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 Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 2
Tensors p-order tensor in n-space: linear transformation between vector spaces T i 1 ,..,i p , ∀ j ∈ 1 , .., p, 1 ≤ i j ≤ n Special cases: 0th order: scalars 1st order: vectors 2nd order: matrices In Visualization “tensors” are mostly 2 nd order tensors CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 3
Tensors 2nd order tensors map vectors to vectors Symmetric / antisymmetric T t = ± T with R n , T ~ u · T t ~ ∀ ~ u, ~ u · ~ v = ~ v ∈ I v Represented * by matrices in cartesian basis (*) tensors exist independently of any matrix representation CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 4
Tensors Eigenvalues, eigenvectors u ⇥ = ⌥ ⇤ � � I R, ⇤ ⌥ 0 , T ⌥ u = �⌥ u Real symmetric tensors: eigenvalues are real and eigenvectors are orthogonal ~ e i · ~ e j = � ij Sorted eigenvalues λ 1 ≥ λ 2 ≥ ... ≥ λ n Invariants: quantities (function of the tensor value) independent of reference frame: eigenvalues and functions thereof ( e.g. , trace, determinant) CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 5
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 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 6
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 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 6
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 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 6
Tensors Anisotropy characterizes tensor shape Example: ink diffusion Kleenex Newspaper CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 7
Anisotropy λ = 1 Eigenvalues: X ¯ λ 1 ≥ λ 2 ≥ λ 3 λ i 3 i Deviatoric: T = ¯ λ I + D Partial anisotropy: λ 1 − λ 2 linear: c l = λ 1 + λ 2 + λ 3 2( λ 2 − λ 3 ) planar: c p = λ 1 + λ 2 + λ 3 3 λ 3 spherical: c s = λ 1 + λ 2 + λ 3 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 8
Tensors Eigenvectors: non-oriented directional info. u = �⇧ u ⇒ T ( µ ⇧ u ) = � ( µ ⇧ u ) T ⇧ Have no intrinsic norm u = �⌅ u ⇒ T ( − ⌅ u ) = � ( − ⌅ u ) T ⌅ Have no intrinsic orientation Eigenvectors ≠ vectors! Tensor visualization requires combined visualization of eigenvectors and eigenvalues CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 9
Symmetric Tensor Glyphs A 2 nd order symmetric 3D tensor is fully characterized by its 3 real eigenvalues (shape) and associated orthogonal eigenvectors (orientation) CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 10
Symmetric Tensor CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 11
Symmetric Tensor Glyphs CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 12
Symmetric Tensor Glyphs CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 13
Symmetric Tensor Glyphs Shortcomings CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 14
Symmetric Tensor Glyphs Shortcomings CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 14
Superquadrics A. Barr, Superquadrics and angle- preserving transformations , IEEE Computer Graphics and Applications 18(1), 1981 cos α θ sin β φ 0 1 0 ≤ φ ≤ π sin α θ sin β φ q z ( θ , φ ) = B C 0 ≤ θ ≤ 2 π A , @ cos β φ x 2 / α + y 2 / α ⌘ α / β ⇣ + z 2 / β − 1 = 0 . q z ( x , y , z ) = cos β φ 0 1 0 ≤ φ ≤ π − sin α θ sin β φ q x ( θ , φ ) = B C 0 ≤ θ ≤ 2 π A , @ cos α θ sin β φ y 2 / α + z 2 / α ⌘ α / β ⇣ + x 2 / β − 1 = 0 . q x ( x , y , z ) = CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 15
���� Superquadric Tensor Glyphs Parameters 𝛽 and 𝛾 are a function of the tensor’s anisotropy measures: α = ( 1 − c p ) γ β = ( 1 − c l ) γ λ 1 − λ 2 c l = c l ≥ c p = ⇒ λ 1 + λ 2 + λ 3 q ( θ , φ ) = q x ( θ , φ ) q ( x , y , z ) = q x ( x , y , z ) 2 ( λ 2 − λ 3 ) c p = λ 1 + λ 2 + λ 3 α = ( 1 − c l ) γ β = ( 1 − c p ) γ 3 λ 3 c l < c p = ⇒ c s = q ( θ , φ ) = q z ( θ , φ ) λ 1 + λ 2 + λ 3 q ( x , y , z ) = q x ( x , y , z ) G. Kindlmann, Superquadric Tensor Glyphs , Joint Eurographics/IEEE VGTC Symposium on Visualization 2004 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 16
Superquadric Tensor Glyphs Superquadric glyphs CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 17
Superquadric Tensor Glyphs G. Kindlmann, Superquadric Tensor Glyphs , Joint Eurographics/IEEE VGTC Symposium on Visualization 2004 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 18
Superquadric Tensor Glyphs G. Kindlmann, Superquadric Tensor Glyphs , Joint Eurographics/IEEE VGTC Symposium on Visualization 2004 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 18
Comparison G. Kindlmann, Superquadric Tensor Glyphs , Joint Eurographics/IEEE VGTC Symposium on Visualization 2004 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 19
Comparison G. Kindlmann, Superquadric Tensor Glyphs , Joint Eurographics/IEEE VGTC Symposium on Visualization 2004 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 19
Comparison G. Kindlmann, Superquadric Tensor Glyphs , Joint Eurographics/IEEE VGTC Symposium on Visualization 2004 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 20
Comparison G. Kindlmann, Superquadric Tensor Glyphs , Joint Eurographics/IEEE VGTC Symposium on Visualization 2004 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 20
Symmetric Tensor Glyphs Color-coding can be used to facilitate the interpretation of the orientation e.g. , e max mapped to R=|x|, G=|y|, B=|z| CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 21
Comparison CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 22
Symmetric Tensor CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 23
Symmetric Tensor Glyph Glyphs for general symmetric tensors? Eigenvalues can be positive or negative � 1 3 , 1 3 , 1 � √ √ √ 3 �������� λ 1 − λ 2 = λ 2 − λ 3 ��� ������� + λ 3 0 > λ 3 λ 2 = λ 3 λ 3 = 0 0 � 1 2 , 1 � < λ 2 > 0 2 , 0 √ √ λ 3 λ 2 < 0 λ 1 = λ 2 + λ 2 + λ 1 λ (1 , 0 , 0) = 2 λ − 3 λ 2 = 0 � 1 3 , 1 3 , − 1 � √ √ √ 3 ��������� � 1 2 , 0 , − 1 � λ √ √ 3 2 − = λ 1 � 1 3 , − 1 3 , − 1 � ��� √ √ √ 3 λ + λ 1 = 1 λ 1 > 0 ��� ��� ��� λ λ 2 = λ 3 − 2 λ 1 < 0 0 (0 , 0 , − 1) = λ 1 ��������� ����������� λ 1 = λ 2 ��� � 0 , − 1 2 , − 1 � λ 1 − λ 2 = λ 2 − λ 3 ����������� √ √ 2 �������� ��������� + λ 2 ������� ������� − λ 3 � � − 1 3 , − 1 3 , − 1 √ √ √ 3 CS53000 / Spring 2020 : Introduction to Scientific Visualization. March 5, 2020 11. Tensor Field Techniques 24 ��� ��� ��� ���
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