CDS 301

Spring, 2013

CH3: Data Representation

Jan. 29, 2013 - Feb. 12, 2013

Jie Zhang Copyright ©

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

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

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

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:

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

Continuous Dataset

f) C, (D,cD1. D: Function domain

2. C: Function co-domain

3. f: Function itself

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

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

Sampling and Reconstruction

fff i

~

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

Jan. 29, 2013 Stopped Here

(To be continued)

Jan. 31, 2013

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

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

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

Reconstruction

function basis called is where

)()(~

}...2,1{},

i

1

xfxf

Ni{f

N

i

ii

i

or interpolation function

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

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

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

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.

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

cell=(p1,p2,p3,p4)

D: (x,y,z)

cell=(v1,v2,v3,v4)

D: (r,s,t) and t=0

Basis Function

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

)),,((),,(~

)),,((),,(),,(

),,(),,(

),,(),,(

CH3.4 Cell Types

Vertex s=0

1),(

}{

0

1

1

sr

vc

Line s=1

2

12

1211

1

2

1

1

21

||||

)()(),,()(

)(

1)(

},{

pp

ppppzyxTr

rr

rr

vvc

Line (cont.)

Actual line along the X-axis

12

12

12

21

12

1

2211 ,,

xx

xxf

xx

xxff

xx

xxr

xpxpxp

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

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

Line (cont.)

Actual line in X-Y plane

2

12

2

12

121121

222111

)()(

))(())((

),(),,(),,(

yyxx

yyyyxxxxr

yxpyxpyxp

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

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.

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

Jan. 31, 2013 Stopped Here

(To be continued)

February 05, 2013

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

)),,((),,(~

)),,((),,(),,(

),,(),,(

),,(),,(

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

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

Volume Visualization

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

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

Hexahedron (cont.) s=3

2

18

181

2

14

141

2

12

121

1

||||

)()(

||||

)()(

||||

)()(

),,(),,(

pp

ppppt

pp

pppps

pp

ppppr

tsrzyxT

Effect of Reconstruction

Geometry:

Constant

Geometry:

Linear

Lighting:

Constant

Staircase

shading

Flat

Shading

Lighting:

Linear

--------- Smooth

(Gouraud)

shading

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

Homework #2 1. Write a portable Matlab program to create the

following two images. Do not use the existing GUI.

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

Uniform Grid

2-D 3-D

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

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

Rectilinear Grid

2-D 3-D

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

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

Structured Grid

Circular domain Curved Surface 3D volume

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…}

Unstructured Grid

Feb. 05, 2013 Stopped Here

(To be continued)

February 12, 2013

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

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

Scalar Attributes

•E.g., temperature, density,

RR

RR

3

2

:

or ,:

f

f

Vector Attributes

E.g.,

•Normal direction

•Force

•velocity

•A vector has a magnitude and orientation

3

2

RR

RR

3

2

:

or ,:

f

f

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

Tensor Attributes

•E.g. curvature of a 2-D surface

•E.g., diffusivity, conductivity, stress

Tensor

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)

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

RGB Cube R

G

B

yellow

magenta

Cyan

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

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

HSV Color Wheel

HSV Color Wheel

(V=1) http://www.codeproject.com/KB/miscctrl/CPicker/ColorWheel.jpg

S=0

S=1

Color, Light, Electromagnetic

Radiation

http://www.dnr.sc.gov/ael/personals/pjpb/lecture/spectrum.gif

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)

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)

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.