d scalar/vector/tensor
TRANSCRIPT
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3D+time (n<4) Scalar/vector/tensor
Source: VIS, University of Stuttgart
Information data nD (n>3)
Scientific data
Heterogeneous
Data we are discussing in the class
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Scalar Data Analysis
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Scalar Fields
• The approximation of certain scalar function in physical space f(x,y,z).
• Discrete representation.
• Visualization primitives: colors, transparency, iso-contours (2D), iso-surfaces (3D), 3D textures.
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3D Surface
• 3D surfaces can be considered some sub-sets corresponding to certain iso-values of some scalar functions (e.g. a distance field).
• Therefore, analyzing the characteristics of the surfaces can help understand the scalar function they represent.
Source: http://www.cs.utah.edu/~angel/vis/project3/
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Shape Analysis
• What features do people care about?– Computing orientation and range of the shape
– Computing extrema (protrusion)
– Finding handles and holes
– Finding ridges and valleys or other feature lines
– Computing skeletal representation
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Sub-Topics
• Compute bounding box
• Compute Euler Characteristic
• Estimate surface curvature
• Line description for conveying surface shape
• Morse function and surface topology--Reeb
graph
• Scalar field topology--Morse-Smale complex
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Bounding Box
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Shape Descriptors
• We need centers and variances of all points on
the surface.
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Shape Descriptors
• We need centers and variances of all points on
the surface.
There are different definitions of centers
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Center of Mass
),( ii yx
),( jj yx
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Center of Mass
• Center of Mass
),( ii yx
),( jj yx
),( 11
N
y
N
xN
i i
N
i i ∑∑ ==
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Geometric Center
• Geometric center
),( ii yx
),( jj yx
)2
maxmin,
2
maxmin( iiii yyxx ++
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Centers
• Geometric center
• Center of Mass
• Which one is better and why?
),( 11
N
y
N
xN
i i
N
i i ∑∑ ==
)2
maxmin,
2
maxmin( iiii yyxx ++
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Variance
• What are the dominant directions?
),( ii yx
),( jj yx
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Variance
• What are the dominant directions?
• How are they defined?
• Principle components analysis
),( ii yx
),( jj yx
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Variance
• What are the dominant directions?
),( ii yx
),( jj yx
−−−−
−−−−=
∑∑
∑∑
iyiyi
iyixi
iyixi
ixixi
cycycycx
cycxcxcxV
))(())((
))(())((Covariance matrix
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Variance
• What are the dominant directions?
• Major, minor eigenvectors give the directions
),( ii yx
),( jj yx
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Variance
),( ii yx
),( jj yx
[ ]ykxkyk
xk
ykykxk
ykxkxk cycxcy
cx
cycycx
cycxcx−−
−−
=
−−−−−−2
2
)())((
))(()(
),( kk yxp =
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Variance
),( ii yx
),( jj yx
[ ]ykxkyk
xk
ykykxk
ykxkxk cycxcy
cx
cycycx
cycxcx−−
−−
=
−−−−−−2
2
)())((
))(()(
),( kk yxp =
• It is a voting scheme
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Bounding Boxes
),( ii yx
),( jj yx
• Axis-aligned bounding box
• Oriented bounding box
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Oriented Bounding Boxes
),( ii yx
),( jj yx
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Oriented Bounding Boxes
),( ii yx
),( jj yx
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Oriented Bounding Boxes
),( ii yx
),( jj yx
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Oriented Bounding Boxes
),( ii yx
),( jj yx
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Oriented Bounding Boxes
),( ii yx
),( jj yx
•How do you extend this to 3D?
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Compute Euler Characteristics
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Surface Representation
• Consider a discrete polygonal representation
– V – number of vertices
– E – number of edges
– F – number of faces
Euler characteristics L = V-E+F
Euler characteristic is a topological invariant, a
number that describes a topological space's
shape or structure regardless of the way it is
bent. (Wiki)
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Elementary Collapse on Edges
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Elementary Collapse on Edges
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Elementary Collapse on Edges
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Elementary Collapse on Edges
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Elementary Collapse on Edges
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Elementary Collapse on Edges
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Elementary Collapse on Edges
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Elementary Collapse on Edges
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Elementary Collapse on Edges
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Elementary Collapse on Edges
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Elementary Collapse on Edges
V=E for a closed and simple planar curve.
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Elementary Collapse on Edges
V=E for a closed and simple planar curve.
What about 3D surfaces ?
Need to consider the merging of the faces!
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Elementary Collapse for Faces
V=6
E=7
F=2
L= ?
V=6
E=6
F=1
L= ?
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Euler Characteristic of a Cube
V=1
E=0
F=1
L= ?
V=8
E=12
F=6
L= ?
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Dual of a Hexahedron
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Dual of a Hexahedron
Does this dual change the Euler characteristic?
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What Do Different Euler
Characteristics Tell us?
L=?
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What Do Different Euler
Characteristics Tell us?
L=0
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What Do Different Euler
Characteristics Tell us?
What is its Euler characteristic?
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What Do Different Euler
Characteristics Tell us?
What is its Euler characteristic?
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What Do Different Euler
Characteristics Tell us?
L = -2!
What about 3D surfaces?
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What Do Different Euler
Characteristics Tell us?
For 3D surfaces, L=V-E+F=?
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Acknowledge
• Part of the materials are provided by
– Prof. Eugene Zhang at Oregon State University