Direct Volume Rendering

Joe Michael Knissjmk@cs.utah.edu

http://www.cs.utah.edu/~jmk

Scientific Computing and Imaging InstituteUniversity of Utah

Outline

1. What is Volume Rendering?

2. Shading models

3. Current hardware methods

4.Visualization 2001 presentation

Volume Rendering

Volume Rendering is a visualization/graphics technique which directly renders data using a reasonable approximation to a physical light transport model.

What?

Overview• Data acquisition

• Reconstruction

• Sample & interpolate

• Transfer function

• Lighting

• Blend/integrate

Volume Rendering

Spiral CT

Filtered back-projection

Tri-linear Interp.

3D T.F.

Light & shadow

Back to Front

OverviewVolume Rendering

Acquisition type Reconstruction

•CT (X-ray)

•MRI

•Simulation

•Inverse Radon

•Inverse Fourier

•None?

Others: PET, SPECT, EEG, MEG, Geological, Atmospheric

InterpolationOverview

Sampled Function

How do we recover a function that is at least C0 continuous??

Real Function

?

Interpolation

• Estimate values between samples

• Reconstruct continuous function

Overview

box

linear

sinc

Catmul-Rom

C0

C0 C1

C00

Interpolation

• Estimate values between samples

• Reconstruct continuous function

Overview

box

linear

sinc

Catmul-Rom

Bad

OK Good

Best

Quality

Interpolation

• Estimate values between samples

• Reconstruct continuous function

Overview

box

linear

sinc

Catmul-Rom

Easiest

Easy OK

Hard

Use

InterpolationOverview

Sampled Function

Laura Bush

InterpolationOverview

Box

What is the value here?

InterpolationOverview

Box

What is the value here?

InterpolationOverview

Box

What is the value here?

InterpolationOverview

Box == Nearest Neighbor

Reconstructed function

InterpolationOverview

Linear

What is the value here?

InterpolationOverview

Linear

What is the value here?

InterpolationOverview

Linear

What is the value here?

InterpolationOverview

Linear == Connect the dots

Reconstructed function

InterpolationOverview

Catmull-Rom == Rolling Hills

Reconstructed function

InterpolationOverview

Sinc == Roller Coaster

Reconstructed function

But what about the ringing??

InterpolationOverview

InterpolationOverview

box

linear

sinc

Catmul-Rom

Bad

OK Better

Hmm…

Quality

In reality:

Interpolation

• Bi-linear

• Tri-linear

What is the value here?

• Bi-linear = 3 linear interpolations, 2 axes

• Tri-linear = 7 linear interpolations, 3 axes

Interpolation

•1st along x

•2nd along y

•1st along x

• Bi-linear = 3 linear interpolations, 2 axes

• Tri-linear = 7 linear interpolations, 3 axes

Interpolation

•1st along x

•2nd along y

•1st along x

•2nd along y

• Bi-linear = 3 linear interpolations, 2 axes

• Tri-linear = 7 linear interpolations, 3 axes

Interpolation

•1st along x

•2nd along y

•1st along x

•2nd along y

•3rd along z

Transfer Function

• Assign optical properties to data– Color– Opacity

Transfer function

Transfer Function

• Assign optical properties to data– Color– Opacity

Transfer function

x

Transfer Function

• Assign optical properties to data– Color– Opacity

Transfer function

T(x)

RenderingOverview

Volume DataEye

Image plane

Projection

RenderingOverview

Volume DataEye

Image plane

Ray Casting

But how do we get the final color??

r0r1

RenderingOverview

Solution: Integrate

r0 r11

0

)(r

r

dxxT

r0r1

T(x)

RenderingOverview

Solution: Sum (Riemann)

r0r1

r0 r11

0

)(r

r

dxxT xxTn

i

i 0

)(

RenderingOverview

r0 r1

1

0

)(r

r

dxxT xxTn

i

i 0

)(

Emissive What about occlusion??

n

i

it0

RenderingOverview

r0 r1

Absorption• Use alpha channel for opacity

• Values should approach ZERO

Exponential-1 curve

0

1

)(expr

r

dxxA

RenderingOverview

r0 r1

0

1

)(expr

r

dxxA

n

i

i

n

i

i xxxAxxxA00

)(exp)(exp

n

i

ia0

RenderingOverview

r0 r1

Emission & Absorption

dsdxxAsTr

s

r

r

10

1

)(exp*)(

RenderingOverview

Emission & Absorption

r0 r1

N

i

N

ij

ii at0 1

*

dsdxxAsTr

s

r

r

10

1

)(exp*)(

RenderingOverview

Emission & Absorption

r0 r1

))))((...(( 0011 Batatat nnnn

cout = ci + ai*cin

N

i

N

ij

ii at0 1

* =

RenderingOverview

Emission & Absorption

r0 r1

cout = ci + ai*ci-1

Where did the light come from??

LightingOverview

Eye

Image planer0

r1

Light

LightingOverview

Light

Sample ri

The volume occludes some of the light

N

i

N

ij

i

L

k

ki aat0 1

*

0

*

dsdxxAdttAsTr

s

r

r

l

s

00

1

0

)(exp*)(exp*)(

LightingOverview

Sample ri (s)First scattering/ shadows

l0

LightingOverview

It still doesn’t look like a surface.

Standard shading models need a surface normal

LightingOverview

Solution: use the gradient as the normal

linear

linear derivative

14

10

5

3

LightingOverview

Solution: use the gradient as the normal

linear

linear derivative

14

10

5

3

dx = 3-10 = -7

dy = 5-14 = -9

LightingOverview

Solution: use the gradient as the normal 14

10

5

3

normallight

eyereflection

LightingOverview

Diffuse14

10

5

3

nl

Color = C*(n l)C = object color

LightingOverview

Diffuse + specular14

10

5

3

nl

er

Color = C*(n l) + LC*(r l)P

LC = light color P = specular power

LightingOverview

Diffuse + specular & shadow

Color = [C*(n l) + LC*(r l)P ] * S

S = shadow amount

LightingOverview

Diffuse + specular & shadow + Ambient

Color = [C*(n l) + LC*(r l)P ] * S + C*A

A = Ambient contribution

RenderingOverview

Volume DataEye

Image plane

Ray Casting

r0r1

• Cast a ray through each pixel

•Intersect with volume

•Numerically integrate & return color

RenderingHardware

Volume DataEye

Image plane

Graphics Hardware•Polygons – Proxy geometry

•Textures – Data & interpolation

•Blending operations – Numerical integration

Slices

HardwareSlices

View direction

1 slice

5 slices

20 slices 45 slices 85 slices 170 slices

Hardware

Data texture

Grad texture

Shadow texture

T.F. texture

Shade textureTexture

combiner

Color & opacity

Data texture

Polygon slicing

through texture

HardwareSlicesFrame buffer

Blend each slice with previously rendered

Projection

References

• Nelson Max, Optical Models for Direct Volume Rendering, IEEE Visualization and Computer Graphics (v1 #2 June 1995)

• Mark Levoy, Display of Surfaces from Volume Data, IEEE Computer Graphics and Applications (v8 #3 May 1988)

• Kenneth R. Castleman, Digital Image Processing, Prentice Hall (c1996 ISBN:0-13-211467-4)