Fast High Accuracy Volume Rendering Thesis Defense May 2004 Kenneth Moreland Ph.D. Candidate Sandia...

Fast High Accuracy Volume Rendering

Thesis Defense

May 2004

Kenneth MorelandPh.D. Candidate

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration

under contract DE-AC04-94AL85000.


2May 2004

Overview

• Problem Description• Previous Work• Contributions

– Adaptive Transfer Function Sampling

– Linear Luminance

– Partial Pre-Integration

– Linear Opacity

• Results• Conclusions


3May 2004

Overview





– Linear Opacity



4May 2004

Problem Description

• Given: A 3D field of scalar information, typically represented as:– A finite set of points in three space with associated

scalar values.

– A connectivity graph.


5May 2004

Problem Description

• Goal: Transform scalars to colors/opacities, render 3D model.

Transfer Function

Scalars

ColorsOpacities


6May 2004

What Is Available

• Traditional/commodity 3D graphics hardware– Fast, powerful, and now flexible.

– Only (directly) support 0, 1, and 2 dimensional primitives (i.e. points, lines, and polygons).

• Special purpose volume rendering hardware– Constrained functionality; rectilinear grids only.

– Smaller economy of scale.• Development/fabrication costs distributed less.• Longer time span between generations.


7May 2004

Using Commodity Graphics Hardware

• Naïve approach: render cell faces as translucent polygons.


8May 2004

Using Commodity Graphics Hardware

• Naïve approach: render cell faces as translucent polygons.

• Result: “unrealistic” hollow cells.


9May 2004

Why is it Wrong?

• 2D polygons only capture surfaces.– Volumes absorb/emit light differently.


10May 2004

Overview





– Linear Opacity



11May 2004

Light Transport


12May 2004

Light Transport


13May 2004

A Light Transport Model

• The Particle Model– Sabella, 1988

0I

0I

dΔ


14May 2004

A Light Transport Model

• Ultimately, as light passes though a disk of size Δd, some percentage energy is absorbed, while some fixed amount is added.

€

dI

dz= L(z)τ (z) − τ (z)I(z)


15May 2004

The Volume Rendering Equation

€

I(d) = I0e− τ ( t )dt

0

d

∫

Attenuationof IncomingLight

1 2 4 3 4 + L(s)τ (s)e− τ (t )dt

s

d

∫ ds0

d

∫Emitted Light (Including Self Attenuation)

1 2 4 4 4 3 4 4 4


16May 2004

The Volume Rendering Equation

• This equation must be solved for every pixel.– In practice, we do piecewise integration, so we may

have to solve 100’s of times or more per pixel.

• Has no closed form.– Must solve for specific L and functions.


17May 2004

Solution: Linear

• We can do a first order approximation through cells with linear interpolation.

• The volume rendering equation can be solved with linear functions for L and , but…


18May 2004

Solution: Linear

€

I0e−d

τ b +τ f2 + L f − Lbe

−dτ b +τ f

2 + Lb − L f( ) 1

d τ b −τ f( )

e

d

2 τ b −τ f( )τ f

2

π2 erf τ b

d

2 τ b −τ f( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟− erf τ f

d

2 τ b −τ f( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥

( )=dI


19May 2004

Solution: Linear

€

I d( ) =

€

I0e−d


−dτ b +τ f

2 + Lb − L f( ) 1

d τ f −τ b( )

e

d

2 τ b −τ f( )τ f

2

π2 erfi τ b

d

2 τ f −τ b( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟− erfi τ f

d

2 τ f −τ b( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥

fb <

fb >

€

I0e−d


−dτ b +τ f

2 + Lb − L f( ) 1

d τ b −τ f( )

e

d

2 τ b −τ f( )τ f

2

π2 erf τ b

d

2 τ b −τ f( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟− erf τ f

d

2 τ b −τ f( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥


20May 2004

Solution: Linear

( )=dI

€

I0e−τd + Lb

1τd − 1

τd e−τd − e−τd

( ) + L f 1+ 1τd e

−τd − 1τd( ) fb =

€

I0e−d


−dτ b +τ f

2 + Lb − L f( ) 1

d τ f −τ b( )

e

d

2 τ b −τ f( )τ f

2

π2 erfi τ b

d

2 τ f −τ b( )

⎛ ⎝ ⎜


d

2 τ f −τ b( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥

fb <

fb >

€

I0e−d


−dτ b +τ f

2 + Lb − L f( ) 1

d τ b −τ f( )

e

d

2 τ b −τ f( )τ f

2

π2 erf τ b

d

2 τ b −τ f( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟− erf τ f

d

2 τ b −τ f( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥


21May 2004

( )=dI

€

I0e−τd + Lb

1τd − 1

τd e−τd − e−τd

( ) + L f 1+ 1τd e

−τd − 1τd( ) fb =

€

I0e−d


−dτ b +τ f

2 + Lb − L f( ) 1

d τ f −τ b( )

e

d

2 τ b −τ f( )τ f

2

π2 erfi τ b

d

2 τ f −τ b( )

⎛ ⎝ ⎜


d

2 τ f −τ b( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥

fb <

fb >

€

I0e−d


−dτ b +τ f

2 + Lb − L f( ) 1

d τ b −τ f( )

e

d

2 τ b −τ f( )τ f

2

π2 erf τ b

d

2 τ b −τ f( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟− erf τ f

d

2 τ b −τ f( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥

Solution: LinearThree Cases

Many Terms

NumericallyUnstable


22May 2004

Linear “Approximation”

• Plug in average luminance and attenuation.


23May 2004

Overview





– Linear Opacity



24May 2004

Overview





– Linear Opacity



25May 2004

Transfer Function

• In the real world, a cloud is parameterized with material properties (luminance and density).

• In scientific visualization, a volume can parameterized by any number of scalars (pressure, temperature, vorticity, density, etc.).

• These scalars are mapped to material properties via a transfer function.

• The scalar (f) often varies linearly, but the transfer function (TL and T) does not.

€

L(s) = TL f (s)( )

€

(t) = Tτ f (t)( )


26May 2004

Transfer Function Sampling

• We cannot solve the volume rendering integral for general transfer functions, so we sample.

0 0.5 1

0

0.5

1


27May 2004



0 0.5 1

0

0.5

1


28May 2004



0 0.5 1

0

0.5

1(0.5)


29May 2004



0 0.5 1

0

0.5

1(0.5) 0

0.5

1(0.5)


30May 2004

Transfer Function Aliasing


31May 2004

Adaptive Transfer Function Sampling

• Constrain the transfer function to be piecewise linear [Williams98].

• The function has linear segments joined at control points.

• Between the control points, the properties change linearly.


32May 2004


• If none of the scalars in a cell are a control point of the transfer function, then the transfer function varies linearly.

• Solution: clip cells at control points.– When the scalars vary linearly, the locus of points

for a given scalar is a plane.

• Rather than clip cells geometrically, clip ray fragments.


33May 2004


• Clipping parameters can be determined from the surface scalars relative to the isosurface scalar.


34May 2004

Overview





– Linear Opacity



35May 2004

Linear Luminance

• A common approach for finding a closed form for the volume rendering integral [Max90, Shirley90] is to hold the luminance constant.

• This simplifies the equation, but introduces error in the color.

• Instead, let us analyze the volume rendering integral with linearly varying luminance.


36May 2004

Linear Luminance

€

I(d) = I0e− τ ( t )dt

0

d

∫ + L(s)τ (s)e− τ (t )dt

s

d

∫ ds0

d

∫

€

L s( ) = Lb 1− sD( ) + L f sD

€

I(d) = I0e− τ ( t )dt

0

d

∫

€

+ Lb 1− sD( ) + L f sDτ (s)e− τ ( t )dt

s

d

∫ ds0

d

∫


37May 2004

Linear Luminance

• After lots of calculus…

€

I d( ) = I0e− τ t( )dt

0

D

∫ + L f 1−1

De

− τ t( )dts

D

∫ ds0

D

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟

€

+Lb1

De

− τ t( )dts

D

∫ ds0

D

∫ − e− τ t( )dt

0

D

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟


38May 2004

Linear Luminance

• After lots of calculus…

€

I d( ) = I0e− τ t( )dt

0

D

∫ + L f 1−1

De

− τ t( )dts

D

∫ ds0

D

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟

€

+Lb1

De

− τ t( )dts

D

∫ ds0

D

∫ − e− τ t( )dt

0

D

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟

Notice theRepetition


39May 2004

Substitute

• Now we just need to solve for and .

Let

€

≡e− τ t( )dt

0

D

∫

€

≡1

De

− τ t( )dts

D

∫ ds0

D

∫

€

I D( ) = I0ζ + L f 1− Ψ( ) + Lb Ψ −ζ( )


40May 2004

Overview





– Linear Opacity



41May 2004

, linear

• Easy enough to solve.– Not too bad to compute.

€

D,τ b ,τ f( ) = e−Dτ b +τ f

2


42May 2004

, linear

• Not so easy to solve/compute.• But, we can build a 3D table.

– Calculate values for all applicable (D,b,f) triples.

€

D,τ b ,τ f( ) =1

De

− τ b 1− tD( )+τ f

tD( )dt

s

D

∫ ds0

D

∫


43May 2004

Smaller Tables

• 3D tables work, but– take lots of space

– are not very cache coherent

• We could afford much more fidelity in a 2D table.• Consider what happens when we change the

limits of the integrals to range from 0 to 1.


44May 2004

Smaller Tables

• Next, we distribute D within the inner integral.

• Notice that this is an algebraic manipulation, not an approximation.

€

D,τ b ,τ f( ) =1

De

− τ b 1− tD( )+τ f

tD( )dt

s

D

∫ ds0

D

∫

€

= e−D τ b 1−t( )+τ f t( )dt

s

1

∫ ds0

1

∫

€

bD,τ fD( ) = e− τ bD 1−t( )+τ f Dt( )dt

s

1

∫ ds0

1

∫


45May 2004

Partial Pre-Integration

• Because part of the equation is stored in a table, I dub this technique partial pre-integration.

Problem: The domain is infinite. goes to zero as bD or fD goes to , but not fast enough.


46May 2004


• Solution: change the variables used to index .

€

γ≡ DτD+1


47May 2004


Average Partial Pre-Integration

[Williams98]


48May 2004

Overview





– Linear Opacity



49May 2004

Linear Attenuation is Not Always Best

• A user-intuitive transfer function editor presents opacity in the range from completely transparent to completely opaque.– But attenuation ranges

from 0 to infinity.• It also allows the user to

vary the visible opacity linearly.– But linear changes in

attenuation result in exponential changes in visible opacity.


50May 2004

Attenuation versus Opacity

• Attenuation relates to the density of the volume.• Opacity is the fraction of light the volume

occludes.• The relationship between the two [Wilhelms91]:

€

α =1− e−τ

€

=−ln 1−α( )


51May 2004

, linear α

• Solving results in an unwieldy equation.• Rather than try to compute , use approximation.

€

≈1− 1− 12 α b +α f( )( )

D


52May 2004

, linear α

• That’s really close, but there is a tapering at the corners.

• A cross section reveals a parabola-like structure.• Fit a parabola to it to get really close.

€

≈1− 1− 12 α b +α f( ) − 0.108165 α b +α f( )

2

( )D


53May 2004

, linear α

with linear α has no closed form.• Approximate with average α.

€

α ≈12 α b +α f( )


54May 2004

, linear α

• That’s good, but we can do better.

• If αf is greater than αb, should be smaller.

– Weight accordingly.

€

α ≈0.27α b + 0.73α f


55May 2004

Linear α Approximations

Average Initial Approx

Improved Approx


56May 2004

Overview





– Linear Opacity



57May 2004

Results, Speed

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Blunt Fin Oxygen Post Delta Wing

Frame Rate (Hz)

Average Color and Attenuation

Linear Color and Attenuation

Partial Pre Integration

Average Color and Opacity

Linear Color, Average Opacity

Linear Color and Opacity Approx


58May 2004

Results, Accuracy

Average Partial Pre-Integration [Williams98]

D = 0.001

D = 0.1

D = 1


59May 2004

Results, Accuracy

Average Approx 1 Approx 2

D = 0.001

D = 0.1

D = 1


60May 2004

Spatial Visualization

• Uniform errors are unlikely to make a difference in visual perception.


61May 2004

Spatial Visualization

• Spatial transitions in color are very noticeable.

- -+

- -+

- -+


62May 2004

Cell Boundaries

• At cell boundaries, approximation errors can change when the color does not.


63May 2004

Results, Cell Boundaries

Uni

form

Att

enua

tion

Bac

kO

paqu

eF

ront

Opa

que

Average Partial Pre-Integration [Williams98]

Uniform

Lum

inanceB

ack Lit

Front L

it


64May 2004

Results, Cell Boundaries

Uni

form

Att

enua

tion

Bac

kO

paqu

eF

ront

Opa

que

Uniform

Lum

inanceB

ack Lit

Front L

it

Average Approx 1 Approx 2


65May 2004

Overview





– Linear Opacity



66May 2004

Conclusions

• Adaptive transfer function sampling functional but slower than expected.– Due to heavy reliance on fragment processor.

– Still competitive with other rendering solutions.

• Partial pre-integration works well.– Visually indistinguishable from [Williams98].

– Over 10X faster.

• Approximations for linear opacity work well.– Nearly as fast as [Wilhelms91].

– No visual artifacts.


67May 2004

fin

Fast High Accuracy Volume Rendering Thesis Defense May 2004 Kenneth Moreland Ph.D. Candidate Sandia...

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