SCALAR VISUALIZATION. OUTLINE Visualizing scalar data A number of the most popular scalar...
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Transcript of SCALAR VISUALIZATION. OUTLINE Visualizing scalar data A number of the most popular scalar...
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 Importing2. Data Filtering3. Data Mapping4. 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
RR
RR
RR
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)fc(c
}{cC
i
,...N,ii
maxmin
21
Where
LUMINANCE COLORMAP•Use grayscale to represent scalar value
Luminance Colormap
)y-10(x 44
ef •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 COLORMAPImplementation
void c(float f, float & R, float & G, float &B){
const float dx=0.8f=(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);
}
3RR :c DvD: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
})(|{)( xpsDpxC •A contour line is also called an isoline
•In 3-D dataset, a contour is a 2-D surface, called isosurface
CONTOURING
Cartograph
CONTOURINGOne 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 linesGradient 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 (vi, vj)•If yes, the isoline intersects the edge at a point q, which uses linear interpolation
ij
ijji
vvvvpvvp
q
)()(
•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
hs
Dxxnxsxxm
DDm
),()()(
,:
HEIGHT PLOTS
Height plot over a planar 2-D surface
HEIGHT PLOTS
Height plot over a nonplanar 2-D surface
VECTOR VISULIZATION
Divergence and VorticityVector Glyphs Vector Color CodingDisplacement PlotsStream ObjectsTexture-Based Vector VisualizationSimplified Representation of Vector Fields
VECTOR FUNCTION
) D-2:case(simpler
D)-3in (usually
22
33
RRf:
RRf:
VECTOR VERSUS SCALAR
),,(
),,(),,(
V
VV
),,(
ˆˆˆ :Vector
z
y
x
zyxf
zyxfzyxf
V
or
VVVV
or
kVjViVV
V
z
y
x
zyx
zyx
),,(
s :Scalar
zyxfss
EXAMPLE IN 2-D
xy
),(),(
VV
:Vector
y
x
yxfyxf
V
V
V
y
x
yxs
es
yxfsyx
:exp
),(s :Scalar
)( 22
GRADIENT OF A SCALAR
)(
)(
)(
22
22
22
2
2
2D:Exp
),,(
vectora isscalar a ofGradient
yxy
yxx
yx
yeysV
xexsV
es
zs
ys
xssV
GRADIENT OF A SCALAR
(1,1) (0,1), (0,0),at ector gradient v theDraw2s and 1,s 0,s linecontour Draw
1
1
2D:Exp
ysV
xsV
yxs
y
x
DIVERGENCE OF A VECTOR
3D)in volumeand 2Din (area Γby enclosed area theis|Γ|
3D)in surface closed and 2Din curve (closed
cehypersurfa closed is Γ
dsnVV )(||
1lim0
DIVERGENCE OF A VECTOR•Divergence computes the flux that the vector field transports through the imaginary boundary Γ, as Γ0•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
xV
xV
xV zyx
V
source :divergence Positive211V
y)(x,V
:Exp
Free Divergence000V
x)(y,V
:Exp
sink :divergence Negative211V
(-x,-y)V
:Exp
DIVERGENCE OF A VECTOR
VORTICITY OF A VECTOR
3D)in volumeand 2Din (area by enclosed area theis|Γ|
3D)in surface and 2Din (curve cehypersurfa closed is
)(||
1lim0
sdVV
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
))(,( xVkxxl
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 GLYPHSub-sampled by a factor of 2(128 X 128)
Original (256 X 256)
Problem with a denseRepresentation using glyph: (1) clutter(2) miss-representation
VECTOR GLYPH
RandomSub-sampling
Is better
VECTOR GLYPH: 3DSimulation box: 128 X 85 X 42; or 456,960 data point
100,000 glyphsProblem: visual occlusion
VECTOR GLYPH: 3D
Simulation box: 128 X 85 X 42; or 456,960 data point10,000 glyphs: less occlusion
VECTOR GLYPH: 3D
Simulation box: 128 X 85 X 42; or 456,960 data point100,000 glyphs, 0.15 transparency: less occlusion
VECTOR GLYPH: 3D
Simulation box: 128 X 85 X 42; or 456,960 data point3D 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
ppwhere
dtpV(τp
pS
t
Streamlines
All lines are traced up to the same maximum time TSeed 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
tVpp
where
tipVdtpV(τp
iii
t
it
11
/
00
)()()
nintegratioEuler
•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 codedVector 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 SΓ 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 blurringor weighting:
Ppoint seed of streamline : texturenoise :
)(
)(
)()),(()(
2
Lk(s)S(p,s)N
esk
dssk
dsskspSNpT
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.