Lecture 2 : Visualization Basics
Bong-Soo Sohn
School of Computer Science and Engineering
Chung-Ang University
Surface Graphics
• Objects are explicitely defined by a surface or boundary representation (explicit inside vs outside)
• This boundary representation can be given by:– a mesh of polygons :
– a mesh of spline patches
Surface Graphics : Pros and Cons
• Pros : – fast rendering algorithms are available– acceleration in special hardware is relatively easy and cheap (many $200
game boards)– use OpenGL to specify rendering parameters– surface realism can be added via texture mapping
• Cons :– discards the interior of the object and just maintains the object’s shell– does not facilitate real-world operations such cutting, slicing, disection– does not enable artificial viewing modes such as semi-transparencies, X-
ray– surface-less phenomena such as clouds, fog, gas are hard to model and
represent
Volume Graphics
• Maintains a 3D image representation that is close to the underlying fully-3D object (but discrete)
Volume Graphics : Pros and Cons
• Pros : – can achieve a level of realism (and ‘hyper-realism’) that is
unmatched by surface graphics– allows easy and natural exploration of volumetric datasets
• Cons :– has high rendering complexity
Rendering of the inside of a human colon
surface rendered volume rendered
Volumetric Image (3D image, volume)
• it is a 3D array of point samples, called voxels (volume elements)• the point samples are located at the grid points• the process of generating a 2D image from the 3D volume is called
volume rendering
Basics on Differentiation (of Scalar and Vector Function)
• Refer to Prof. Han-Wei Shen’s Notes.
• Useful for understanding images and gradients
Data Acquisition
• Scanned/Sampled Data– CT/MRI/Ultrasound– Electron Microscopy
• Computed/Simulated Data
• Modeled/Synthetic Data
Time-Varying Data•
• Time-Varying Data from Scanning
Evolutionary Morphing
Imaging Scanners
• Scanners can yield both domains and functions on domains– Scanners yielding domains
• Point Cloud Scanners: 300μ-800μ• CT, MRI: 10μ-200μ• Light microscopy: 5μ-10μ• Electron microscopy: < 1μ• Ultra microscopy like Cyro EM 50Å-100Å
Imaging Techniques
• Computed Tomography (CT)– Measures spacially varying X-ray attenuation coefficient
– Each slice 1-10mm thick
– High resolution , low noise
– Good for high density solids
• Magnetic Resonance Imaging (MRI)– Measures distribution of mobile hydrogen nuclei by quantifying relaxation
times
– Moderate noise
– Works well with soft tissue
• Ultrasound– Handheld probe
– Inexpensive, fast, and real-time
– High noise with moderate resolution
Various Data Characteristics
• Static • Scalar• Meshed• Dense
• Time varying data• Vector , Tensor• Meshless• Sparse
Data Format
• Mesh (Grid) Type– Regular– Rectilinear– Unstructured– Meshless
• Mesh type conversion– Meshless to meshed
Mesh Types
• Mesh taxonomy– Regular static meshes:
• There is an indexing scheme, say i,j,k, with the actual positions being determined as i*dx, j*dy, k*dz.
• If dx=dy=dz, then,– In 2-D, we get a pixel, and in 3-D, a voxel.
dy
dx
A 2-D regular rectilinear grid
Mesh Types
– Irregular static meshes:• Rectilinear:
– Individual cells are not identical but are rectangular, and connectivity is related to a rectangular grid
dx, dy are not constant in grid,but connectivity is similar in topologyto regular grids.
A 2-D regular rectilinear grid
Mesh types (contd)
• Curvilinear:– Sometimes called structured grids as the cells are
irregular cubes – a regular grid subjected to a non-linear transformation so as to fill a volume or surround an object.
A 2-D curvilinear grid
Mesh Types (contd) • Unstructured:
– Cells are of any shape (tetrahedral), hexahedra, etc with no implicit connectivity
• Hybrid:– Combination of curvilinear and unstructured grids.
– Dynamic (Time-varying) meshes
Triangulations (Delaunay) & Dual Diagrams (Voronoi)
Union of ballsTriangulation & DualNerve sub-complex
Meshless Data Meshed Data
Particle Data to Meshes
Weighted point P = ( p, wp ) where
Power distance from
with is the Euclidean distance
A
pd wp ,
ppd wxpxpx 2||||)( to
2|||| xp
x p|||| xp
)(xp
pw
pw
Power Diagram ( PD ) of a weighted point set
Tiling of space into convex regions where ith region ( tile ) are the set of points in nearest to pi in the power distance metric.
Regular Triangulation ( RT ) Dual of Power Diagram ( PD ) with an edge of RT for each Bisector Plane of PD
Bisector Plane which matches power distance.
l
l1 l2
p1 p21pw
2pw
21 2
221
21 pppp wlwl
d
Particle Data to Meshes
Atomic Centers CPK CPK Alpha-Shape
Solvent Accessible Surface (SAS) Power Diagram of SAS Solvent Excluded Surface (SES)
Molecular Surfaces(Solvent Excluded Surface)
SES = spherical patches + toroidal patches +concave patches
Field Data
• Scalartemperature, pressure, density, energy, change, resistance,
capacitance, refractive index, wavelength, frequency & fluid content.
• Vector velocity, acceleration, angular velocity, force, momentum, magnetic
field, electric field, gravitational field, current, surface normal
• Tensorstress, strain, conductivity, moment of inertia and electromagnetic field
• Multivariate Time Series
Interpolation
• Interpolation/Approximation are often used to approximate the data on the domain
• In other words, it constructs a continuous function on the domain
Linear Interpolation on a line segment
p0 p p1
The Barycentric coordinates α = (α0 α1) for any point p
on line segment <p0 p1>, are given by
)),(
),(,
),(
),((
10
0
10
1
ppdist
ppdist
ppdist
ppdist
which yields p = α0 p0 + α1 p1
and fp = α0 f0 + α1 f1
ff1f0fp
Linear interpolation over a triangle
p0
p1 p p2
For a triangle p0,p1,p2, the Barycentric coordinates
α = (α0 α1 α2) for point p, )
),,(
),,(,
),,(
),,(,
),,(
),,((
210
10
210
20
210
21
ppparea
ppparea
ppparea
ppparea
ppparea
ppparea
2
0ii pp
2
0ii fpfp
Linear interpolant over a tetrahedron
Linear Interpolation within a • Tetrahedron (p0,p1,p2,p3)
α = αi are the barycentric coordinates of p
p3
p
p0 p2
p1
3
0ii pp
fp0
fp1
fp3
fp2
3
0ii fpfp
fp
Other 3D Interpolation
• Unit Prism (p1,p2,p3,p4,p5,p6)
p1
p2 p3
p p4
p5 p6
))(1()(6
43
3
1 iiii ptptp
Note: nonlinear
Other 3D Interpolation
• Unit Pyramid (p0,p1,p2,p3,p4)
p0
p1 p2 p p3
p4
)))1()(1())1(()(1( 43210 pssptpssptuupp
Note: nonlinear
Trilinear Interpolation• Unit Cube (p1,p2,p3,p4,p5,p6,p7,p8)
– Tensor in all 3 dimensions
p1 p2
p3 p4
p
p5 p6
p7 p8
)))1()(1())1(()(1(
)))1()(1())1(((
8765
4321
pssptpssptu
pssptpssptup
Trilinear
interpolant
Comparison
• Bicubic vs Bilinear vs nearest point
Resampling
• Used in image resize or data type conversion
– Rectilinear to rectilinear
– Unstructured to rectilinear
Rendering
• Isocontouring (Surface Rendering)– Builds a display list of isovalued lines/surfaces
• Volume Rendering– 3D volume primitives are transformed into 2D discrete pixel space
Volume Rover Demo
Isosurface Visualization
• Isosurface (i.e. Level Set ) : – C(w) = { x | F(x) - w = 0 }( w : isovalue , F(x) : real-valued function )
< ocean temperature function > < two isosurfaces (blue,yellow) >
isosurfacing <medical>
<bio-molecular>• Surface Topology :– Property that is invariant to continuous deformation
(without cutting or gluing), e.g. donut & cup
Isocontouring
2. Isocontouring [Lorensen and Cline87,…]
• Popular Visualization Techniques for Scalar Fields
• Definition of isosurface C(w) of a scalar field F(x)
C(w)={x|F(x)-w=0} , ( w is isovalue and x is domain R3 )
( Isocontour in 2D function: isovalue=0.5 )
• Marching Cubes for Isosurface Extraction
1. Dividing the volume into a set of cubes
2. For each cubes, triangulate it based on the 2^8(reduced to 15) cases
0.7 0.6 0.75 0.4
0.40.80.40.6
0.4 0.3 0.35 0.25
1.0 0.8 0.4 0.3
0.7 0.6 0.75 0.4
0.40.80.40.6
0.4 0.3 0.35 0.25
1.0 0.8 0.4 0.3
0.7 0.6 0.75 0.4
0.40.80.40.6
0.4 0.3 0.35 0.25
1.0 0.8 0.4 0.3
Cube Polygonization Template
Volume Rendering
1. Volume Rendering [Drebin88,…]
• Popular Visualization Techniques for Scalar Fields
• Hardware Acceleration ( 3D Texturing ) [Westermann98]
C , C II’
I’= C C + (1- C)I
C : colorC: opacity
Light traversal from back to front
1. Slicing along the viewing direction
2. Put 3D textures on the slice
3. Interactive color table manipulation
<emission> <incoming light>
<produced by CCV vistool>
Transfer Function
• Mapping from density to (color, opacity)
Medical applications
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