Lecture 2 : Visualization Basics

1
Lecture 2 Visualization Basics

• Bong-Soo Sohn
• School of Computer Science and Engineering
• Chung-Ang University

2
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

3
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 objects 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

4
Volume Graphics

• Maintains a 3D image representation that is close
  to the underlying fully-3D object (but discrete)

5
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
volume rendered
surface rendered
6
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

7
Basics on Differentiation (of Scalar and Vector
Function)

• Refer to Prof. Han-Wei Shens Notes.
• Useful for understanding images and gradients

8
Data Acquisition

• Scanned/Sampled Data
• CT/MRI/Ultrasound
• Electron Microscopy
• Computed/Simulated Data
• Modeled/Synthetic Data

9
Time-Varying Data

• Time-Varying Data from Scanning

10
Evolutionary Morphing
11
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 lt 1µ
• Ultra microscopy like Cyro EM 50Å-100Å

12
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

13
Various Data Characteristics

• Static
• Scalar
• Meshed
• Dense

• Time varying data
• Vector , Tensor
• Meshless
• Sparse

14
Data Format

• Mesh (Grid) Type
• Regular
• Rectilinear
• Unstructured
• Meshless
• Mesh type conversion
• Meshless to meshed

15
Mesh Types

• Mesh taxonomy
• Regular static meshes
• There is an indexing scheme, say i,j,k, with the
  actual positions being determined as idx, jdy,
  kdz.
• If dxdydz, then,
• In 2-D, we get a pixel, and in 3-D, a voxel.

dx
A 2-D regular rectilinear grid
dy
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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 topology to regular grids.
A 2-D regular rectilinear grid
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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
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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

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Triangulations (Delaunay) Dual Diagrams
(Voronoi)
Meshless Data ? Meshed Data

• Union of balls
• Triangulation Dual
• Nerve sub-complex

20
Particle Data to Meshes
A
Weighted point P ( p, wp ) where
p
x
Power distance from
with is the Euclidean distance
21
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.
p1
p2
l1
l2
l
Bisector Plane which matches power distance.
Regular Triangulation ( RT ) Dual of Power
Diagram ( PD ) with an edge of RT for each
Bisector Plane of PD
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Particle Data to Meshes
Atomic Centers
CPK

CPK Alpha-Shape
Solvent Accessible Surface (SAS)
Power Diagram of SAS Solvent Excluded
Surface (SES)
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Molecular Surfaces(Solvent Excluded Surface)
toroidal patches
concave patches
spherical patches
SES
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Field Data

• Scalar
• temperature, 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
• Tensor
• stress, strain, conductivity, moment of inertia
  and electromagnetic field
• Multivariate Time Series

25
Interpolation

• Interpolation/Approximation are often used to
  approximate the data on the domain
• In other words, it constructs a continuous
  function on the domain

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Linear Interpolation on a line segment

• p0 p p1
• The Barycentric coordinates a (a0 a1) for any
  point p on line segment ltp0 p1gt, are given by

f
f1
fp
f0
which yields p a0 p0 a1 p1 and
fp a0 f0 a1 f1
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Linear interpolation over a triangle

• p0
• p1 p p2
• For a triangle p0,p1,p2, the Barycentric
  coordinates
• a (a0 a1 a2) for point p,

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Linear interpolant over a tetrahedron

• Linear Interpolation within a
• Tetrahedron (p0,p1,p2,p3)
• a ai are the barycentric coordinates of
  p
• p3
• p
• p0 p2
• p1

fp3
fp
fp2
fp0
fp1
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Other 3D Interpolation

• Unit Prism (p1,p2,p3,p4,p5,p6)
• p1
• p2 p3
• p p4
• p5 p6

Note nonlinear
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Other 3D Interpolation

• Unit Pyramid (p0,p1,p2,p3,p4)
• p0
• p1 p2 p p3
• p4

Note nonlinear
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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

Trilinear interpolant
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Comparison

• Bicubic vs Bilinear vs nearest point

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Resampling

• Used in image resize or data type conversion
• Rectilinear to rectilinear
• Unstructured to rectilinear

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Rendering

• Isocontouring (Surface Rendering)
• Builds a display list of isovalued lines/surfaces
• Volume Rendering
• 3D volume primitives are transformed into 2D
  discrete pixel space

35
Volume Rover Demo
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Isosurface Visualization

• Isosurface (i.e. Level Set )
• C(w) x F(x) - w 0
• ( w isovalue , F(x) real-valued function )

isosurfacing
ltmedicalgt
lt ocean temperature function gt
lt two isosurfaces (blue,yellow) gt
ltbio-moleculargt

• Surface Topology
• Property that is invariant to continuous
  deformation (without cutting or gluing), e.g.
  donut cup

37
Isocontouring

• Popular Visualization Techniques for Scalar
  Fields

2. Isocontouring Lorensen and Cline87,

• Definition of isosurface C(w) of a scalar field
  F(x)
• C(w)xF(x)-w0 , ( w is isovalue and x is
  domain R3 )

1.0
1.0
1.0
0.8
0.4
0.3
0.8
0.4
0.3
0.8
0.4
0.3
0.7
0.6
0.75
0.4
0.7
0.6
0.75
0.4
0.7
0.6
0.75
0.4
0.4
0.4
0.4
0.8
0.4
0.6
0.8
0.4
0.6
0.8
0.4
0.6
0.4
0.4
0.4
0.3
0.25
0.3
0.25
0.3
0.25
0.35
0.35
0.35
( Isocontour in 2D function isovalue0.5 )

• Marching Cubes for Isosurface Extraction
• Dividing the volume into a set of cubes
• For each cubes, triangulate it based on the
  28(reduced to 15) cases

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Cube Polygonization Template
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Volume Rendering

• Popular Visualization Techniques for Scalar
  Fields

1. Volume Rendering Drebin88,
C color ?C opacity
C , ?C
I
I
Light traversal from back to front
I C ?C (1- ?C)I
ltemissiongt
ltincoming lightgt
ltproduced by CCV vistoolgt

• Hardware Acceleration ( 3D Texturing )
  Westermann98

1. Slicing along the viewing direction
2. Put 3D textures on the slice
3. Interactive color table manipulation

40
Transfer Function

• Mapping from density to (color, opacity)

41
Medical applications