Superquadric Glyphs for Symmetric Second- Order Tensors

Transcription

1 Superquadric Glyphs for Symmetric Second- Order Tensors Thomas Schultz, Gordon L. Kindlmann Computer Science Dept, Computation Institute University of Chicago Symmetric Tensor Representations [Kindlmann 24] R R -1 Eigenvalues tell us about Positive Definiteness

2 Symmetric Tensor Representations Topic of this talk R R -1 Tensor is indefinite Why care about indefinite tensors? Indefinite Symmetric Tensors arise in many fields: Hessians Stress Tensors Rate-of-deformation Tensors Geometry Tensors

3 Principles for Tensor Glyph Design Faithful and expressive visualization requires: Preservation of Symmetry: Glyph should have same symmetries as the tensor Continuity: Disambiguity: Principles for Tensor Glyph Design Natural for a wide range of applications: Invariance under scaling: Invariance under projection to eigenplanes:

4 Glyph Coloring We color each point x on the glyph by sign(x T Dx) + red - blue Satisfies! Preservation of Symmetry! Continuity! Disambiguity Our Fundamental Glyph Equation scaling rotation normalized eigenvalues basis shape

5 Explicit Ellipses Dx for x 1 " Symmetry Preservation " Disambiguity Implicit Ellipses x T D -2 x 1 sign preserving # Unbounded Surfaces

6 Exp-Ellipses exp(d)x for x 1 " Disambiguity Reynolds Glyphs (x T D -2 x)x for x 1 " Invariance under Projection # Occlusions in 3D

7 New Glyph for 2D symmetric tensor Meet disambiguity, symmetry preservation req. Glyph shape shows eigenvalue sign differences Convex indicates same sign (positive-definite, negative-definite) Concave indicates different sign (indefinite) From 2D to 3D tensor glyphs Understand space of 3D symmetric tensor shapes 2D (, ) Scaffold (prototype) 3D glyphs with 2D glyphs λ 2 (+,+) λ 1 (+, ) 3D? Find mapping from 3D tensor shapes to 3D superquadrics

8 Space of 3D tensor shape Eigenvalue sorting!1 "!2 "!3 Lune of tensor shape (+,+)!1 "!2!2 "!3!1 "!2 "!3 λ1 (, ) (+, ) +!3 +!3 +!3 +!1 +!2 +!1 +!2 +!1 +!2 +!1 +!1 +!1 +!2 +!2 +!2 -!3 -!3 -!3 > < λ 1 λ 3 λ 2 λ 3 λ 2 λ 3 λ1 λ3 λ1 λ3 λ 3 λ 2 λ 3 λ 1 λ 2 λ 1 Positive Definite λ 1 λ 2 λ 3 > λ 1 λ 2 λ 3 < Indefinite (mixed eval signs) λ 1 > λ 1 < Negative Definite Space of 3D tensor shape Mercator projection!1!2!3 >!1!2!3 < Kindlmann Superquadric Tensor Glyphs, Vissym 4

9 λ 3 1 λ 3 2 λ λ λ1 λ 2 λ3 λ + 1 λ 2 λ 2 λ1 λ 3 λ 1 λ 2 λ 3 λ3 λ3 +λ1 λ1 λ1 λ3 λ1 > λ1 < λ1 λ1 λ 3 λ1 v 3 λ + λ +λ λ3 λ < λ Cubic projection unfold, shear # unit square of tensor shape 1 λ 1 λ 2 λ 3 λ3 λ3 > λ3 < λ1 > λ3 Space of 3D tensor shape λ1 λ 2 λ3 u λ3 > λ3 λ3 > λ3 < < λ1 3D (+,+)? λ 3 λ1 λ3 λ1 λ 2 Find mapping from 3D tensor shapes to 3D superquadrics λ3 Scaffold (prototype) 3D glyphs with 2D glyphs λ1 (+, ) (, ) λ1 λ 1 λ 2 λ 3 λ1 Understand space of 3D tensor shapes 2D λ 1 λ 2 λ 3 From 2D to 3D tensor glyphs λ1 > λ1 <

10 From 2D to 3D tensor glyphs Understand space of 3D tensor shapes 2D (, ) λ 2 (+,+) λ 1 (+, ) 3D > <? λ1 λ3 λ3 λ3 λ1 λ3 λ1 λ3 λ3 λ3 λ1 λ1 λ1 λ3 > λ1 λ3 < λ1 > λ1 < Scaffold (prototype) 3D glyphs with 2D glyphs Find mapping from 3D tensor shapes to 3D superquadrics Invariance under eigenplane projection! orthogonal 2D glyphs form scaffold for 3D glyphs Can use these 3D shapes to guide next step...

11 From 2D to 3D tensor glyphs Understand space of 3D tensor shapes 2D (, ) λ 2 (+,+) λ 1 (+, ) 3D > <? λ1 λ3 λ3 λ3 λ1 λ3 λ1 λ3 λ3 λ3 λ1 λ1 λ1 λ3 > λ1 λ3 < λ1 > λ1 < Scaffold (prototype) 3D glyphs with 2D glyphs Find mapping from 3D tensor shapes to 3D superquadrics 4 $ %

12 4 $ %!i hides 4 discontinuity $ 2 1 slight shape problem 1 %

13 Meet the glyphs... (+,+,+) (+,+,-) (+,-,-) (-,-,-) Halos: improve visibility, colormap of tensor trace The Glyph Rendering Pipeline 4. Scale and rotate (modelview matrix) 1. Map! i to (u,v) 2. Map (u,v) to ($,%) 3. Generate Base Glyph 5. Color code sign (fragment shader)

14 Using a Palette of Base Geometry Precompute and re-use superquadric geometry on a grid in ($,%) space Implementation & Performance Geometry-based implementation Performance: 3 full-resolution glyphs 18x1 viewport 25fps (incl. halos) on NVIDIA Quadro FX 18 Most code is in Teem; please see tutorial on

15 Why not use Reynolds Glyphs? Colored Reynolds Glyphs clearly show negative eigenvalues unless you re viewing them from an unfortunate direction. Why Prefer Superquadrics? Superquadrics remain legible from a wide range of directions.

16 What About Traceless Tensors? Existing Traceless Superquadric Glyphs [Jankun-Kelly/Mehta, Vis6] Our new Superquadric Glyphs Hessians vs. Geometry Tensors Geometry Tensors Hessians See how geometry tensors are related to Hessians G (I-nn T )H(I-nn T )/ g

17 Inspecting Smoothing Image Filters Original Data Gaussian Smoothing Total Variation Flow Equal Remaining Variance What are Differences? Inspecting Smoothing Image Filters Original Data Gaussian Smoothing Total Variation Flow Let s look at Hessians

18 Inspecting Smoothing Image Filters Original Data Gaussian Smoothing Total Variation Flow Gaussian Filtering Smoothes Hessians. Leads to smearing out into flat regions and cancellation effects. Inspecting Smoothing Image Filters Original Data Gaussian Smoothing Total Variation Flow TV flow creates flat regions and sharp edges between them.

19 Results, turbulent jet flow LIC Results, turbulent jet flow LIC, modulate contrast by velocity

20 Results, turbulent jet flow Rate-of-Deformation Tensor ( v + T v)/2 Ellipses with! # exp(!) Results, turbulent jet flow Rate-of-Deformation Tensor ( v + T v)/2 New glyphs, s( D ) D

21 Results, turbulent jet flow Rate-of-Deformation Tensor ( v + T v)/2 New glyphs, s( D ) D 1/2 Results, double point stress field Can now see tensors underlying hyperstreamlines s( D ) D 1/1

22 Conclusions Presented a new set of tensor glyphs Can visualize all symmetric 3D 2nd-order tensors Glyphs design guided by principles Need disambiguity, avoid misleading symmetries Eigenplane projection invariance! scaffolds Future: still no satisfactory glyphs for: General 2nd-order (non-symmetric) tensors Rotations Acknowledgements Funding: DAAD Flow data: Wolfgang Kollmann, UC Davis Symmetry and Continuity discussion: 29 Dagstuhl Scientific Visualization Seminar 9251 Thank you