Review of Texture-based Methods - UHchengu/Teaching/Fall2012/Lecs/Lec_11_12.pdf · Review of Texture-based Methods ... locate a set of pixels hit by the streamline ... det 0 =%’−&10

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

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)

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

IBFV

Image source: van Wijk

http://www.win.tue.nl/~vanwijk/ibfv/

http://www.win.tue.nl/

Texture-based Methods on Surfaces

Surface LIC (Detlev Stalling, ZIB, Germany)

IBFVS ISA

Volumetric Texture

7

http://cs.swan.ac.uk/~csbob/te

aching/csM07-vis/

Arrows vs. Streamlines vs. Textures

Streamlines: selective

Arrows: simple

Textures: 2D-filling

http://cs.swan.ac.uk/

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.

Vector Field Visualization:

Feature-based

What features are in flows

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

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

Helmholtz decomposition

� = ∇" + � × #

Hodge decomposition

� = �" + � × # + $

Curl (or rotation) free Divergence free

Curl (or

rotation) free

Divergence

freeHarmonic

Helmholtz decomposition

curl-free divergence-freeneither

Vortices

Source:

http://www.onera.fr/cahierdelabo/english/aerod_ind03.htm

Post et al. STAR report 2003

Post et al. STAR report 2003

http://www.onera.fr/cahierdelabo/english/aerod_ind03.htm

Separation Flows

Separation flow on delta wing surface [Tricoche et al. AIAA 2004]

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.

(�*, *)

(�+, +)

(�,, ,)

(��*, �*)

(��+, �+)

(��,, �,)

- �, =� ��

=%� + & + �� + ' + �

��*�*

=%�* + &* + ��* + '* + �

Assume a piecewise linear vector field

We have

……

Solve for a linear system to get the coefficients

Vector Field Topology:

Introduction

Examples

• 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

Let us focus on steady vector fields at this moment

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

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

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.

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

Repellor and Attractor Manifolds

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

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.

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

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

Poincaré Index

• Poincaré index: sources

Poincaré Index

• Poincaré index: saddles

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

−+=

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


Source: A topology simplification method for 2D vector fields. Xavier Tricoche, Gerik

Scheuermann, & Hans Hagen


Source: A topology simplification method for 2D vector fields. Xavier Tricoche, Gerik

Scheuermann, & Hans Hagen

21

heI

−+=

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).

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)

• 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

Periodic Orbit Extraction

• Poincaré-Bendixson theorem:

– If a region contains a limit set and no critical

point, it contains a closed orbit


• Detect closed cell cycle

• Check for flow exit along boundary

• Find exact position with Poincaré map(fixed point)

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

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

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.

Vector Field Visualization:

Computing Topology

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

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












ECG Construction

• Two steps pipeline

– Extract fixed points and periodic orbits

– Compute connections between these features

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

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

Finding Regions of Recurrence

[Kalies et al. 2006]


• No guarantees to find all the cycles

• In practice, it tends to work well

• And it is fast

• Can detect embedded orbits

Synthetic field

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

Compute Connections

• Fixed point and periodic orbit extraction

Compute Connections

• Separatrices computation

Compute Connections

• Repelling periodic orbit to a fixed point or

orbit

Compute Connections

• Repelling periodic orbit to a fixed point or

orbit

Compute Connections

• Attracting periodic orbit to a fixed point

Applications (1)

• Feature-aware streamline placement

– First extract topology, then use it as the initial set

of streamlines to compute seeds for later

placement

Applications (2)

• CFD simulation on gas engine

• Velocity extrapolated to the boundary

105K polygons

56 fixed points

9 periodic orbits

31.58s on analysis

Applications (3)

• CFD simulation on diesel engine


886K polygons

226 fixed points

52 periodic orbits

29.15s on analysis

Application (4)

• CFD simulation on cooling jacket


Any Issues of This Method?

Instability of ECG (1)

Case 1: different sampling

−−−=

1

))1()(1(),(

2rkrrrV θ


Case 2: noise in the data

−−=

)1(),(

xxkx

yyxV


Case 3: different numerical integration

schemes

−−

=)1(

),( 2y

xyyxV












Morse Decomposition Results

• Stable


• Stable


• Stable

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)

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)

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]

Flow

combinatorialization

Strongly connected

component extracting

Constructing a

quotient graph

Computing MCG

A Pipeline of Morse Decomposition

79

Flow

combinatorialization

Strongly connected

component extracting

Constructing a

quotient graph

Computing MCG

A Pipeline of Morse Decomposition

80

Geometry Based Flow Combinatorialization

τ-Map Based Flow Combinatorialization

Why τ-Map is Better?

Why Is MCG Stable?

Results – Analytic Data

Results – Gas Engine

Geometry-based ττττ=0.3ττττ=0.1

Results – Diesel Engine

ττττ=0.3Geometry-based

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.












Simplification

Reduce flow complexity so that people can focus

on the more important structure

Multi-level feature representation

90

Refinement

Automatic vector field simplification

Vector Field Data Compression

Source: [Theisel et al. Eurographics 2003]

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.

Acknowledgment

Thanks materials from

• Prof. Eugene Zhang, Oregon State University

• Prof. Joshua Levine, Clemson University

• Prof. Zhanping Liu, Kenturky State University

Documents

Review of Texture-based Methods - UHchengu/Teaching/Fall2012/Lecs/Lec_11_12.pdf · Review of Texture-based Methods ... locate a set of pixels hit by the streamline ... det 0 =%’−&10