CIS 49306930 902 SCIENTIFIC VISUALIZATION TENSOR FIELD VISUALIZATION
- Slides: 53
CIS 4930/6930 -902 SCIENTIFIC VISUALIZATION TENSOR FIELD VISUALIZATION Paul Rosen Assistant Professor University of South Florida Slide credit X. Tricoche
• Tensor basics Tensor glyphs Hyperstreamlines DTI visualization • • • OUTLINE •
• TENSORS p-ranked tensor in n-space: linear transformation between vector spaces • Special cases: 0 th order (rank): scalars 1 st order: vectors 2 nd order: matrices In Visualization “tensors” are mostly 2 nd order tensors • • •
• TENSORS 2 nd order tensors map vectors to vectors Symmetric / antisymmetric Tt = ±T with • • Represented* by matrices in cartesian basis • (*) tensors exist independently of any matrix representation
• • • TENSORS Eigenvalues, eigenvectors Real symmetric tensors: eigenvalues are real and eigenvectors are orthogonal • Sorted eigenvalues. Invariants: quantities (function of the tensor value) that do not change with the reference frame • Eigenvalues and all functions of the eigenvalues Trace (sum), determinant (product), FA, mode, … • • .
• EXAMPLES • Forces • stress: cause of deformation • strain: deformation description • Derivative • Jacobian: 1 st-order derivative of a vector field • Hessian: 2 nd-order derivative of a scalar field • Diffusion tensor field
• • TENSORS Anisotropy characterizes tensor shape Example: ink diffusion • Kleene x Newspap er
• • 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 • •
• • SYMMETRIC TENSOR GLYPHS A 2 nd order symmetric 3 D tensor is fully characterized by its 3 real eigenvalues (shape) and associated orthogonal eigenvectors (orientation)
• SYMMETRIC TENSOR GLYPHS
• SYMMETRIC TENSOR GLYPHS
• SYMMETRIC TENSOR GLYPHS
• SYMMETRIC TENSOR GLYPHS • Shortcomings
• SYMMETRIC TENSOR GLYPHS • Shortcomings
• SUPERQUADRICS • A. BARR, SUPERQUADRICS AND ANGLE- PRESERVING TRANSFORMATIONS,
• • SUPERQUADRIC TENSOR GLYPHS Parameters �� and �� are a function of the tensor’s anisotropy measures: with • G. KINDLMANN, SUPERQUADRIC TENSOR GLYPHS, • JOINT EUROGRAPHICS/IEEE VGTC SYMPOSIUM ON VISUALIZATION
• SUPERQUADRIC TENSOR GLYPHS • Superquadric glyphs
• SUPERQUADRIC TENSOR GLYPHS • G. KINDLMANN, SUPERQUADRIC TENSOR GLYPHS, • JOINT EUROGRAPHICS/IEEE VGTC SYMPOSIUM ON VISUALIZATION
• SUPERQUADRIC TENSOR GLYPHS • G. KINDLMANN, SUPERQUADRIC TENSOR GLYPHS, • JOINT EUROGRAPHICS/IEEE VGTC SYMPOSIUM ON VISUALIZATION
• COMPARISON • G. KINDLMANN, SUPERQUADRIC TENSOR GLYPHS, • JOINT EUROGRAPHICS/IEEE VGTC SYMPOSIUM ON VISUALIZATION
• COMPARISON • G. KINDLMANN, SUPERQUADRIC TENSOR GLYPHS, • JOINT EUROGRAPHICS/IEEE VGTC SYMPOSIUM ON VISUALIZATION
• COMPARISON • G. KINDLMANN, SUPERQUADRIC TENSOR GLYPHS, • JOINT EUROGRAPHICS/IEEE VGTC SYMPOSIUM ON VISUALIZATION
• • 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| •
• COMPARISON
• SYMMETRIC TENSOR GLYPHS
• SYMMETRIC TENSOR GLYPH Glyphs for general symmetric tensors? Eigenvalues can be positive or negative • 1 1 1 √ , √ 3 3 3 tive posi i t e dein λ 3 ( λ 2 = λ 2 −− +λ 3 = 2 0 λ 1 λ 3 0 = λ 3 ( √ 2 , √ >0 λ 3 <0 λ 3 1 2 , =λ λ 2 = − λ 3 (1, 0, 0) 1 λ 1 0 > 0 λ 2 < λ 2 +λ 1 λ 2 ( = 1 1 , 1 , √ 3 −√ 3 0 inde ( , −√ 1 2 inite 0, 1 − √ , − √ 1 2 2 λ 3 0 λ 1 = λ 2 −− +λ 2 = = ( >0 λ 1 (0, 0, 1) <0 − λ 1 − λ 3 1 λ 2 λ 1 = − λ 2 λ +λ 1 1 1 √ , − √ 3 3 3 λ 3 2 λ 3 ( =− 1 1 0, √ λ 1 • ( −√ 3 1 , −√ 1 3 tive nega i t e dein
• SYMMETRIC TENSOR GLYPH (0, 4, 2) ) (1, 0 (1, 2) ) , 2 (1 2) (0, 4) (1, ( , 4. 0 ) , 1 (1 2) (0, ) , 2 (1 ) 2) (1, 0 ) 4) (1, v ) β ( 2. 0 , ) ) , 1 (1 1. 0 (1, 0 ) (d) (α , β ) hybrid superquadric (a, � W) = (0, 4, 2) 0. 0 α 1. 0 base glyph tensor glyph regular superquadric (a, �) = (0, 4) u • T. SCHULTZ, G. KINDLMANN, SUPERQUADRIC GLYPHS FOR SYMMETRIC SECOND-ORDER TENSORS, IEEE TVCG 16 (6) (IEEE VISUALIZATION
• (a) Glyphs on vertical cutting plane RESULTS (b) Superquadric tensor glyphs; s(∥D∥) ∝∥D∥ (c) Superquadric tensor glyphs; s(∥D∥) ∝∥D∥ 1/2 • T. SCHULTZ, G. KINDLMANN, SUPERQUADRIC GLYPHS FOR SYMMETRIC SECOND-ORDER TENSORS, IEEE TVCG 16 (6) (IEEE VISUALIZATION
• GLYPH PACKING • Distribute (discrete) glyphs over continuous domain in data-driven way • Reveal underlying continuous structures • Remove artifacts caused by sampling bias G. KINDLMANN AND C. -F. WESTIN, DIFFUSION TENSOR VISUALIZATION WITH GLYPH PACKING, IEEE VISUALIZATION 2006
• Regular grid GLYPH PACKING Glyph packing • G. KINDLMANN AND C. -F. WESTIN, DIFFUSION TENSOR VISUALIZATION WITH GLYPH PACKING, IEEE VISUALIZATION 2006
• Regular grid GLYPH PACKING Glyph packing • G. KINDLMANN AND C. -F. WESTIN, DIFFUSION TENSOR VISUALIZATION WITH GLYPH PACKING, IEEE VISUALIZATION 2006
• • HYPERSTREAMLINES Method for symmetric 2 nd order tensor fields in 3 D • • Identify eigenvector fields w. r. t. associated eigenvalues •
• HYPERSTREAMLINES Tensor field lines (2 D/3 D): curve everywhere tangential to a given eigenvector field • • • R. R. DICKINSON, A UNIFIED APPROACH TO THE DESIGN OF VISUALIZATION SOFTWARE FOR THE ANALYSIS OF FIELD PROBLEMS, SPIE PROCEEDINGS VOL. 1083,
• HYPERSTREAMLINES Remark: numerical integration using e. g. Runge-Kutta is faced with the problem of maintaining orientation consistency • • R. R. DICKINSON, A UNIFIED APPROACH TO THE DESIGN OF VISUALIZATION SOFTWARE FOR THE ANALYSIS OF FIELD PROBLEMS, SPIE PROCEEDINGS, VOL. 1083,
• 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 . • T. DELMARCELLE, L. HESSELINK, VISUALIZATION OF SECOND ORDER TENSOR FIELDS AND MATRIX DATA, IEEE VISUALIZATION 1992
• • 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 • • T. DELMARCELLE, L. HESSELINK, VISUALIZATION OF SECOND ORDER TENSOR FIELDS AND MATRIX DATA, IEEE VISUALIZATION 1992
• HYPERSTREAMLINES • T. DELMARCELLE, L. HESSELINK, VISUALIZATION OF SECOND ORDER TENSOR FIELDS AND MATRIX DATA, IEEE VISUALIZATION 1992
• HYPERSTREAMLINES
• HYPERSTREAMLINES
• HYPERSTREAMLINES • 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
• 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 • •
• BRAIN STRUCTURE - FIBER TRACKS
• DT MRI VISUALIZATION
• WHITE MATTER TRACTS • PARK, WESTIN, AND KIKINIS, BWH, HARVARD MEDICAL SCHOOL, 2003
• DIFFUSION IN BIOLOGICAL TISSUE Motion of water through tissue Faster in some directions than others • • Kleene x • Newspap er Anisotropy: diffusion rate depends on direction isotropic anisotropi c
• • DIFFUSION MRI OF THE BRAIN Anisotropy high along white matter fiber tracts
• • DIFFUSION MRI OF THE BRAIN Anisotropy high along white matter fiber tracts 2. 1 -0. 2 -0. 1 2. 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 11/13/15 1. 7 0. 1 -0. 1 2. 3 -0. 1 -0. 3
• 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 L. ZHUKOV, A. BARR, ORIENTED TENSOR RECONSTRUCTION: TRACING NEURAL PATHWAYS FROM DIFFUSION TENSOR MRI, IEEE VISUALIZATION 2002
• FIBER TRACING • White matter • L. ZHUKOV, A. BARR, ORIENTED TENSOR RECONSTRUCTION: TRACING NEURAL PATHWAYS FROM DIFFUSION TENSOR MRI, IEEE VISUALIZATION 2002
• FIBER TRACING • White matter • L. ZHUKOV, A. BARR, ORIENTED TENSOR RECONSTRUCTION: TRACING NEURAL PATHWAYS FROM DIFFUSION TENSOR MRI, IEEE VISUALIZATION
• FIBER TRACING • Heart • L. ZHUKOV, A. BARR, HEART FIBER RECONSTRUCTION FROM DIFFUSION TENSOR MRI, IEEE VISUALIZATION 2003
- Glyphs
- Texas rules of evidence 902
- Cuifar
- Rotation
- Scientific visualization tutorial
- How is a scientific law different from a scientific theory?
- Scientific inquiry vs scientific method
- Field dependent and field independent
- Electric field and magnetic field difference
- Magnetic field
- Electric field and magnetic field difference
- Waveguide cutoff frequency
- Field dependent vs field independent
- Database field types and field properties
- Field dependent vs field independent
- Tensor product
- Supraspinatus taping
- Tensor basics
- Strain tensor matrix
- Auris interna anatomy
- Multiple view geometry
- Reciprocal lattice concept
- Tensor notation
- Thermal expansion tensor
- Tensor calculus
- Tensor factorization
- Arcus tendineus m solei
- Playground.tensorflow
- Tensor naprężeń
- Inverse metric tensor
- Maxilla
- Tensor tympani
- Musculo amigdalogloso
- @ebbiya
- What is tensor in machine learning
- Tensor vaginae femoris
- Destressing of lwr with rail tensor
- Neural tensor network
- Impermeability tensor
- Tensor contraction engine
- Extensor: an accelerator for sparse tensor algebra
- Strain tensor
- Musculo aductor mayor
- Tensor notation
- Tensor network
- Tensor fascia latae
- Corpus os sphenoidale
- Atl transformation language
- Tensor de reynolds
- Shear stress convention
- Tuber ischiadicum
- Inertia tensor of cone
- Incisura terminalis ear
- Tensor meaning in physics