Tensor Field Techniques Lecture 11 March 5, 2020 Outline Basics - - PowerPoint PPT Presentation

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Tensor Field Techniques Lecture 11 March 5, 2020 Outline Basics - - PowerPoint PPT Presentation

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CS53000 - Spring 2020 Introduction to Scientific Visualization Tensor Field Techniques Lecture 11 March 5, 2020 Outline Basics of tensor algebra Tensor glyphs Hyperstreamlines DTI visualization CS53000 / Spring 2020 : Introduction to

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CS53000 - Spring 2020

Introduction to Scientific Visualization

Lecture

Tensor Field Techniques

March 5, 2020

11

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CS53000 / Spring 2020 : Introduction to Scientific Visualization.

  • 11. Tensor Field Techniques

March 5, 2020

Outline

Basics of tensor algebra Tensor glyphs Hyperstreamlines DTI visualization

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CS53000 / Spring 2020 : Introduction to Scientific Visualization.

  • 11. Tensor Field Techniques

March 5, 2020

Tensors

p-order tensor in n-space: linear transformation between vector spaces Special cases:

0th order: scalars 1st order: vectors 2nd order: matrices

In Visualization “tensors” are mostly 2nd order tensors

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Ti1,..,ip, ∀j ∈ 1, .., p, 1 ≤ ij ≤ n

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  • 11. Tensor Field Techniques

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Tensors

2nd order tensors map vectors to vectors Symmetric / antisymmetric with Represented* by matrices in cartesian basis

(*) tensors exist independently of any matrix representation

4

Tt = ±T

∀~ u, ~ v ∈ I Rn, T~ u · ~ v = ~ u · Tt~ v

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  • 11. Tensor Field Techniques

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Tensors

Eigenvalues, eigenvectors Real symmetric tensors: eigenvalues are real and eigenvectors are orthogonal Sorted eigenvalues Invariants: quantities (function of the tensor value) independent of reference frame:

eigenvalues and functions thereof (e.g., trace, determinant)

5

⇤ I R, ⇤⌥ u ⇥= ⌥ 0, T⌥ u = ⌥ u

λ1 ≥ λ2 ≥ ... ≥ λn

~ ei · ~ ej = ij

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  • 11. Tensor Field Techniques

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Examples

Forces

stress: cause of deformation strain: deformation description

Derivative

Jacobian: 1st-order derivative of a vector field Hessian: 2nd-order derivative of a scalar field

Diffusion tensor field

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CS53000 / Spring 2020 : Introduction to Scientific Visualization.

  • 11. Tensor Field Techniques

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Examples

Forces

stress: cause of deformation strain: deformation description

Derivative

Jacobian: 1st-order derivative of a vector field Hessian: 2nd-order derivative of a scalar field

Diffusion tensor field

6

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CS53000 / Spring 2020 : Introduction to Scientific Visualization.

  • 11. Tensor Field Techniques

March 5, 2020

Examples

Forces

stress: cause of deformation strain: deformation description

Derivative

Jacobian: 1st-order derivative of a vector field Hessian: 2nd-order derivative of a scalar field

Diffusion tensor field

6

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  • 11. Tensor Field Techniques

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Tensors

Anisotropy characterizes tensor shape Example: ink diffusion

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Kleenex Newspaper

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  • 11. Tensor Field Techniques

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Anisotropy

Eigenvalues: Deviatoric: Partial anisotropy: linear: planar: spherical:

8

cl = λ1 − λ2 λ1 + λ2 + λ3 λ1 ≥ λ2 ≥ λ3 cp = 2(λ2 − λ3) λ1 + λ2 + λ3 ¯ λ = 1 3 X

i

λi

T = ¯ λI + D

cs = 3λ3 λ1 + λ2 + λ3

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  • 11. Tensor Field Techniques

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Tensors

Eigenvectors: non-oriented directional info.

Have no intrinsic norm Have no intrinsic orientation

Eigenvectors ≠ vectors! Tensor visualization requires combined visualization of eigenvectors and eigenvalues

9

T⇧ u = ⇧ u ⇒ T(µ⇧ u) = (µ⇧ u) T⌅ u = ⌅ u ⇒ T(−⌅ u) = (−⌅ u)

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  • 11. Tensor Field Techniques

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Symmetric Tensor Glyphs

A 2nd order symmetric 3D tensor is fully characterized by its 3 real eigenvalues (shape) and associated orthogonal eigenvectors (orientation)

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  • 11. Tensor Field Techniques

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Symmetric Tensor

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Symmetric Tensor Glyphs

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Symmetric Tensor Glyphs

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Symmetric Tensor Glyphs

Shortcomings

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Symmetric Tensor Glyphs

Shortcomings

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Superquadrics

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  • A. Barr,

Superquadrics and angle- preserving transformations, IEEE Computer Graphics and Applications 18(1), 1981

qz(θ,φ) = B @ cosα θsinβ φ sinα θsinβ φ cosβ φ 1 C A , 0 ≤ φ ≤ π 0 ≤ θ ≤ 2π

qz(x,y,z) = ⇣ x2/α +y2/α⌘α/β +z2/β −1 = 0.

qx(θ,φ) = B @ cosβ φ −sinα θsinβ φ cosα θsinβ φ 1 C A , 0 ≤ φ ≤ π 0 ≤ θ ≤ 2π

qx(x,y,z) = ⇣ y2/α +z2/α⌘α/β +x2/β −1 = 0.

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Superquadric Tensor Glyphs

Parameters 𝛽 and 𝛾 are a function of the tensor’s anisotropy measures:

16

cl ≥ cp = ⇒        α = (1−cp)γ β = (1−cl)γ q(θ,φ) = qx(θ,φ) q(x,y,z) = qx(x,y,z) cl < cp = ⇒        α = (1−cl)γ β = (1−cp)γ q(θ,φ) = qz(θ,φ) q(x,y,z) = qx(x,y,z)

cl = λ1 −λ2 λ1 +λ2 +λ3 cp = 2(λ2 −λ3) λ1 +λ2 +λ3 cs = 3λ3 λ1 +λ2 +λ3

  • G. Kindlmann, Superquadric Tensor Glyphs,

Joint Eurographics/IEEE VGTC Symposium on Visualization 2004

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  • 11. Tensor Field Techniques

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Superquadric Tensor Glyphs

Superquadric glyphs

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Superquadric Tensor Glyphs

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  • G. Kindlmann, Superquadric Tensor Glyphs,

Joint Eurographics/IEEE VGTC Symposium on Visualization 2004

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  • 11. Tensor Field Techniques

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Superquadric Tensor Glyphs

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  • G. Kindlmann, Superquadric Tensor Glyphs,

Joint Eurographics/IEEE VGTC Symposium on Visualization 2004

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Comparison

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  • G. Kindlmann, Superquadric Tensor Glyphs,

Joint Eurographics/IEEE VGTC Symposium on Visualization 2004

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  • 11. Tensor Field Techniques

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Comparison

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  • G. Kindlmann, Superquadric Tensor Glyphs,

Joint Eurographics/IEEE VGTC Symposium on Visualization 2004

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  • 11. Tensor Field Techniques

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Comparison

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  • G. Kindlmann, Superquadric Tensor Glyphs,

Joint Eurographics/IEEE VGTC Symposium on Visualization 2004

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  • 11. Tensor Field Techniques

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Comparison

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  • G. Kindlmann, Superquadric Tensor Glyphs,

Joint Eurographics/IEEE VGTC Symposium on Visualization 2004

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Symmetric Tensor Glyphs

Color-coding can be used to facilitate the interpretation of the orientation e.g., emax mapped to R=|x|, G=|y|, B=|z|

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Comparison

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Symmetric Tensor

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Symmetric Tensor Glyph

  • +λ2

+λ3 +λ1 +λ1 +λ2 −λ3

λ2 = 0 λ1 =

λ

2

= − λ

3

λ

1

= − λ

2

λ

1

= − λ

3

λ1 = λ2 λ2 = λ3 λ1 = λ2 λ2 = λ3

λ2 > 0 λ2 < 0

  • λ1 − λ2 = λ2 − λ3
λ1 − λ2 = λ2 − λ3
  • (1, 0, 0)
1 √ 2, 0, − 1 √ 2
  • 1
√ 3, 1 √ 3, − 1 √ 3
  • 1
√ 2, 1 √ 2, 0
  • 1
√ 3, 1 √ 3, 1 √ 3
  • 1
√ 3, − 1 √ 3, − 1 √ 3
  • (0, 0, −1)
  • 0, − 1
√ 2, − 1 √ 2
  • λ3 = 0

λ3 < λ1 > 0

  • λ1 < 0
  • − 1
√ 3, − 1 √ 3, − 1 √ 3
  • λ3

>

Glyphs for general symmetric tensors? Eigenvalues can be positive or negative

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Symmetric Tensor Glyph

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(α, β)

  • α

β

  • v

u

  • T. Schultz, G. Kindlmann, Superquadric Glyphs for Symmetric Second-Order Tensors,

IEEE TVCG 16 (6) (IEEE Visualization 2010)

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Results

26 (a) Glyphs on vertical cutting plane (b) Superquadric tensor glyphs; s(D) ∝ D (c) Superquadric tensor glyphs; s(D) ∝ D1/2

  • T. Schultz, G. Kindlmann, Superquadric Glyphs for Symmetric Second-Order Tensors,

IEEE TVCG 16 (6) (IEEE Visualization 2010)

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Distribute (discrete) glyphs over continuous domain in data-driven way Reveal underlying continuous structures Remove artifacts caused by sampling bias

Glyph Packing

  • G. Kindlmann and C.-F. Westin, Diffusion Tensor Visualization with Glyph Packing, IEEE Visualization 2006

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Glyph Packing

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Regular grid Glyph packing

  • G. Kindlmann and C.-F. Westin, Diffusion Tensor Visualization with Glyph Packing, IEEE Visualization 2006
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Glyph Packing

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Regular grid Glyph packing

  • G. Kindlmann and C.-F. Westin, Diffusion Tensor Visualization with Glyph Packing, IEEE Visualization 2006
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Hyperstreamlines

Method for symmetric 2nd order tensor fields in 3D Identify eigenvector fields w.r.t. associated eigenvalues

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and

x ⇥ I R3 ⇤ T(x) x ⇤⇥ λ1(x) λ2(x) λ3(x) x ⇥ ⇧ ei(x), T(x) = i(x)⇧ ei(x)

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Hyperstreamlines

Tensor field lines (2D/3D): curve everywhere tangential to a given eigenvector field

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  • R. R. Dickinson,

A Unified Approach to the Design of Visualization Software for the Analysis of Field Problems, SPIE Proceedings, Vol. 1083, 1989

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Hyperstreamlines

Remark: numerical integration using e.g. Runge- Kutta is faced with the problem of maintaining

  • rientation consistency

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  • R. R. Dickinson,

A Unified Approach to the Design of Visualization Software for the Analysis of Field Problems, SPIE Proceedings, Vol. 1083, 1989

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Hyperstreamlines

Method

Compute tensor field line along major eigenvector Sweep geometric primitive representing two other eigenvalues and eigenvectors

Ellipse stretched along eigenvectors by eigenvalues Cross depicting eigenvectors + eigenvalues

Color coding on geometric primitive determined by

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  • T. Delmarcelle, L. Hesselink,

Visualization of Second Order Tensor Fields and Matrix Data, IEEE Visualization 1992

⇥ e1

λ1

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Hyperstreamlines

Remarks

Eigenvectors are orthogonal: cross section always orthogonal to tensor field line Eigenvalues mapped to length of edges in cross section: problems with negative eigenvalues

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  • T. Delmarcelle, L. Hesselink,

Visualization of Second Order Tensor Fields and Matrix Data, IEEE Visualization 1992

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Hyperstreamlines

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  • T. Delmarcelle, L. Hesselink,

Visualization of Second Order Tensor Fields and Matrix Data, IEEE Visualization 1992

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Fabric-like Texture

Visualize tensor through distortion of basic fabric pattern

36 CS530 / Spring 2019 : Introduction to Visualization. Tensor Fields 03/25

Fabric-like Textures

  • Visualize tensor through distortion of basic fabric pattern

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Hotz, I.; Feng, L.; Hamann, B.; Joy, K. Tensor-fields Visualization using a Fabric like Texture on Arbitrary two-dimensional Surfaces. Mathematical Foundations of Scientific Visualization, Computer Graphics Springer , 2006

CS53000 / Spring 2020 : Introduction to Scientific Visualization .

  • 11. Tensor Field Techniques

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Fabric-like Texture

37 CS53000 / Spring 2020 : Introduction to Scientific Visualization .

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Hyperstreamlines

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Hyperstreamlines

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Hyperstreamlines

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Extension of Hultquist’s stream surfaces to eigenvector fields

  • T. Delmarcelle, L. Hesselink,

Visualization of Second Order Tensor Fields and Matrix Data, IEEE Visualization 1992

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Diffusion Tensor Imaging

Diffusion Tensor (DT)-MRI measures anisotropic (directional) diffusion properties of biological tissue (e.g., brain) Diffusion tensor is symmetric positive definite (positive eigenvalues) Objective: use tensor information to reconstruct the path of tissue fibers Problems: (very) noisy data + isotropy

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Brain Structure - Fiber Tracks

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DT MRI Visualization

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White Matter Tracts

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Park, Westin, and Kikinis, BWH, Harvard Medical School, 2003

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Diffusion in Biological Tissue

Motion of water through tissue Faster in some directions than others

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  • Anisotropy: diffusion rate depends on direction

isotropic anisotropic Newspaper Kleenex

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Diffusion Tensor Visualization

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Diffusion Tensor Visualization

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Diffusion MRI of the Brain

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Anisotropy high along white matter fiber tracts

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Diffusion MRI of the Brain

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Anisotropy high along white matter fiber tracts

1.7 0.1 -0.1 0.1 2.3 -0.3

  • 0.1 -0.3 0.3
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Diffusion MRI of the Brain

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Anisotropy high along white matter fiber tracts

3.7 0.3 -0.8 0.3 0.6 -0.1

  • 0.8 -0.1 0.8

1.7 0.1 -0.1 0.1 2.3 -0.3

  • 0.1 -0.3 0.3
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Diffusion MRI of the Brain

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Anisotropy high along white matter fiber tracts

2.1 -0.1 -0.2

  • 0.1 2.0 -0.0
  • 0.2 -0.0 2.1

3.7 0.3 -0.8 0.3 0.6 -0.1

  • 0.8 -0.1 0.8

1.7 0.1 -0.1 0.1 2.3 -0.3

  • 0.1 -0.3 0.3
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Fiber Tracing

Moving Least Squares: Apply Gauss filter mask whose support is determined by current path orientation and local anisotropy Trace fiber path along filtered eigenvector

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  • L. Zhukov, A. Barr,

Oriented Tensor Reconstruction: Tracing Neural Pathways from Diffusion Tensor MRI, IEEE Visualization 2002

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Fiber Tracing

White matter

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  • L. Zhukov, A. Barr,

Oriented Tensor Reconstruction: Tracing Neural Pathways from Diffusion Tensor MRI, IEEE Visualization 2002

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Fiber Tracing

White matter

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  • L. Zhukov, A. Barr,

Oriented Tensor Reconstruction: Tracing Neural Pathways from Diffusion Tensor MRI, IEEE Visualization 2002

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Fiber Tracing

Heart

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  • L. Zhukov, A. Barr,

Heart Fiber Reconstruction from Diffusion Tensor MRI, IEEE Visualization 2003

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Tensor Mode

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linear

mode = 1

  • rthotropic

mode = 0

planar

mode = -1

λ2 λ3 λ1

mode = -1 mode = 0 mode = +1

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Tensor Mode

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linear

mode = 1

  • rthotropic

mode = 0

planar

mode = -1

λ2 λ3 λ1

mode = -1 mode = 0 mode = +1

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Tensor Mode

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linear

mode = 1

  • rthotropic

mode = 0

planar

mode = -1

λ2 λ3 λ1

mode = -1 mode = 0 mode = +1

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Tensor Mode

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linear

mode = 1

  • rthotropic

mode = 0

planar

mode = -1

λ2 λ3 λ1

mode = -1 mode = 0 mode = +1

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Slice Inspection: RGB(e1) (original data)

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Slice Inspection: RGB(e1)

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Slice Inspection: FA

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Slice Inspection: | FA|

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Δ

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Slice Inspection: ridge strength: max(0, λ3)

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Slice Inspection: g.e3 (modulated by strength)

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Slice Inspection: sqrt((g.e3)2 + (g.e2)2)

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3-D Results: coronal fibers

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3-D Results: ridge surfaces

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3-D Results: valley surfaces

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3-D Results: valley surfaces with

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3-D Results: brainstem fibers

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3-D Results: brainstem ridge surfaces

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3-D Results: brainstem valley surfaces

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3-D Results: combined results

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Ridge Lines of FA in Brain

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425 lines with length > 15mm Sort based on average ridge strength

1.5 x 1.5 x 1.5 mm

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Ridge Lines of FA in Brain

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1.5 x 1.5 x 1.5 mm

FOR = Fornix, CB = Cingulum Bundles Sort based on average ridge strength

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Line Creases of White Matter Anisotropy

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CBL CBR FOR ILF ICP ML CST

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  • G. Kindlmann et al., Sampling and Visualizing Creases with Scale-space particles, IEEE

Visualization 2009

DTI CT