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Visualization and Computer Graphics LabJacobs University
6. Flow Field Topology
320581: Advanced Visualization 613
Visualization and Computer Graphics LabJacobs University
Motivation
• An abstraction of flow field behavior is to partition the domain into areas of uniform flow behavior.
320581: Advanced Visualization 614
Visualization and Computer Graphics LabJacobs University
Flow Topology
sinksource
saddle
critical points
separating structure
320581: Advanced Visualization 615
Visualization and Computer Graphics LabJacobs University
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.
Visualization and Computer Graphics LabJacobs University
6.1 2D Critical Points
320581: Advanced Visualization 617
Visualization and Computer Graphics LabJacobs University
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.
320581: Advanced Visualization 618
Visualization and Computer Graphics LabJacobs University
Repulsion
• Re(λ1) > 0 and Re(λ2) > 0:– Im(λ1) = Im(λ2) = 0:
Repelling node
– Im(λ1) = - Im(λ2) ≠ 0 (rotational component)Repelling focus
320581: Advanced Visualization 619
Visualization and Computer Graphics LabJacobs University
Attraction
• Re(λ1) < 0 and Re(λ2) < 0:– Im(λ1) = Im(λ2) = 0:
Attracting node
– Im(λ1) = - Im(λ2) ≠ 0 (rotational component)Attracting focus
320581: Advanced Visualization 620
Visualization and Computer Graphics LabJacobs University
Saddle point and center
• Re(λ1) < 0 and Re(λ2) > 0:Saddle point
• Re(λ1) = Re(λ2) = 0:Center
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Visualization and Computer Graphics LabJacobs University
Results
Visualization and Computer Graphics LabJacobs University
6.2 3D Critical Points
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Visualization and Computer Graphics LabJacobs University
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
320581: Advanced Visualization 624
Visualization and Computer Graphics LabJacobs University
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
Visualization and Computer Graphics LabJacobs University
7. Diffusion Tensor Visualization
Visualization and Computer Graphics LabJacobs University
7.1 Diffusion Tensor Imaging
320581: Advanced Visualization 627
Visualization and Computer Graphics LabJacobs University
Motivation
• Goal:– Elucidating internal structure within a human brain
• Application:– Brain surgery planning
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Visualization and Computer Graphics LabJacobs University
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
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Visualization and Computer Graphics LabJacobs University
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.
320581: Advanced Visualization 630
Visualization and Computer Graphics LabJacobs University
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.
Visualization and Computer Graphics LabJacobs University
7.2 Color Coding
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Visualization and Computer Graphics LabJacobs University
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.
320581: Advanced Visualization 633
Visualization and Computer Graphics LabJacobs University
Color coding
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Visualization and Computer Graphics LabJacobs University
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).
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Visualization and Computer Graphics LabJacobs University
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.
320581: Advanced Visualization 636
Visualization and Computer Graphics LabJacobs University
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.
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Visualization and Computer Graphics LabJacobs University
Color coding of anisotropic regions
Visualization and Computer Graphics LabJacobs University
7.3 Elliptic Glyphs
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Visualization and Computer Graphics LabJacobs University
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.
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Visualization and Computer Graphics LabJacobs University
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
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Visualization and Computer Graphics LabJacobs University
Linear anisotropic diffusion
• Case 1: λ1 >> λ2, λ3
Prolate case (cigar-shaped)
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Visualization and Computer Graphics LabJacobs University
Planar anisotropic diffusion
• Case 2: λ1 ≈ λ2 >> λ3
Oblate case (pancake-shaped)
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Visualization and Computer Graphics LabJacobs University
Isotropic diffusion
• Case 3: λ1 ≈ λ2 ≈ λ3
Spherical case (ball-shaped)
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Visualization and Computer Graphics LabJacobs University
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.
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Visualization and Computer Graphics LabJacobs University
Elliptic tensor glyphs: barycentric space
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Visualization and Computer Graphics LabJacobs University
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.
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Visualization and Computer Graphics LabJacobs University
Glyph-based tensor visualization
Visualization and Computer Graphics LabJacobs University
7.4 Superquadric Glyphs
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Visualization and Computer Graphics LabJacobs University
Motivation
• Elliptic glyphs have the problems that depending on the viewing angle significantly different ellipsoids look identical.
320581: Advanced Visualization 650
Visualization and Computer Graphics LabJacobs University
Elliptic glyphs: worst case scenario
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Visualization and Computer Graphics LabJacobs University
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.
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Visualization and Computer Graphics LabJacobs University
Superquadrics
beta
alpha
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Visualization and Computer Graphics LabJacobs University
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.
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Visualization and Computer Graphics LabJacobs University
Linear anisotropic case
• Use a long cylinder with the following set-up:
320581: Advanced Visualization 655
Visualization and Computer Graphics LabJacobs University
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.
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Visualization and Computer Graphics LabJacobs University
Isotropic case
• Use a sphere with the following set-up:
This is the same as for elliptic glyphs.
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Visualization and Computer Graphics LabJacobs University
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.
320581: Advanced Visualization 658
Visualization and Computer Graphics LabJacobs University
Superquadric tensor glyphs: barycentric space
320581: Advanced Visualization 659
Visualization and Computer Graphics LabJacobs University
Worst case scenario revisisted
320581: Advanced Visualization 660
Visualization and Computer Graphics LabJacobs University
Ellipsoids
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Visualization and Computer Graphics LabJacobs University
Superquadrics
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Visualization and Computer Graphics LabJacobs University
Superquadric glyphs with optimized spacing