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