al. TVCG08, TVCG11a]
– PC Morse decomposition [Szymczak EuroVis11] [Szymaczak and Zhang
TVCG12][Szymaczak TVCG12]
• Combinatorial topology
– Combinatorial vector field [Forman 98]
– Combinatorial 2D vector field topology [Reininghaus et al. TopoInVis09, TVCG11]
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2D Vector Field Topology
• Differential topology
– Topological skeleton [Helman and Hesselink 1989; CGA91]
– Entity connection graph [Chen et al. TVCG07]
• Discrete topology
– Morse decomposition [Conley 78] [Chen et al. TVCG08, TVCG11a]
– PC Morse decomposition [Szymczak EuroVis11] [Szymaczak and Zhang
TVCG12][Szymaczak TVCG12]
• Combinatorial topology
– Combinatorial vector field [Forman 98]
– Combinatorial 2D vector field topology [Reininghaus et al. TopoInVis09, TVCG11]
Sink
Source
Saddle
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2D Vector Field Topology
• Differential topology
– Topological skeleton [Helman and Hesselink 1989; CGA91]
– Entity connection graph [Chen et al. TVCG07]
• Discrete topology
– Morse decomposition [Conley 78] [Chen et al. TVCG08, TVCG11a]
– PC Morse decomposition [Szymczak EuroVis11] [Szymaczak and Zhang
TVCG12][Szymaczak TVCG12]
• Combinatorial topology
– Combinatorial vector field [Forman 98]
– Combinatorial 2D vector field topology [Reininghaus et al. TopoInVis09, TVCG11]
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ECG Construction
• Two steps pipeline
– Extract fixed points and periodic orbits
– Compute connections between these features
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Fixed Point Extraction
• Cell-wise
– First, locate the cells that contain fixed points
• Using the unique characteristics of the Poincaré index
around a fixed point instead of solving a linear system
– Second, solve for the position of the fixed point
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Efficient Periodic Orbit Extraction
• Observation: periodic orbits are located in certain
regions of recurrent flow by the definitions of limit
sets.
• Basic idea: first extract these regions; then locate
periodic orbits inside only these regions.
• Advantage: 1) fast; 2) extract periodic orbits not
connected with fixed points; 3) extensible to
surface vector fields
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Finding Regions of Recurrence
[Kalies et al. 2006]
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Periodic Orbit Extraction
• No guarantees to find all the cycles
• In practice, it tends to work well
• And it is fast
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• Can detect embedded orbits
Synthetic field
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Extract Topology
– Fixed point extraction
– Periodic orbit identification
– Compute connections
• Separatrix computation (emitting from saddles)
• Other connectivity
– A source and a sink
– A source/sink with a periodic orbit
– A periodic orbit with other periodic orbit
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Compute Connections
• Fixed point and periodic orbit extraction
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Compute Connections
• Separatrices computation
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Compute Connections
• Repelling periodic orbit to a fixed point or
orbit
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Compute Connections
• Repelling periodic orbit to a fixed point or
orbit
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Compute Connections
• Attracting periodic orbit to a fixed point
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Applications (1)
• Feature-aware streamline placement
– First extract topology, then use it as the initial set
of streamlines to compute seeds for later
placement
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Applications (2)
• CFD simulation on gas engine
• Velocity extrapolated to the boundary
105K polygons
56 fixed points
9 periodic orbits
31.58s on analysis
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Applications (3)
• CFD simulation on diesel engine
• Velocity extrapolated to the boundary
886K polygons
226 fixed points
52 periodic orbits
29.15s on analysis
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Application (4)
• CFD simulation on cooling jacket
• Velocity extrapolated to the boundary
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Any Issues of This Method?
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Instability of ECG (1)
Case 1: different sampling
−−−=
1
))1()(1(),(
2rkrrrV θ
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Instability of ECG (2)
Case 2: noise in the data
−−=
)1(),(
xxkx
yyxV
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Instability of ECG (3)
Case 3: different numerical integration
schemes
−−
=)1(
),( 2y
xyyxV
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2D Vector Field Topology
• Differential topology
– Topological skeleton [Helman and Hesselink 1989; CGA91]
– Entity connection graph [Chen et al. TVCG07]
• Discrete topology
– Morse decomposition [Conley 78] [Chen et al. TVCG08, TVCG11a]
– PC Morse decomposition [Szymczak EuroVis11] [Szymaczak and Zhang
TVCG12][Szymaczak TVCG12]
• Combinatorial topology
– Combinatorial vector field [Forman 98]
– Combinatorial 2D vector field topology [Reininghaus et al. TopoInVis09, TVCG11]
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Morse Decomposition Results
• Stable
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Morse Decomposition Results
• Stable
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Morse Decomposition Results
• Stable
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Morse Decomposition
• A Morse decomposition of surface X for the flow is a
finite collection of disjoint compact invariant sets,
called Morse sets.
• The computation result of a Morse decomposition is a
directed graph called Morse connection graph (MCG)
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Morse Connection Graph (MCG)
• An MCG
M(X,ϕ,P,>)={M(p)|p∈(P,>)}– is an acyclic directed graph, whose nodes P
are Morse sets, the set of directed edges is a
strict partial order >
– such that for any x∉ ∪p∈P M(p), there exist
p>q in P and α(x) ⊂ M(p) and ω(x) ⊂ M(q)
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77
MCG Visualization
MCG
– Always exists
– Not unique
– Use Morse neighborhoods that are triangle strips to
visualize the corresponding Morse sets
MCG�ECG- ECG can be computed from MCG
[Chen et al. 2007]
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Flow
combinatorialization
Strongly connected
component extracting
Constructing a
quotient graph
Computing MCG
A Pipeline of Morse Decomposition
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79
Flow
combinatorialization
Strongly connected
component extracting
Constructing a
quotient graph
Computing MCG
A Pipeline of Morse Decomposition
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80
Geometry Based Flow Combinatorialization
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τ-Map Based Flow Combinatorialization
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Why τ-Map is Better?
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Why Is MCG Stable?
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Results – Analytic Data
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Results – Gas Engine
Geometry-based ττττ=0.3ττττ=0.1
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Results – Diesel Engine
ττττ=0.3Geometry-based
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Results - Timing
Dataset name Number of triangles
Number of Morse sets
Time for flow combinatorialization (seconds)
Time for computing MCG (seconds)
Gas engine (ττττ =0.1) 26,298 50 27.8 7.9
Gas engine (ττττ =0.3) 26,298 57 75.4 1.2
Diesel engine (ττττ =0.3) 221,574 200 1101.3 37.7
All the results are obtained in a 3.6 GHz PC with 3GB RAM.
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2D Vector Field Topology
• Differential topology
– Topological skeleton [Helman and Hesselink 1989; CGA91]
– Entity connection graph [Chen et al. TVCG07]
• Discrete topology
– Morse decomposition [Conley 78] [Chen et al. TVCG08, TVCG11a]
– PC Morse decomposition [Szymczak EuroVis11] [Szymaczak and Zhang
TVCG12][Szymaczak TVCG12]
• Combinatorial topology
– Combinatorial vector field [Forman 98]
– Combinatorial 2D vector field topology [Reininghaus et al. TopoInVis09, TVCG11]
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Simplification
Reduce flow complexity so that people can focus
on the more important structure
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Multi-level feature representation
90
Refinement
Automatic vector field simplification
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Vector Field Data Compression
Source: [Theisel et al. Eurographics 2003]
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Additional Reading
• Frits H. Post, Benjamin Vrolijk, Helwig Hauser, Robert S.
Laramee and Helmut Doleisch. The State of the Art in Flow
Visualization: Feature Extraction and Tracking. Computer
Graphics Forum, 22 (4): pp. 1-17, 2003.
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Acknowledgment
Thanks materials from
• Prof. Eugene Zhang, Oregon State University
• Prof. Joshua Levine, Clemson University
• Prof. Zhanping Liu, Kenturky State University