Interactive Design and Visualization of Tensor Fields on Surfaces

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Interactive Design and Visualization of Tensor
Fields on Surfaces

• Eugene Zhang James Hays Greg Turk
• Oregon State University Carnegie Mellon
  University Georgia Tech

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Tensors in Nature
Image courtesy of pics4learning.com
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Vectors and Tensors

• Vectors one-way street
• Tensors two-way street

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Vectors and Tensors

• When representing natural directions in images
  and shapes, tensor fields provide a larger
  vocabulary of visual elements
• Tensor field features cannot always be modeled by
  any continuous vector field
• Tensor fields can be used to mimic any given
  vector field
• What happens if we insist on using vector fields?

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Vectors and Tensors
tensor-based edge field
vector-based edge field
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Vectors and Tensors
tensor-based edge field
vector-based edge field
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Vectors and Tensors
?
?
?
?
?
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Tensor Field Design

• Many graphics applications require 2D
  second-order symmetric tensor fields
• painterly rendering (edge field)
• pen-and-ink sketch of surfaces (curvature
  tensor)
• anisotropic remeshing (curvature tensor)
• tensor field visualization

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Tensor Field Design

• Different tensor fields lead to different effects

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Tensor Field Design

• Unwanted degenerate points cause visual artifacts

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Tensor Field Design

• Tensor field design on surfaces
• creates second-order symmetric tensor fields
• from scratch or by modifying an existing field
• requires relatively little user effort
• provides control over tensor field topology
• number and location of degenerate points

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Background on Tensors

• A second-order symmetric tensor
• are also eigenvectors
• We focus on the directional information in a
  tensor

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Background on Tensors

• Deviator-isotropic decomposition

isotropic tensor
deviator

• A tensor has the same set of eigenvectors as its
  deviator
• Isotropic tensors and deviators form
  complementary linear subspaces

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Background on Tensors

• A tensor field is a continuous tensor-valued
  function
• Our interest
• use tensor fields to describe and control
  directional information in images or shapes

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Background on Tensors

• Tensor lines
• Degenerate points
• Separatrices
• Topology

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Background on Tensors

• Fundamental types of degenerate points

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Background on Tensors

• Tensor index wedges

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Background on Tensors

• Tensor index trisectors

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Video
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Related Work

• Tensor field visualization and analysis (a rather
  incomplete list)
• Tensor field topology
• Delmarcelle and Hesselink 1993
• Tricoche and Scheuermann 2003
• Tensor field smoothing
• Alliez et al. 2003
• Vector and tensor field visualization
• Delmarcelle and Hesselink 1993
• Carbal and Leedom 1993,
• Zheng and Pang 2003
• van Wijk 2002, 2003

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Related Work

• Vector design systems
• Special-purpose
• Texture synthesis Praun et al. 2000, Turk
  2001, Wei and Levoy 2001
• Fluid simulation Stam 2003
• Vector field visualization van Wijk 2002, 2003
• Hair modeling Yu 2001
• Pen-and-ink sketch Salisbury et al. 1997
• General-purpose
• Planar Rockwood and Bunderwala 2001, Treisel
  2002
• Surfaces Zhang et al. 2004

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Tensor Field Design

• Major issues
• Feature specification
• Degenerate point control
• Surface tensor field design (details omitted)
• Interactive visual feedback (details omitted)

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Our Approach

• A two-stage algorithm
• Initialization feature specification
• Iterative editing degenerate point control
• A similar pipeline was used in the vector field
  design system of Zhang et al. 2004

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Initialization

• Each element is extended to a global basis field
• Global basis fields are summed along with an
  input field
• zero
• numerical estimation of a meaningful tensor field

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Initialization

• Design elements (basic)

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Initialization

• Design elements (higher-order)

3rd
4th
5th
6th
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Initialization

• Design elements (higher-order)

-6rd
-5th
-4th
-3rd
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Editing Topological Editing

• Degenerate point movement and pair cancellation
• These operations are at the core of our system

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Topological Editing

• Topological editing on a tensor field is
  difficult
• No algorithm on degenerate point movement
• Algorithms on pair cancellation assume the
  degenerate point pair are the closest neighbors
  to each other
• Efficient algorithms exist in vector field domain
• Can we borrow algorithms from vector field domain
  and apply them to tensor fields?

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Conversions

• Given a domain D
• Let V(D) and T(D) be the set of continuous vector
  and deviate tensor fields over D
• We define the following mapping
• m T(D) V(D)

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Conversions

• is a degenerate point for a tensor field T
  if and only if it is a singularity for
  V(D)
• The tensor index is half of the vector (Poincaré)
  index

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Conversion
vector
tensor
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Conversions
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Conversions
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Conversions
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Back to Topological Editing

• Our approach
• convert the tensor field into a vector field
  using ,
• perform topological editing to the vector field,
• convert the edited vector field to a tensor field
  using
• Step 2 can be implemented using any existing
  vector field editing algorithm
• We use algorithms that are based on Conley index
  theory (Zhang et al 2004)

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Adapting to Surfaces

• Details omitted in the interest of time

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Applications

• Painterly rendering
• Pen-and-ink sketch

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Painterly Rendering

• Feature designs (vector vs. tensor)

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Painterly Rendering

• Feature designs (vector vs. tensor)

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Pen-and-ink Sketch

• Numerical estimation vs. design

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Conclusion

• Natural directions in images and shapes are
  better described by tensor fields

tensor-based edge field
vector-based edge field
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Conclusion

• Tensor field design is important in graphics
• Different effects can be achieved by using
  different fields
• Problems in numerical estimation can be corrected

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Summary

• Our design system
• first tensor field design system
• can generate a wide variety of tensor fields
• uses design elements of any order
• provides control over degenerate points
• works on surfaces (details omitted)
• Conversions crucial in our system
• A high-quality interactive visualization
  algorithm (details omitted)

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Future Interests

• Control over other aspects of tensor field
  topology
• separatrices and periodic orbits
• Extension to other domains
• volume
• Other applications
• anisotropic remeshing
• Higher-order tensors

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Acknowledgement

• Irfan Essa
• Konstantin Mischaikow
• Spencer Reynolds for A/V help
• Andrzej Szymczak and Cyberware for 3D models
• Pics4learning.com for images

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Thank you!