Spatial Data: 3D Scalar Fields CSC444 Recap: 2D contouring - - PowerPoint PPT Presentation

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Spatial Data: 3D Scalar Fields CSC444 Recap: 2D contouring - - PowerPoint PPT Presentation

Nov 15, 2023 106 likes •607 views

Spatial Data: 3D Scalar Fields CSC444 Recap: 2D contouring https://www.e-education.psu.edu/geog486/node/1873 Recap: 2D contouring Cases + - Case Polarity Rotation Total No Crossings x2 2 (x2 for Singlet x2 x4 8 polarity) x2

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Spatial Data: 3D Scalar Fields

CSC444

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Recap: 2D contouring

https://www.e-education.psu.edu/geog486/node/1873

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Recap: 2D contouring Cases

No Crossings Case Polarity Rotation Total x2 2 Singlet x2 8 x4 Double adjacent x2 4 x2 (4) Double Opposite x2 2 x1 (2) (x2 for polarity) 16 = 24

+

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3D Contouring

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3D Contouring

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Splitting 3D space into simple shapes

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Cube into tetrahedra

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Cube into tetrahedra

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Cube into tetrahedra

1 tetrahedron,

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Cube into tetrahedra

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Cube into tetrahedra

2 tetrahedra

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Cube into tetrahedra

1 cube splits into 6 tetrahedra

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Cube into tetrahedra

1 cube splits into 6 tetrahedra… but also into 5 tetrahedra!

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Cube into tetrahedra

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Marching Tetrahedra

3 cases, “obvious”

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Marching Tetrahedra

3 cases, “obvious”

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3D Contouring

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3D Contouring

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`

3D Contouring

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Spatial Data: Vector Fields

http://hint.fm/wind

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Experimental Flow Vis

von Kármán vortex street, depending on Reynolds number

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http://envsci.rutgers.edu/~lintner/teaching.html

Guadalupe Island

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Mathematics of Vector Fields

v : Rn → Rn

Function from vectors to vectors

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Spatial Data: Vector Fields

https://www.youtube.com/watch?v=nuQyKGuXJOs

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A simple vector field: the gradient

https://www.youtube.com/watch?v=v0_LlyVquF8

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Vector fields can be more complicated

http://www.math.umd.edu/~petersd/241/html/ex27b.html v(x, y) = (cos(x + 2y), sin(x − 2y))

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Glyph Based Techniques

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Hedgehog Plot: Not Very Good

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Hedgehog Plot: Not Very Good

From Laidlaw et al.’s “Comparing 2D Vector Field Visualization Methods: A User Study”, TVCG 2005

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Uniformly-placed arrows: Not Very Good Either

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Jittered Hedgehog Plot: Better

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Space-filling scaled glyphs

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Streamline-Guided Placement

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Streamline-Guided Placement

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Streamlines

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Streamlines

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Streamlines

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Curves everywhere tangent to the vector field

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Curves everywhere tangent to the vector field

x0(t) = vx(x(t), y(t)) y0(t) = vy(x(t), y(t))

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Visualization via streamlines

  • Pick a set of seed points
  • Integrate streamlines from those points
  • How do we compute this?
  • https://cscheid.net/writing/data_science/
  • des/index.html
  • Which seed points?
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Uniform placement

Turk and Banks, Image-Guided Streamline Placement SIGGRAPH 1996

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Density-optimized placement

Turk and Banks, Image-Guided Streamline Placement SIGGRAPH 1996

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Density-optimized placement

Turk and Banks, Image-Guided Streamline Placement SIGGRAPH 1996

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Image-Based Vector Field Visualization

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Line Integral Convolution

Cabral and Leedom, Imaging Vector Fields using Line Integral

  • Convolution. SIGGRAPH 1993

http://www3.nd.edu/~cwang11/2dflowvis.html

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Line Integral Convolution

Given a vector field compute streamlines average source of noise along streamlines Result

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Line Integral Convolution

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Advantages

  • “Perfect” space usage
  • Flow features are very

apparent

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Downsides

  • No perception of velocity!
  • No perception of direction!