SCALAR VISUALIZATION

SCALAR VISUALIZATION

OUTLINE

• Visualizing scalar data

• A number of the most popular scalar visualization techniques

• Color mapping

• Contouring

• Height plots

•

Color mapping

•

Design of effective colormaps

•

Contouring in 2D and 3D

•

Height plots

VISUALIZATION PIPELINE

1. Data Importing 2. Data Filtering 3. Data Mapping 4. Date Rendering

SCALAR FUNCTION

10]) [Chap.

ion Visualizat

Volume

Slicing, ,

Isosurface e.g.,

D, - (3

:

) contouring mapping,

color plot,

- height e.g.,

D, - (2

:

histogram) D,

- (1

:

3 2

R R

R R

R R







f

f

f

COLOR MAPPING

•color look-up table

•Associate a specific color with every scalar value

•The geometry of Dv is the same as D

N )

if (N-i)f

c(

c

} {c C

i

,...N ,

i i

max min

2 1

Where

 



LUMINANCE COLORMAP

•Use grayscale to represent scalar value

Luminance Colormap )

y -10(x^{4 4}

e

f 

•Most scientific data (through measurement, observation, or simulation) are intrinsically

grayscale, not color

Legend

RAINBOW COLORMAP

•Red: high value; Blue: low value

•A commonly used colormap

Luminance Map Rainbow Colormap

RAINBOW COLORMAP

•Construction

•f<dx: R=0, G=0, B=1

•f=2: R=0, G=1, B=1

•f=3: R=0, G=1, B=0

•f=4: R=1, G=1, B=0

•f>6-dx: R=1, G=0, B=0

RAINBOW COLORMAP

Implementation

void c(float f, float & R, float & G, float &B) {

const float dx=0.8

f=(f<0) ? 0: (f>1)? 1 : f //clamp f in [0,1]

g=(6-2*dx)*f+dx //scale f to [dx, 6-dx]

R=max(0, (3-fabs(g-4)-fabs(g-5))/2);

G=max(0,(4-fabs(g-2)-fabs(g-4))/2);

B=max(0,(3-fabs(g-1)-fabs(g-2))/2);

}

R

R : c

Dv D

: c





COLORMAP: DESIGNING ISSUES

•Choose right color map for correct perception

•Grayscale: good in most cases

•Rainbow: e.g., temperature map

•Rainbow + white: e.g., landscape

•Blue: sea, lowest

•Green: fields

•Brown: mountains

•White: mountain peaks, highest

RAINBOW COLORMAP

http://atmoz.org/img/weatherchannel_national_temps.png

EXP: EARTH MAP

http://www.oera.net/How2/PlanetTexs/EarthMap_2500x1250.jpg

EXP: SUN IN GREEN-WHITE

COLORMAP

EXP: CORONAL LOOP

http://media.skyandtelescope.com/images/SPD+on+CME+image+5+--+TRACE.gif

COLOR BANDING EFFECT

Caused by a small number of colors in a look-up table

CONTOURING

•A contour line C is defined as all points p in a dataset D that have the same scalar value, or isovalue s(p)=x

} )

(

| {

)

( x p D s p x

C   

•A contour line is also called an isoline

•In 3-D dataset, a contour is a 2-D surface, called isosurface

CONTOURING

Cartograph

CONTOURING

One contour at s=0.11

S > 0.11 S < 0.11

Contouring and Color Banding

CONTOURING

7 contour lines

Contouring and Colormapping:

Show (1) the smooth

variation and (2) the

specific values

PROPERTIES OF CONTOURS

•Indicating specific values of interest

•In the height-plot, a contour line corresponds with the interaction of the graph with a horizontal plane of s value

PROPERTIES OF CONTOURS

•The tangent to a contour line is the direction of the function’s minimal (zero) variation

•The perpendicular to a contour line is the direction of the function’s maximum variation: the gradient

Contour lines Gradient vector

CONSTRUCTING CONTOURS

V=0.48

Finding line segments within cells

CONSTRUCTING CONTOURS

•For each cell, and then for each edge, test whether the isoline value v is between the attribute values of the two edge end points (v_i, v_j)

•If yes, the isoline intersects the edge at a point q, which uses linear interpolation

i j

i j

j i

v v

v v

p v

v q p







 (  ) ( )

•For each cell, at least two points, and at most as many points as cell edges

•Use line segments to connect these edge-intersection points within a cell

•A contour line is a polyline.

CONSTRUCTING CONTOURS

V=0.37: 4 intersection points in a cell -> Contour ambiguity

IMPLEMENTATION: MARCHING SQUARES

•Determining the topological state of the current cell with respect to the isovalue v

•Inside state (1): vertex attribute value is less than isovalue

•Outside state (0): vertex attribute value is larger than isovalue

•A quad cell: (S₃S₂S₁S₀), 2⁴=16 possible states

•(0001): first vertex inside, other vertices outside

•Use optimized code for the topological state to construct independent line segments for each cell

•Merge the coincident end points of line segments originating from neighboring grid cells that share an edge

Topological State of a Quad Cell

Implementation: Marching Squares

Topological State of a hex Cell

Implementation: Marching Cube

Marching cube generates a set of polygons for each contoured cell:

triangle, quad, pentagon, and hexagon

CONTOURS IN 3-D

•In 3-D scalar dataset, a contour at a value is an isosurface

Isosurface for a value corresponding to the skin tissue of an MRI scan 128³ voxels

CONTOURS IN 3-D

Two nested isosurface:

the outer isosurface is transparent

HEIGHT PLOTS

•The height plot operation is to “warp” the data domain

surface along the surface normal, with a factor proportional to the scalar value

s

h s

D x

x n x s x

x m

D D

m











), (

) ( )

(

,

: 

HEIGHT PLOTS

Height plot over a planar 2-D surface

HEIGHT PLOTS

Height plot over a nonplanar 2-D surface

VECTOR VISULIZATION



Divergence and Vorticity



Vector Glyphs



Vector Color Coding



Displacement Plots



Stream Objects



Texture-Based Vector Visualization



Simplified Representation of Vector Fields

VECTOR FUNCTION

) D -

2 : case (simpler

D) -

3 in (usually

2 2

3 3

R R

f:

R R

f:





VECTOR VERSUS SCALAR

















 

























) , , (

) , , (

) , , (

V V V

) ,

, (

ˆ ˆ ˆ

: Vector

z y x

z y x f

z y x f

z y x f

V or

V V

V V

or

k V j

V i

V V

V

z y x z y

x

z y

x









) , , (

s : Scalar

z y x f s

s



EXAMPLE IN 2-D

 

 



 

 

 



 

 

 



 

x y )

, (

) , ( V V

: Vector

y x

y x f

y x V f

V

V

y







y x

s

e s

y x f s

y x













: exp

) , (

s : Scalar

) ( ^{2 2}

GRADIENT OF A SCALAR

) (

) (

) (

2 2

2 2 2

2

2 2 2D

: Exp

) ,

, (

vector a

is scalar a

of Gradient

y x y

y x x

y x

y ye V s

x xe V s

e s

z s y

s x

s s V















 

 



 

 













 





GRADIENT OF A SCALAR

(1,1) (0,1),

(0,0), at

ector gradient v

the Draw

2 s

and 1,

s 0, s

line contour

Draw

1 1 2D :

Exp







 

 

 

 





y V s

x V s

y x

s

y x

DIVERGENCE OF A VECTOR

3D) in

volume and

2D in (area

Γ by enclosed

area the

is

| Γ

|

3D) in

surface closed

and

2D in curve (closed

ce hypersurfa closed

is Γ

ds n

V

V ( )

|

| lim 1

0  





_  ^





   

DIVERGENCE OF A VECTOR

•Divergence computes the flux that the vector field transports through the imaginary boundary Γ, as Γ0

•Divergence of a vector is a scalar

•A positive divergence point is called source, because it indicates that mass would spread from the point (in fluid flow)

•A negative divergence point is called sink, because it

indicates that mass would get sucked into the point (in fluid flow)

•A zero divergence denotes that mass is transported without compression or expansion.

DIVERGENCE OF A VECTOR

x V x

V x

V



 



 



 



 V 

source :

divergence Positive

2 1

1 V

y) (x, V

: Exp











 



Free Divergence

0 0

0 V

x) (y, V

: Exp











 



sink :

divergence Negative

2 1

1 V

(-x,-y) V

: Exp















 



DIVERGENCE OF A VECTOR

VORTICITY OF A VECTOR

3D) in

volume and

2D in (area

by enclosed

area the

is

| Γ

|

3D) in

surface and

2D in (curve

ce hypersurfa closed

is





)

| (

| lim 1

0

V d s

V   









 





VORTICITY OF A VECTOR

•Vorticity computes the rotation flux around a point

•Vorticity of a vector is a vector

•The magnitude of vorticity expresses the speed of angular rotation

•The direction of vorticity indicates direction perpendicular to the plane of rotation

•Vorticity signals the presence of vortices in vector field

VORTICITY OF A VECTOR

Color:

Glyph:

VECTOR GLYPH

)) (

,

( x x k V x l      

x

V 

•Vector glyph mapping technique associates a vector glyph (or icon) with the sampling points of the vector dataset

•The magnitude and direction of the vector attribute is indicated by the various properties of the glyph: location, direction, orientation, size and color

VECTOR GLYPH

Line glyph,

or hedgehog glyph

Sub-sampled by a factor of 8

(32 X 32)

Original (256 X 256)

Velocity Field of a 2D Magnetohydrodynamic Simulation

VECTOR GLYPH

Velocity Field of a 2D Magnetohydrodynamic Simulation Line glyph,

or hedgehog glyph

Sub-sampled by a factor of 4

(64 X 64)

Original (256 X 256)

VECTOR GLYPH

Sub-sampled by a factor of 2

(128 X 128)

Original (256 X 256)

Problem with a dense Representation using glyph:

(1) clutter

(2) miss-representation

VECTOR GLYPH

Random Sub-sampling

Is better

VECTOR GLYPH: 3D

Simulation box: 128 X 85 X 42; or 456,960 data point 100,000 glyphs

Problem: visual occlusion

VECTOR GLYPH: 3D

Simulation box: 128 X 85 X 42; or 456,960 data point 10,000 glyphs: less occlusion

VECTOR GLYPH: 3D

Simulation box: 128 X 85 X 42; or 456,960 data point 100,000 glyphs, 0.15 transparency: less occlusion

VECTOR GLYPH: 3D

Simulation box: 128 X 85 X 42; or 456,960 data point 3D velocity isosurface

VECTOR GLYPH

•Glyph method is simple to implement, and intuitive to interpretation

•High-resolution vector datasets must be sub-sampled in order to avoid overlapping of neighboring glyphs.

•Glyph method is a sparse visualization: does not represent all points

•Occlusion

•Subsampling artifacts: difficult to interpolate

•Alternative: color mapping method is a dense visualization

VECTOR COLOR CODING

•Similar to scalar color mapping, vector color coding is to associate a color with every point in the data domain

•Typically, use HSV system (color wheel)

•Hue is used to encode the direction of the vector, e.g., angle arrangement in the color wheel

•Value of the color vector is used to encode the magnitude of the vector

•Saturation is set to one

2-D Velocity Field of the MHD simulation:

Orientation, Magnitude

Vector Color Coding

2-D Velocity Field of the MHD simulation:

Orientation only; no magnitude

Vector Color Coding

VECTOR COLOR CODING

•Dense visualization

•Lacks of intuitive interpretation; take time to be trained to interpret the image

STREAM OBJECTS

•Vector glyph plots show the trajectories over a short time of trace particles released in the vector fields

•Stream objects show the trajectories for longer time intervals for a given vector field

STREAMLINES

•Streamline is a curved path over a given time interval of a trace particle passing through a given start location or

seed point

point seed

the ,

) 0 (

) (

)

T]}

[0, ),

( {

0 0

p p

where

dt p

V (τ

p

p S

t





 





















Streamlines

All lines are traced up to the same maximum time T Seed points (gray ball) are uniformly sampled Color is used to reinforce the vector magnitude

STREAMLINES: ISSUES

•Require numerical integration, which accumulates errors as the integration time increases

t V

p p

where

t i p V dt

p V (τ

p

i i

i

t

t i

























1 1

/

0

( )

( )

)

n integratio Euler

 



 

 



•Euler integration: fast but less accurate

•Runge-Kutta integration: slower but more accurate

•Need to find optimal value of time step Δt

•Choose number and location of seed points

•Trace to maximum time or maximum length

•Trace upstream or downstream

•Saved as a polyline on an unstructured grid

Stream tubes

Tracing

downstream: the seed points are on a regular grid

•Add a circular cross section along the streamline curves, making the lines thicker

Stream tubes

Tracing

upstream: the arrow heads

are on a regular grid

Stream Objects in 3-D

Input:

128 X 85 X 42

Undersampling:

10 X 10 X 10 Opacity 1

Maximum Length

Stream Objects in 3-D

Input:

128 X 85 X 42

Undersampling:

3 X 3 X 3 Opacity 1

Maximum Length

Stream Objects in 3-D

Input:

128 X 85 X 42

Undersampling:

3 X 3 X 3 Opacity 0.3

Maximum Time

Stream Objects in 3-D

Stream tubes Seed area at the

flow inlet

Stream Ribbons

Two thick

Ribbons Vorticity is color coded

Vector Glyth

STREAM RIBBONS

•A stream ribbon is created by launching two stream lines from two seed points close to each other. The surface

created by the lines of minimal length with endpoints on the two streamlines is called a stream ribbon

STREAM SURFACE

•Given a seed curve Γ, a stream surface S_Γ is a surface that contains Γ and its streamlines

•Everywhere tangent to the vector field

•Flow can not cross the surface

•Stream tube is a particular case of a stream surface: the seed curve is a small closed curve

•Stream ribbon is also a particular case of a stream surface: the seed curve is a short line

TEXTURE-BASED VECTOR VIS.

•Discrete or sparse visualizations can not convey

information about every point of a given dataset domain

•Similar to color plots, texture-based vector visualization is a dense representation

•The vector field (direction and magnitude) is encoded by texture parameters, such as luminance, color, graininess, and pattern structure

Vector magnitude:

Color

Vector direction:

Graininess

TEXTURE-BASED VECTOR VIS.

LIC principle: Line Integrated

Convolution Principle

TEXTURE-BASED VECTOR VIS.

function blurring

of width :

function blurring

or weighting :

P point seed

of streamline :

texture noise

: ) (

) (

) ( )) , ( ) (

(

2

L k(s) S(p,s) N

e s

k

ds s k

ds s k s p S p N

T

s

L L L

L









 



TEXTURE-BASED VECTOR VIS.

•LIC is a process of blurring or filtering the texture (noise) image along the streamlines

•Due to blurring, the pixels along a streamline are getting smoothed; the graininess of texture is gone

•However, between neighboring streamlines, the

graininess of texture is preserved, showing contrast.