ch3: data representationsolar.gmu.edu/.../ch3_data_representation.pdf · ch3.1 continuous data...
TRANSCRIPT
![Page 1: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/1.jpg)
CDS 301
Spring, 2013
CH3: Data Representation
Jan. 29, 2013 - Feb. 12, 2013
Jie Zhang Copyright ©
![Page 2: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/2.jpg)
Outline
3.1. Continuous Data
3.2. Sampled Data
3.3. Discrete Datasets
•Basis function for reconstruction
3.4. Cell Types
•point, line, triangle, quad, tetrahedron, hexahedron
•Transformation between world cell and reference
cell
3.5. Grid Types
•Uniform, rectlinear, structured, Unstructured
3.6. Attributes
•Scalar, Vector, Color, Tensor, Non-numerical
3.7. Computing Derivatives of Sampled Data
3.8. Implementation
3.9. Advanced Data Representation
![Page 3: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/3.jpg)
CH3.1 Continuous Data versus
Discrete Data
•Continuous Data (Intrinsic)
•Most scientific quantities are continuous in nature
•Scientific Visualization, or scivis
•Discrete Data (Intrinsic)
•E.g., text, images and others that can not be
interpolated or scaled
•Information Visualization, or infovis
•Continuous data, when represented by computers, are
always in discrete form
•These are often called “sampled data”
•Originated from continuous data
•Intended to approximate the continuous quantity
![Page 4: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/4.jpg)
Continuous Data
2
2 )()('',
)()(),(
dx
xfdxf
dx
xdfxfxf
a: discontinuous
function
b: first-order
continuous
function
c: high-order
continuous
function Discontinuous:
with jumps or holes
![Page 5: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/5.jpg)
Continuous Data
f is a function: d-geometrid dimension, and c values
D: function domain
C: function co-domain
)....,()....,(
C
D
CD:
2121 dc
c
d
xxxfyyy
f
R
R
Continuous data can be modeled as:
![Page 6: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/6.jpg)
Continuous Data
||f(p)-f(x)||
then
||p-x||
ifsuch that
0 0,
Cauchy criterion of continuity
Graphically, a function is continuous if the graph of the
function is a connected surface without “holes” or “jumps”
In words, small changes in the input always result in small
changes in the output
For all ε, there
exists a δ
![Page 7: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/7.jpg)
Continuous Dataset
f) C, (D,cD1. D: Function domain
2. C: Function co-domain
3. f: Function itself
![Page 8: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/8.jpg)
Dimensions
•Geometric dimension: d
• the space into which the function domain D is embedded
•It is always 3 in the usual Euclidean space: d=3
•Topological dimension: s
•The function domain D itself
•A line or curve: s=1, d=3
•A plane or curved surface: s=2, d=3
•It is determined by how many independent variables in
the function
•Dataset dimension refers to the topological dimension
•Function values in the co-domain are called dataset attributes
•Attribute dimension: dimension of the function co-domain
![Page 9: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/9.jpg)
CH3.2 Sampling and
Reconstruction
To prepare for the visualization of the continuous dataset:
1. Sampling
• For a given continuous data, produce a set of sampled
data points
• as an input to the computer
2. Reconstruction
• For a given sampled dataset, recover an approximate
version of the original continuous data
• The approximation shall be of piecewise continuous
functions
Computer can not “see” continuous data domain,
because of limited spatial resolution
![Page 10: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/10.jpg)
Sampling and Reconstruction
fff i
~
![Page 11: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/11.jpg)
Sampled Dataset
Sampled dataset
f) C, (D,D
})({ k
iiii },{Φ}, {f}, {CpsD
Continuous dataset
1. p: sampled points
2. c: cell or shape that groups neighboring points
3. f: sampled values
4. Φ: basis function or interpolation function
![Page 12: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/12.jpg)
Jan. 29, 2013 Stopped Here
(To be continued)
![Page 13: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/13.jpg)
Jan. 31, 2013
![Page 14: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/14.jpg)
Review
Sampled dataset
f) C, (D,D
})({ k
iiii },{Φ}, {f}, {CpsD
Continuous dataset
1. p: sampled points
2. c: cell or shape that groups neighboring points
3. f: sampled values
4. Φ: basis function or interpolation function
![Page 15: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/15.jpg)
Sampling •Sampled data is provided only at the vertices of the
cells of a given grid
•Sample points (pi): where the data domain is sampled
•Cell (ci): primitive shapes formed by a set of points as
the vertices of the cell, e.g., polygons
•Grid (or mesh)(D): the union of cells that completely
covers the topological domain of the data.
D
0
}....,{
,
21
ii
ji
ni
d
i
cU
ji,cc
pppc
p RPoint
Cell
Grid
![Page 16: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/16.jpg)
Sampling (on the domain D)
A data domain is sampled in a grid that contains a set of
cells defined by the sample points as vertices
Point,
Cell,
Grid
![Page 17: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/17.jpg)
Reconstruction
function basis called is where
)()(~
}...2,1{},
i
1
xfxf
Ni{f
N
i
ii
i
or interpolation function
![Page 18: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/18.jpg)
Reconstruction 1-D example:
•sinuous function
•10 points sampling
•Cell type: line segment
•Cell vertices: two vertices per cell
2/)()(
value,same thehave wouldcell e within thpoints All
vertices) twoof (because 2/1
function, basisconstant is If
)()(
}p,{pc
21
2
1
21j
ffxf
(x)
xfxf
j
i
i
iij
staircase
![Page 19: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/19.jpg)
Reconstruction
Constant basis function
i
icx
cNx
,0
x,/1{)(
i0
It has virtually no
computational cost,
but provide a poor,
staircase like
approximation of the
original function
![Page 20: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/20.jpg)
Reconstruction
Linear basis function
rfrfrf
rr
rr
21
1
2
1
1
)1()(
,)(
,1)(
For 1-D line
P1 P2
0 r 1
reference cell
![Page 21: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/21.jpg)
Basis Function
Basis function shall be orthonormal
1. Orthogonal: only vertex points within the
same cell have contribution to the
interpolated value
2. Normal: the sum of the basic functions of all
the vertices shall be unity.
![Page 22: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/22.jpg)
Basis Function (Example) Linear basis function: indicated by “1” in the superscript
For 2-D quad
srsr
rssr
srsr
srsr
)1(),(
),(
)1(),(
)1)(1(),(
1
4
1
3
1
2
1
1
![Page 23: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/23.jpg)
cell=(p1,p2,p3,p4)
D: (x,y,z)
cell=(v1,v2,v3,v4)
D: (r,s,t) and t=0
Basis Function
![Page 24: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/24.jpg)
Coordinate Transformation •Basis function is defined in reference cell
•Reference cell: axis-aligned unit cell, e.g., unit
square in 2-D, unit line in 1-D
•Data are sampled at actual (world) cells
•Mapping between actual cell and reference cell
N
i
ii zyxTfzyxf
zyxTtsrzyx
zyxTtsr
tsrTzyx
1
1
1
1
)),,((),,(~
)),,((),,(),,(
),,(),,(
),,(),,(
![Page 25: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/25.jpg)
CH3.4 Cell Types
![Page 26: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/26.jpg)
![Page 27: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/27.jpg)
Vertex s=0
1),(
}{
0
1
1
sr
vc
![Page 28: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/28.jpg)
Line s=1
2
12
1211
1
2
1
1
21
||||
)()(),,()(
)(
1)(
},{
pp
ppppzyxTr
rr
rr
vvc
![Page 29: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/29.jpg)
Line (cont.)
Actual line along the X-axis
12
12
12
21
12
1
2211 ,,
xx
xxf
xx
xxff
xx
xxr
xpxpxp
![Page 30: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/30.jpg)
Example
Question:
Considering the following two cells, C1 and C2, of the line-
segment type in 1-D function domain: C1={P1,P2}, C2={P2, P3},
where P1=15, P2=30, and P3=45 respectively. The sampling
values at the three sampling points are f1=0.7, f2=3.4 and f3=5.0,
respectively. You are asked to reconstruct the data values using
interpolation at points PA=22 and PB=40
(1) Find fA and fB using a constant basis function
(2) Find fA and fB using a linear basis function
![Page 31: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/31.jpg)
Example Answer:
(1) In the case of constant basis function
(2) In the case of linear basis function
2.4
1.2
B
A
f
f
5.4
0.2
B
A
f
f
![Page 32: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/32.jpg)
Line (cont.)
Actual line in X-Y plane
2
12
2
12
121121
222111
)()(
))(())((
),(),,(),,(
yyxx
yyyyxxxxr
yxpyxpyxp
![Page 33: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/33.jpg)
Quad s=2
2
14
141
2
12
121
1
4
1
3
1
2
1
1
4321
||||
)()(
||||
)()(
)1(),(
),(
)1(),(
)1)(1(),(
},,,{
pp
pppps
pp
ppppr
srsr
rssr
srsr
srsr
vvvvc
![Page 34: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/34.jpg)
Example
Question:
Considering an actual cell of quad type in 3-D space, C1={P1,
P2, P3, P4}, where P1=[1.0, 1.0, 1.0] and P2=[2.0, 1.0, 1.0],
P3=[2.0, 2.0 ,1.5], and P4=[1.0, 2.0, 1.8], The functional values of
the four vertex points are f1=1.0, f2=1.2, f3=1.5 and f4=1.5. You
are asked to reconstruct the functional value at point PA, where
PA=[1.5, 1.2, 1.3]
(1)To find fA, use constant basis function
(2)To find fA, use linear basic function.
![Page 35: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/35.jpg)
Example Answer:
(1) In the case of constant basis function
(2) In the case of linear basis function
30.1Af
23.1Af
See scanned note for details
![Page 36: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/36.jpg)
Jan. 31, 2013 Stopped Here
(To be continued)
![Page 37: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/37.jpg)
February 05, 2013
![Page 38: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/38.jpg)
Review •Basis function is defined in reference cell
•Reference cell: axis-aligned unit cell, e.g., unit
square in 2-D, unit line in 1-D
•Data are sampled at actual (world) cells
•Mapping between actual cell and reference cell
N
i
ii zyxTfzyxf
zyxTtsrzyx
zyxTtsr
tsrTzyx
1
1
1
1
)),,((),,(~
)),,((),,(),,(
),,(),,(
),,(),,(
![Page 39: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/39.jpg)
Triangle s=2
2
13
131
2
12
121
1
1
3
1
2
1
1
321
||||
)()(
||||
)()(
),(),,(
),(
),(
1),(
},,{
pp
pppps
pp
ppppr
srzyxT
ssr
rsr
srsr
vvvc
![Page 40: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/40.jpg)
Cell Types
1. The cell types of quads and triangles are well
suited for surface visualization.
2. But for volume visualization (s=3), topologically
3-D cells (s=3) are needed
![Page 41: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/41.jpg)
Volume Visualization
![Page 42: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/42.jpg)
Tetrahedron s=3
2
14
141
2
13
131
2
12
121
1
1
4
1
3
1
2
1
1
4321
||||
)()(
||||
)()(
||||
)()(
),,(),,(
),(
),(
),(
1),(
},,,{
pp
ppppt
pp
pppps
pp
ppppr
tsrzyxT
tsr
ssr
rsr
tsrsr
vvvvc
![Page 43: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/43.jpg)
Hexahedron s=3
strsr
rstsr
tsrsr
tsrsr
trsr
trssr
tsrsr
tsrsr
vvvvvvvvc
)1(),(
),(
)1(),(
)1)(1(),(
)1)(1(),(
)1(),(
)1)(1(),(
)1)(1)(1(),(
},,,,,,,{
1
8
1
7
1
6
1
5
1
4
1
3
1
2
1
1
87654321
![Page 44: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/44.jpg)
Hexahedron (cont.) s=3
2
18
181
2
14
141
2
12
121
1
||||
)()(
||||
)()(
||||
)()(
),,(),,(
pp
ppppt
pp
pppps
pp
ppppr
tsrzyxT
![Page 45: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/45.jpg)
Effect of Reconstruction
Geometry:
Constant
Geometry:
Linear
Lighting:
Constant
Staircase
shading
Flat
Shading
Lighting:
Linear
--------- Smooth
(Gouraud)
shading
![Page 46: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/46.jpg)
Effect of Reconstruction
Staircase Shading Flat Shading Smooth Shading
Note: all three images are based on (1) the same data sampling, (2)
the same data mapping, but (3) different rendering
![Page 47: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/47.jpg)
Homework #2 1. Write a portable Matlab program to create the
following two images. Do not use the existing GUI.
![Page 48: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/48.jpg)
CH 3.5 Grid Types
•Grid is the pattern of cells in the data domain
•Grid is also called mesh
•Uniform grid
•Rectilinear grid
•Structured grid
•Unstructured grid
![Page 49: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/49.jpg)
Uniform Grid
2-D 3-D
![Page 50: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/50.jpg)
Uniform Grid •The simplest grid type
•Domain D is usually an axis-aligned box
•Line segment, when s=1
•Rectangle, when s=2
•Parallelepiped, when for s=3
•Sample points are equally distributed on every axis
•Structured coordinates: the position of the sample
points in the data domain are simply indicated by d
integer coordinates (n1,..nd); d=1, 2 or 3
•Simple to implement
•Efficient to run (storage, memory and CPU)
•Numerical simulations are mostly run on uniform grid
![Page 51: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/51.jpg)
Uniform Grid •Data points, even s=3, can be simply stored in the
increasing order of the indices, that is, an 1-D array
•Lexicographic order
213121
121
12
121
2
1
1
1
3,d If
mod
2,d If
)(
NNnNnni
)N (ni n
i/Nn
, orNnni
Nnnid
k
k
l
lk
![Page 52: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/52.jpg)
Rectilinear Grid
2-D 3-D
![Page 53: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/53.jpg)
Rectilinear Grid
•Domain D is also an axis-aligned box
•However, the sampling step is not equal
•It is not as simple or as efficient as the uniform grid
•However, improving modeling power
![Page 54: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/54.jpg)
Structured Grid
•Further relaxing the constraint, a structured grid can be
seen as the free deformation of a uniform or rectilinear
grid
•The data domain can be non-rectangular
•It allows explicit placement of every sample points
•The matrix-like ordering of the sampling points are
preserved
•Topology is preserved
•But, the geometry has changed
![Page 55: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/55.jpg)
Structured Grid
Circular domain Curved Surface 3D volume
![Page 56: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/56.jpg)
Unstructured Grid •It is allowed to define both sample points and cells explicitly
•The most general and flexible grid type
•However, it needs to store
•The coordinates of all sample points pi
•For each cell, the set of vertex indices ci={vi1,…viCi), and
for all cells {c1,c2…}
![Page 57: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/57.jpg)
Unstructured Grid
![Page 58: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/58.jpg)
Feb. 05, 2013 Stopped Here
(To be continued)
![Page 59: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/59.jpg)
February 12, 2013
![Page 60: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/60.jpg)
CH 3.6 Attributes
•Attribute data are the set of sample values of a sampled
dataset
•Attribute = {fi}
})({ k
iiii },{Φ}, {f}, {CpsD
Sampled dataset
![Page 61: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/61.jpg)
Attribute Types
•Scalar Attribute
•Vector Attribute
•Color Attribute: c=3
•Tensor Attributes c = (3 x 3) matrix
•Non-Numerical Attributes
1
C
c
cR
3or ,2
C
cc
cR
![Page 62: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/62.jpg)
Scalar Attributes
•E.g., temperature, density,
RR
RR
3
2
:
or ,:
f
f
![Page 63: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/63.jpg)
Vector Attributes
E.g.,
•Normal direction
•Force
•velocity
•A vector has a magnitude and orientation
3
2
RR
RR
3
2
:
or ,:
f
f
![Page 64: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/64.jpg)
Tensor Attributes
•A high-dimensional generalization of vectors
VV
VyVyVxV
VVVVVV
VVVVVV
VVVVVV
BzAzByAzBxAz
BzAyByAyBxAy
BzAxByAxBxAx
),,(
,,
,,
,, ,
BAVVVTensor
Vector
Scalar
•A tensor describes physical quantities that depend on
both position and the direction at the position
•Vector and scalar describes physical quantities that
depend on position only
![Page 65: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/65.jpg)
Tensor Attributes
•E.g. curvature of a 2-D surface
•E.g., diffusivity, conductivity, stress
Tensor
![Page 66: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/66.jpg)
Non-numerical Attributes
•E.g. text, image, voice, and video
•Data can not be interpolated
•Therefore, the dataset has no basis function
•Domain of information visualization (infovis)
![Page 67: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/67.jpg)
Color Attributes
•A special type of vector attributes with dimension c=3
•RGB system: convenient for hardware and implementation
•R: red
•G: green
•B: blue
•HSV system: intuitive for human user
•H: Hue
•S: Saturation
•V: Value
![Page 68: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/68.jpg)
RGB Cube R
G
B
yellow
magenta
Cyan
![Page 69: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/69.jpg)
RGB System
•Every color is represented as a mix of “pure” red, green and
blue colors in different amount
•Equal amounts of the three colors determines gray shades
•RGB cube’s main diagonal line connecting the points (0,0,0)
and (1,1,1) is the locus of all the grayscale value
•(0,0,0) - dark
•(1,1,1) - white
•(0.5,0.5,0.5) - intermediate gray
•(1,0,0) - red
•(0,1,0) - green
•(1,1,0) -yellow
•(1,0,1) - magenta
•(0,1,1) - cyan
![Page 70: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/70.jpg)
HSV System
•Hue: distinguish between different colors of different
wavelengths, from red to blue
•Human eyes are sensitive to visible wavelengths
•Saturation: represent the color of “purity”, or how much hue
is diluted with white
•S=1, pure, undiluted color
•S=0, whitened
•Value: represent the brightness, or luminance
•V=0, always dark
•V=1, brightest color for a given H and S
![Page 71: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/71.jpg)
HSV Color Wheel
HSV Color Wheel
(V=1) http://www.codeproject.com/KB/miscctrl/CPicker/ColorWheel.jpg
S=0
S=1
![Page 72: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/72.jpg)
Color, Light, Electromagnetic
Radiation
http://www.dnr.sc.gov/ael/personals/pjpb/lecture/spectrum.gif
![Page 73: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/73.jpg)
RGB to HSV •All values are in [0,1]
max=max(R,G,B)
min=min(R,G,B)
diff=max-min
•V = max
•largest RGB component
•S = diff/max
•For hue H, different cases
•H = (G-B)/diff if R=max
•H =2+(B-R)/diff if G=max
•H =4+(R-G)/diff if B=max
•then H=H/6
•H=H+1 if H < 0
Exp: Full Green Color
(R,G,B)=(0,1,0)
(H,S,V)=(1/3, 1,1)
Exp: Yellow Color
(R,G,B)=(1,1,0)
(H,S,V)=(1/6, 1, 1)
![Page 74: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/74.jpg)
HSV to RGB
huecase = {int} (h*6)
frac = 6*h – huecase
lx= v*(1-s)
ly= v*(1-s*frac)
lz= v*(1-s(1-frac))
huecase =6 (0<h<1/6): r=v, g=lz, b=lx
huecase =1 (1/6<h<2/6): r=ly, g=v, b=lx
huecase =2 (2/6<h<3/6): r=lx, g=v, b=lz
huecase =3 (3/6<h<4/6): r=lx, g=ly, b=v
huecase =4 (4/6<h<5/6): r=lz, g=lx, b=v
huecase =5 (5/6<h<1): r=v, g=lx, b=ly
Exp: Full Green Color
(H,S,V)=(1/3,1,1)
(R,G,B)=(0,1,0)
Exp: Yellow Color
(H,S,V)=(1/6,1,1)
(R,G,B)=(1,1,0)
![Page 75: CH3: Data Representationsolar.gmu.edu/.../ch3_data_representation.pdf · CH3.1 Continuous Data versus Discrete Data •Continuous Data (Intrinsic) •Most scientific quantities are](https://reader033.vdocument.in/reader033/viewer/2022043011/5fa481b435300420a834f613/html5/thumbnails/75.jpg)
End
of Chap. 3
Note:
•The class has covered section 3.1 to 3.6.
•We have skipped section 3.7, 3.8, and 3.9. You may
want to study these section by yourselves.