Line Integral Convolution
- A global method to visualize vector fields
Scientific Visualization
Flow Visualization
computer graphics & visualization
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Line Integral Convolution
- Line Integral Convolution (LIC)
- Visualize dense flow fields by imaging its integral curves
- Cover domain with a random texture (so called ‚input
texture‘
texture‘, usually stationary white noise)
- Blur (convolve) the input texture along the path lines
computer graphics & visualization
Line Integral Convolution
- Idea of Line Integral Convolution (LIC)
- Global visualization technique
- Start with random texture
- Smear out along stream lines
using a specified filter kernel
- Look of 2D LIC images
- Intensity distribution along path lines shows high
correlation
- No correlation between neighboring path lines
Course WS 05/06 – Scientific Visualization
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Line Integral Convolution
- Algorithm for 2D LIC
- Let t → Φ0(t) be the path line containing the point (x
(x0,y0)
- T(x,y) is the randomly generated input texture
- Compute the pixel intensity as:
L
convolution with
kernel
I ( x0 , y 0 ) = ∫ −L k (t ) ⋅ T (φ0 (t )) dt
- Kernel:
- Finite support [-L,L]
- Normalized
- Often simple box filter
- Often symmetric (isotropic)
kernel
k(t)
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Line Integral Convolution
- Algorithm for 2D
LIC
- Convolve a random
texture along the
stream lines
∫ k (t )dt = 1
L
1
-L
-L
L
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
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1
Line Integral Convolution
Line Integral Convolution
- Fast LIC
- Problems with LIC
Input noise
Vector field
Convolution
- Idea:
∫ k (t )dt = 1
L
kernel
k(t)
1
-L
-L
- New stream line is computed at each pixel
- Convolution (integral) is computed at each pixel
- Slow
- Compute very long stream lines
- Reuse these stream lines for many different pixels
- Incremental computation of the convolution integral
L
Final image
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Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Line Integral Convolution
- Next position I1
- Discretization of convolution integral
TL
- Summation
T0 x0
I0 =
L
1
2L +1
∑T
i = -L
i
x-L
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
kernel
- Stream line x-m,..., x0,..., xn with m,n ≥ L
- Given texture values T-m,...,T0,...,T
,...,Tn
- What are results of convolution: I-m+L,..., I0,..., In-L?
kernel):
- For box filter (constant
L
∑T
i
- Incremental integration:
I j +1 − I j =
∑ (T
L
1
2L +1
i = -L
i + j +1
− Ti + j ) =
1
2L +1
(T
L + j +1
− T- L + j )
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1
2L +1
T0 x0
T-L
x-L
I0 =
L
1
2L +1
∑T
i
i = -L
I1 = I 0 + (TL +1 − T-L ) /(2 L + 1)
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Line Integral Convolution
- Fast LIC: incremental integration for constant
i = -L
k (t ) =
x-m
T-m
T-L
x-m
T-m
1
2L +1
Tn
xn
xL
Assumption: box filter
Assumption: box filter
I0 =
TL
Tn
xn
xL
1
2L +1
computer graphics & visualization
Line Integral Convolution
- Fast LIC: incremental integration
- Fast LIC: incremental integration
k (t ) =
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Line Integral Convolution
- Fast LIC: Algorithm
-
Data structure for output: Luminance/Alpha image
- Luminance = graygray-scale output
- Alpha = number of streamline passing through that pixel
For each pixel p in output image
If (Alpha(p
(Alpha(p) < #min) Then
Initialize streamline computation with x0 = center of p
Compute convolution I(x0)
Add result to pixel p (increment alpha)
For m = 1 to Limit M
Incremental convolution for I(xm) and I(x-m)
Add results to pixels containing xm and x-m (incr. alpha)
End for
End if
End for
Normalize all pixels according to Alpha
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
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2
Line Integral Convolution
- Summary:
- Dense representation of flow fields
- Convolution along stream lines → correlation along
stream lines
- For 2D and 3D flows
- Stationary flows
- Extensions:
-
Line Integral Convolution
- Oriented LIC (OLIC):
- Visualizes orientation (in addition to direction)
- Sparse texture
- Anisotropic convolution kernel
- Acceleration: integrate individual drops and compose
them to final image
Unsteady flows
Animation
Texture advection
1
anisotropic
convolution kernel
-l
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Line Integral Convolution
- Oriented LIC (OLIC)
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Line Integral Convolution
- LIC - Line Integral Convolution
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l
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Line Integral Convolution
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Line Integral Convolution
Vector field =
gradient fields of image
Lic – Applications: Motion blurring by means of biased
triangle filter and variable length LIC
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
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3
Line Integral Convolution
Line Integral Convolution
Lic-Applications: length of convolution integral
with respect to magnitude of vector field
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Lic and color coding of velocity magnitude
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Line Integral Convolution
- Summary:
- Dense representation of flow fields
- Convolution along stream lines → correlation along
stream lines
- For 2D and 3D flows
- Stationary flows
- Extensions:
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Vector Field Topology
- Idea: Do not draw “all”
all” streamlines, but only the
“important”
important” ones
- Show only topological skeletons
- Connection of critical points
- Critical points:
- Points where the vector field vanishes: v = 0
- Points where the vector magnitude goes to zero and the
- Unsteady flows
- Animation
- Texture advection
vector direction is undefined
- Sources, sinks, …
- The critical points are connected
to divide the flow into regions
with similar properties
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
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Vector Field Topology
- How to find critical points
- Cell search (for cells which contain critical points)
points)
- Mark vertices by (+,+), (–, –), (+, –) or (–,+), depending on the
signs of vx and vy
- Determine cells that have vertices where the sign changes in both
both
components
–> these are the cells that contain critical points
- Find the critical points by interpolation
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Vector Field Topology
- How to find critical points within a cell
- Determine intersection of isolines (c=0) of the two
components
- Two bilinear equations (or one quadratic equation) to be
solved
- Critical points are the (real) solutions within the cell boundaries
boundaries
(+,+)
(+,+)
(yes,yes)
(+,–)
(no,yes)
Vy=
0
Vx=
0
(–,–)
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
(+,–)
(+,–)
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4
Vector Field Topology
- How to find critical points (cont.)
Vector Field Topology
- Taylor expansion for the velocity field around a
- How to find critical points within simplex?
critical point rc with (V(r
V(rc)= 0):
- Based on barycentric interpolation
- Solve analytically
v(r ) = v(rc ) + ∇v ⋅ (r − rc ) + O(r − rc )2
≈ J ⋅ (r − rc )
- Alternative method:
- Iterative approach based on 2D / 3D nested intervals
- Recursive subdivision into 4 / 8 subregions if critical point is
contained in cell
- Divide Jacobian into symmetric and antiantisymmetric parts
J = Js + Ja = ((J
((J + JT) + (J
(J - JT))/2
Js = (J
(J + JT)/2
Ja = (J
(J - JT)/2
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
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Vector Field Topology
- The symmetric part can be solved to give real
eigenvalues R and real eigenvectors
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Vector Field Topology
- AntiAnti-symmetric part
J ⋅ d = (J − J )⋅ d =
a
J s rs = R rs
R = R 1, R 2 , R 3
- Eigenvectors rs are an orthonormal set of vectors
- Describes change of size along eigenvectors
- Describes flow into or out of region around critical point
computer graphics & visualization
1
2
T
1
2
⎛ 0
⎜ ∂v
⎜ ∂xy − ∂∂vyx
⎜ ∂v z ∂v x
⎜ ∂x − ∂z
⎝
∂v x
∂y
∂v y ∂v x
∂x ∂z
∂v y
∂z
∂v y
−
0
∂v z
∂y
∂z
− ∂∂vxz ⎞⎟
− ∂∂vyz ⎟ ⋅ d =
⎟
0 ⎟
⎠
1
2
(∇ × v ) × d
- Describes rotation of difference vector d = (r
(r - rc)
- The antianti-symmetric part can be solved to give imaginary
eigenvalues I
J a ra = Ira
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Vector Field Topology
- 2D structure: eigenvalues are (R
(R1, R2) and (I
(I1,I2)
Repelling node
R1, R2 > 0
I1,I2 = 0
Repelling focus
R1, R2 > 0
I1,I2 ≠ 0
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Vector Field Topology
- 2D structure: eigenvalues are (R
(R1, R2) and (I
(I1,I2)
Attracting node
R1, R2 < 0
I1,I2 = 0
Saddle point
R1 * R2 < 0
I1,I2 = 0
Attracting focus
R1, R2 < 0
I1,I2 ≠ 0
Center
R1, R2 = 0
I1,I2 ≠ 0
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I = I1, I 2 , I 3
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5
Vector Field Topology
- Also in 3D
Vector Field Topology
- Mapping to graphical primitives: streamlines
- Some examples
- Start streamlines close to critical points
- Initial direction along the eigenvectors
- End particle tracing at
- Other “real”
real” critical points
- Interior boundaries: attachment or detachment points
- Boundaries of the computational domain
Attracting node
R1, R2 , R3 < 0
I1,I 2,I3 = 0
Center
R1, R2 = 0, R3 > 0
I1,I2 ≠ 0, ,I3 = 0
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
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Vector Field Topology
- Example of a topological graph of 2D flow field
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Vector Field Topology
- Further examples of topologytopology-guided streamline
positioning
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Vector Field Topology
- Summary:
- Draw only relevant streamlines (topological skeleton)
- Partition domain in regions with similar flow features
- Based on critical points
- Good for 2D stationary flows
- Instationary flows?
- 3D?
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
3D Vector Fields
- Most algorithms can be applied to 2D and 3D
vector fields
- Main problem in 3D: effective mapping to
graphical primitives
- Main aspects:
- Occlusion
- Amount of (visual) data
- Depth perception
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
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6
3D Vector Fields
- Approaches to occlusion issue:
- Sparse representations
- Animation
- Color differences to distinguish separate objects
- Continuity
3D Vector Fields
- Missing continuity
- Reduction of visual data:
- Sparse representations
- Clipping
- Importance of semisemi-transparency
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
3D Vector Fields
- Color differences to identify connected structures
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
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3D Vector Fields
- Reduction of visual data
- 3D LIC
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3D Vector Fields
- Improving spatial perception:
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
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3D Vector Fields
- Illumination
- Depth cues
-
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Perspective
Occlusion
Motion parallax
Stereo disparity
Color (atmospheric, fogging)
- Halos
- Orientation of structures
by shading (highlights)
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
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7
3D Vector Fields
- Illuminated streamlines [Zö
[Zöckler et al. 1996]
1996]
- Model: streamline is made of thin cylinders
- Problem
3D Vector Fields
.)
- Illuminated streamlines (cont
(cont.)
- Light vector is split in tangential and normal parts
V ⋅ R = V ⋅ (L N − LT ) = V ⋅ ((L ⋅ N)N − (L ⋅ T)T )
- No distinct normal vector on surface
- Normal vector in plane perpendicular to tangent: normal
space
- Cone of reflection vectors
= (L ⋅ N)(V ⋅ N) − (L ⋅ T)(V ⋅ T)
= 1 − (L ⋅ T) 2 1 − (V ⋅ T) 2 − (L ⋅ T)(V ⋅ T)
= f ((L ⋅ T), (V ⋅ T) )
- Idea: Represent f() by 2D texture
- Access prepre-computed f() during rendering
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
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3D Vector Fields
- Illuminated stream lines
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
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Particle Tracing on Grids
- How do we handle curvilinear grids?
- C-space (computational space) vs. PP-space
(physical space)
x
s
r00
q
r
x
0,1
0,0
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computer graphics & visualization
- Points by Φ
- Vectors by J
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Particle Tracing on Grids
- Transformation of points:
- From CC-space to PP-space: r = Φ(s)
- From PP-space to CC-space: s = Φ-1(r)
- From CC-space to PP-space: v = J ⋅ u
- From PP-space to CC-space: u = J-1 ⋅ v
- J is Jacobi matrix:
x
s
Φ or J
⎛ ∂Φx
⎜
∂p
J=⎜
⎜ ∂Φy
⎜
⎝ ∂p
r00
q
r
x
1,0
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
- Transformation of vectors:
Φ-1 or J-1
0,1
p
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Particle Tracing on Grids
- Particle tracing can be done in CC-space or PP-space
- Transformation of
0,0
1,0
P-space
C-space
P-space
p
∂Φx
∂q
∂Φy
⎞
⎟
⎟
⎟
⎟
∂q ⎠
(2D case)
C-space
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
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8
Particle Tracing on Grids
- Particle tracing in CC-space
- Algorithm
Particle Tracing on Grids
- Transformation of points from CC-space to PP-space
Select start point r in PP-space (seed point)
Find PP-space cell
Transform start point to CC-space s
While (particle in domain) do
Transform v → u at vertices (P → C)
Interpolate u in CC-space
Integrate to new position s in CC-space
If outside current cell then
Clipping
Find new cell
Endif
Transform to P(C → P)
P-space s → r
Draw line segment between latest
particle positions
Endwhile
- r = Φ(s):
- Bilinear (in 2D) or trilinear (in 3D) interpolation between
point location
coordinates of the cell’
cell’s vertices
- Transformation of vectors from
from PP-space to CC-space
- u = J-1 ⋅ v
- Needs inverse of the Jacobian
interpolation
integration
- Numerical computation of elements of the Jacobi
point location
matrix:
- Backward differences (bd
(bd))
- Forward differences (fd
(fd))
- Central differences (cd
(cd))
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Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Particle Tracing on Grids
- Computing Jacobi matrices on curvilinear grids:
- Per cell (central differences): fast, inaccurate
- Per vertex (central differences): slower, higher accuracy
- 4 (in 2D) or 8 (in 3D) per vertex (backward and forward
differences): timetime-consuming, compatible with bi /
trilinear interpolation, accurate
Jcd
Jbdfd
Jcd
Jbdbd
computer graphics & visualization
Particle Tracing on Grids
- Particle tracing in PP-Space
- Algorithm
Select start point r in PP-space (seed point)
point location
Find PP-space cell and local coords
While (particle in domain) do
Interpolate v in PP-space
interpolation
integration
point location
Integrate to new position r in PP-space
Jcd
Jcd
Once per cell
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Jcd
Once per vertex
Jfdbd
Jfdfd
4 per vertex
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Find new position
Draw line segment between latest
particle positions
Endwhile
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Particle Tracing on Grids
- Main problem: Point location / local coordinates in
cell
- Given is point r in PP-space
- What is corresponding position s in CC-space?
- Coordinates s are needed for interpolating v
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Particle Tracing on Grids
- StencilStencil-walk on curvilinear grids
- Given: rold, rnew, sold , Φ(sold) = rold
- Determine snew
Φ-1 or J-1
x
- Solution: StencilStencil-walk algorithm [Bunnig ’89]
- Iterative technique
- Needs Jacobi matrices
s old
Φ or J
q
0,0
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1,0
x
r00
s new
x x
0,1
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rold
rnew
P-space
p
C-space
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9
Particle Tracing on Grids
- StencilStencil-walk on curvilinear grids (cont.
(cont.))
- Transformation of distances Δs = snew–sold , Δr = rnew–rold
- Taylor expansion:
- Therefore
Δr = rnew − rold = Φ( snew ) − Φ( s old ) = JΦ ( s old ) ⋅ Δs + K
Δs ≈ [ JΦ ( s old )]−1 Δr
Φ-1 or J-1
Φ or J
q
0,1
0,0
1,0
Δs
-
r00
s new
x x
-
Δr
x
x
s old
Particle Tracing on Grids
- Idea of stencilstencil-walk:
rold
rnew
Guess start point in CC-space
Compute difference between corresponding position and
target position in PP-space
Improve position in CC-space
Iterate
-1
-1
Φ or J
Δr
x
x
s old
Φ or J
r00
q
rold
r
s
p
x x
0,1
C-space
0,0
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1,0
Δs
P-space
p
C-space
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Particle Tracing on Grids
- StencilStencil-walk algorithm
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Particle Tracing on Grids
- StencilStencil-walk algorithm
Given: target position rtarget in PP-space
initialize : r := rold ; p := p old ;
Δp > ε
while
do
required accuracy ε in CC-space
Guess start point s in CC-space
Do
Transformation to PP-space r = Φ (s)
Difference in PP-space Δr = rtarget – r
Transformation to CC-space Δs = J-1 Δr
If (|Δs| < ε) then exit
s = s + Δs
If (s
(s outside current cell)
cell) then
Set s = midpoint of corresponding neighboring cell
Endif
Repeat
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new
new
P-space
Δ r = rnew − r ;
Δp = [ J Φ (p )]−1 Δ r ;
p = p + Δp;
endwhile
r = Φ (p ) ;
cell_index ( rnew ) = ( ⎣Px ⎦, ⎣Py ⎦ ) ;
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Particle Tracing on Grids
- High convergence speed of stencil walk
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Particle Tracing on Grids
V3
- Inverse
- Typically 33-5 iteration steps in a cell
- Interpolation of velocity in PP-space approach:
distance
weighting
d3
computer graphics & visualization
V2
d2
- Bi / trilinear within cell, based on local coordinates p
- Alternative method: inverse distance weighting → no
rp
stencil walk needed
d1
d0
V1
v =
∑
i
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V0
w ivi ,
where
d i = ri − r p
1
and
if (d i < ε) then w i = 1 ; w j≠i = 0
wi =
d i2
∑
1
j
d
2
i
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10
Particle Tracing on Grids
- Important properties of CC-space integration:
+ Simple incremental cell search
+ Simple interpolation
- Complicated transformation of velocities / vectors
- Important properties of PP-space integration:
+ No transformation of velocities / vectors
- Complicated point location (stencil walk) for bi / trilinear
interpolation
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11