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Graphics II 91.547

Volume Rendering

Session 10

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What do we mean by “volume rendering”?Conventional rendering:

Bicubic Parametric Patches

Polygons

3D Structured Objects

2D Image

I x y( , )

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What do we mean by “volume rendering”?The starting point:

V x y z( , , )Scalar Field:

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Volume Rendering:Options for presentation of the data

Cutaway, or cross section Isosurface

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What do we mean by “volume rendering”?What the samples mean:

Voxel with samplesat vertices.

Voxel with sampleat center.

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What do we mean by “volume rendering”?The process:

V x y z( , , )

V x y z( , , )

Structured 3DModel

3D Scalar Field

3D Scalar Field

2D Scalar Image

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Data volume geometries

Cartesian - (also known as a voxel grid) cubic data elements,axis aligned Preferred

Regular - same as cartesian, except that the cells are rectangular,i.e. different sizes along different axes Most medical data

Rectilinear - aligned to axes, but distance between cells alongeach axis can vary

Structured or curvilinear non rectilinear, but cells are hexahedra rectangular warped to fill a volume or to fit aroundan object Often used in CFD

Unstructured - no geometric constraints imposed

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Early Volume Visualization TechniquesHerman & Liu 1979

V x y z k( , , ) Establish a threshold:

Binary Partitioning of the volume:

2D Slice:

“Boundary” Voxels are considered opaque cubesand rendered by standard lighting model.

Inside

Outside

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Improving the blockiness of the image:The gradient operator

V x y z k( , , )

V x y z

V

x

V

y

V

z

( , , )

0

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V x y z

V x y z V x y z

V x y z V x y z

V x y z V x y zi j k

i j k i j k

i j k i j k

i j k i j k

( , , )

[ ( , , ) ( , , )]

[ ( , , ) ( , , )]

[ ( , , ) ( , , )]

12 1 1

12 1 1

12 1 1

0

Approximating the gradient numerically:

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Projecting from Contour Data:Pizer 1986

y j

y j 1

V x y z k( , , )

Polygon structure generated by “skimming”

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Topological problem withPizer approach: branching structures

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The “Marching Cubes” AlgorithmLorenson & Cline 1987

Inside

Outside

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The “Marching Cubes” Algorithm

V x y z k( , , )

V x y z k( , , )

V x y z k( , , )

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The “Marching Cubes” AlgorithmPossible Vertex States

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The “Marching Cubes” AlgorithmGenerated contour

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Generation of contour from subcontours

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Generation of contour from subcontours:Disconnected regions

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Marching Cubes AlgorithmPossible Polygon Arrangements

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Marching Cubes AlgorithmPossible Polygon Arrangements

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Marching Cubes

Direction of march

Current cube Previously dealt with

New vertex

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Weaknesses of Intermediate Structure Methods

0 Can impose a structure on the data which does not exist, per se.- Selection of a “constant” implies a binary decision on the

existence of an intermediate surface that should be extracted and rendered

- Could just be a gradual transition in density through the medium

0 Not feasible to visualize structures within structures0 Does not handle objects that would intrinsically be

transparent such as fluids, clouds

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Volume rendering by ray castingBlinn 1982/Kajiya 1984

0 Volume made up of small spherical particles that both scatter (reflect) and attenuate light

0 Parallel rays are cast from viewer into the volume.- At each point, the progressive attenuation due to the

particle field is calculated- Light scattered in the eye direction from the light

source(s) is calculated at each point- These values are integrated along the ray and a single

brightness value at the eye is calculated

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Scattering/Attenuation Model

Light Source

D x y z( , , )

t1 t2t

Eye Point

D x t y t z t D t

I x t y t z t I t

( ( ), ( ), ( )) ( )

( ( ), ( ), ( )) ( )

Density:

Illumination:

Light scattered along R in direction of eye from point at t:

R

I t D t P( ) ( ) cos

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Scattering/Attenuation Model

Attenuation of light scattered from point t: exp ( )

D s dst

t

1

Summing the intensity of light arriving at the eye along R fromall of the elements:

B D s ds I t D t P dtt

t

t

t

exp ( ) ( ) ( ) cos 11

2

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Additive ReprojectionLevoy 1988

R

V x y z( , , )

C R k

R k

( , )

( , )

C R( )

Image Plane

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Additive ReprojectionLevoy 1988

Determining the color of a voxel: C i j k( , , )

Calculated using the Phong model, assuming that the normal is given by the gradient: V x y zi j k( , , )

f1 f2 f3 f4

f 1

f 2

f 3

f 4

( )X

V X( )

Determining the opacity: ( , , )i j k

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Additive ReprojectionLevoy 1988

CinCout

C R k

R k

( , )

( , )

C C R k C R k R kout in ( ( , )) ( , ) ( , )1

TransparencyTerm

OpacityTerm

C R C R k R k R ii k

K

k

K

( ) ( , ) ( , ) ( ( , ))

110

kth voxel along R