6. Flow Field Topology · Flow Field Topology. 320581: Advanced Visualization 613 Visualization and...

Visualization and Computer Graphics LabJacobs University

6. Flow Field Topology

320581: Advanced Visualization 613


Motivation

• An abstraction of flow field behavior is to partition the domain into areas of uniform flow behavior.



Flow Topology

sinksource

saddle

critical points

separating structure



Critical points

• In order to determine the separating structure, we need to determine the critical points.

• Critical points are points of vanishing flow magnitude.• In order to characterize the critical points, one

needs to look into the Jacobians.• Assuming linear interpolation, we only need to look

into first-order critical points.


6.1 2D Critical Points



Notations

• Let denote the Jacobian of flow f at point p.

• Let λ1 and λ2 denote the eigenvalues of the Jacobian.• These are complex eigenvalues.• Let Re(λ) and Im(λ) denote the real and the imaginary

part of the complex number λ.



Repulsion

• Re(λ1) > 0 and Re(λ2) > 0:– Im(λ1) = Im(λ2) = 0:

Repelling node

– Im(λ1) = - Im(λ2) ≠ 0 (rotational component)Repelling focus



Attraction

• Re(λ1) < 0 and Re(λ2) < 0:– Im(λ1) = Im(λ2) = 0:

Attracting node

– Im(λ1) = - Im(λ2) ≠ 0 (rotational component)Attracting focus



Saddle point and center

• Re(λ1) < 0 and Re(λ2) > 0:Saddle point

• Re(λ1) = Re(λ2) = 0:Center



Results


6.2 3D Critical Points



3D critical points• Let λ1, λ2, and λ3 denote the eigenvalues of the Jacobian.

• Im(λ1) = Im(λ2) = Im(λ3) = 0 – Re(λ1), Re(λ2), Re(λ3) > 0:

Repelling 3D node– Re(λ1), Re(λ2), Re(λ3) < 0:

Attracting 3D node– Re(λ1), Re(λ2), Re(λ3) have different signs:

3D saddleExample:

2D repelling node3D saddle



3D critical points

• Im(λ1) = 0, Im(λ2) = Im(λ3) ≠ 0 – Re(λ2), Re(λ3) > 0:

Repelling 3D spiral– Re(λ2), Re(λ3) < 0:

Attracting 3D spiral– Re(λ2), Re(λ3) have different signs:

3D spiral saddle


7. Diffusion Tensor Visualization


7.1 Diffusion Tensor Imaging



Motivation

• Goal:– Elucidating internal structure within a human brain

• Application:– Brain surgery planning



Approach

• Nervous tissue consists of fibers.• Fibers constrain the diffusion of water molecules

along the direction of the fibers.• To understand the orientation of the fibers at a point

p, one detects the direction of fastest diffusion at p.

fiberdiffusion



Measuring

• Water molecules can be measured using magnetic resonance imaging (MRI).

• Diffusion can be measured using MRI by applying magnetic fields in discrete directions (so-called diffusion gradients).

• If the directions co-align with a Cartesian system, we get a vector field (one scalar in each direction).

• Since measurements are subject to a lot of noise, more directions are desirable, typically 6 or 12.



Mathematic representation

• The directional diffusion in all directions is captured by a tensor.

• The tensor is given in form of a 3x3 positive symmetric matrix D.

• The entries are computed using a least-squares approach.


7.2 Color Coding



Color coding

• A common way to visualize the data on 2D slices is a color coding of the direction of maximum diffusion.

• Diffusion tensor D has 3 real eigenvalues λ1 > λ2 > λ3, where eigenvector e1 to the largest eigenvalue λ1represents the direction of maximum diffusion.

• Using an RGB color cube, a mapping of the given global Cartesian coordinate system to the RGB axes can be established.



Color coding



Anisotropy

• Since one is interested in the fibers, the resulting image is more comprehensible, if only fibers are color coded.

• Fibers can be detected by looking into isotropy.• Fibers represent anisotropic regions, i.e., the

diffusion in one direction is larger than in the others.• Hence, for color coding, isotropic regions are omitted

by just coloring them black (or transparent).



Fractional Anisotropy

• A typical measure for anisotropy is the so-called fractional anisotropy (FA).

• It is based on the observation that in isotropic regions the 3 eigenvalues are approximately equal.

• The fractional anisotropy is defined by

with being the average of the eigenvalues.



Color coding of anisotropic regions

• Using the definition of fractional anisotropy only those values with a fractional anisotropy larger than a certain threshold are color coded.

• Of course, one can apply this idea to the entire volume data using direct volume rendering and appropriate transfer functions to visualize it.



Color coding of anisotropic regions


7.3 Elliptic Glyphs



Observation

• The diffusion tensor glyphs have a 1-to-1 mapping to the geometric shape of an ellipsoid.

• The 3 eigenvectors represent the 3 axes of the ellipsoid.

• The 3 (positive) eigenvalues represent the 3 radii of the ellipsoid along the 3 axes.



Elliptic shapes

• Let λ1 > λ2 > λ3 be the three eigenvaluesand e1, e2, and e3 be the respective eigenvectors.

• One distinguishes 3 cases of elliptic shapes:– Linear anisotropic diffusion– Planar anisotropic diffusion– Isotropic diffusion



Linear anisotropic diffusion

• Case 1: λ1 >> λ2, λ3

Prolate case (cigar-shaped)



Planar anisotropic diffusion

• Case 2: λ1 ≈ λ2 >> λ3

Oblate case (pancake-shaped)



Isotropic diffusion

• Case 3: λ1 ≈ λ2 ≈ λ3

Spherical case (ball-shaped)



Measurements

• The 3 cases can be measured using the 3 coefficients:– Linear anisotropic diffusion

– Planar anisotropic diffusion

– Isotropic diffusion

• As cl + cp + cs = 1, the 3 coefficients parameterize a barycentric space.



Elliptic tensor glyphs: barycentric space



Glyph-based tensor visualization

• The tensor field can be visualized by rendering the ellipsoids at the respective position in space.

• This is called glyph-based visualization.• The ellipsoids are called glyphs• Glyphs can be colored according to the introduced

color coding scheme.



Glyph-based tensor visualization


7.4 Superquadric Glyphs



Motivation

• Elliptic glyphs have the problems that depending on the viewing angle significantly different ellipsoids look identical.



Elliptic glyphs: worst case scenario



Superqradrics• Idea: Use a glyph representation that changes the shape

with varying tensor properties.• Employ superquadrics as glyphs.• A superquadric is defined implicitly by

• For α=β=1, qz(x,y,z)=0 defines a quadric.• Note that the representation is not symmetric with

respect to its parameterization, i.e., with respect to permutation of the axes.

• qz has a rotational symmetry with respect to the z-axis.• Analogously, we define qx and qy.



Superquadrics

beta

alpha



Barycentric space

• Use a subspace of the space of superquadrics to define a barycentric space for tensor glyphs.

• We have to define glyphs the 3 cases of– linear anisotropic diffusion,– planar anisotropic diffusion, and – isotropic diffusion

and parameterize them such that a barycentric space is spanned.



Linear anisotropic case

• Use a long cylinder with the following set-up:



Planar anisotropic case

• Use a flat cylinder with the following set-up:

Note that the orientation of the cylinder is being flipped when comparing to the linear anisotropic case.



Isotropic case

• Use a sphere with the following set-up:

This is the same as for elliptic glyphs.



Barycentric space• Using these 3 shapes a barycentric space can be spanned

by defining the tensor glyphs as

where γ is a parameter to tune the sharpness of the cylinders’ edges.In particular, for γ = 0, we get the elliptic glyphs, again.



Superquadric tensor glyphs: barycentric space



Worst case scenario revisisted



Ellipsoids



Superquadrics



Superquadric glyphs with optimized spacing

6. Flow Field Topology · Flow Field Topology. 320581: Advanced Visualization 613 Visualization and...

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