The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Practical Logarithmic Shadow Maps

Brandon Lloyd UNC-CHNaga Govindaraju UNC-CHDavid Tuft UNC-CHSteve Molnar NvidiaDinesh Manocha UNC-CH

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Motivation

Interactive shadow computation remains a challenge

increasing scene complexitylarge, open environmentsuser expect high quality

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Shadow maps

Simple two pass algorithmSupports wide range of geometric representationsCheap to render

Disadvantage: Aliasing artifacts at shadow edges

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Warping algorithms

Perspective shadow maps(PSMs) [Stamminger and Drettakis 2002]

Increase sample density where needed by reparametrizing the shadow map

Trapezoidal shadow maps(TSMs) [Martin and Tan 2004]

Light-space perspective shadow maps (LSPSMs) [Wimmer et al. 2004]

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Comparison of parameterizations

4x4 projection matrix Logarithmic

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Main results

Logarithmic parameterization for directional and point lightsError analysis for point lightsHardware architecture enhancements for logarithmic rasterization

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Outline

Related workAliasing errorLogarithmic parameterizationHardware architectureResults

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Partitioning algorithms

Separate shadow maps for different parts of the sceneIncludes cascaded shadow maps

Tiled shadow maps [Arvo 2004]

Adaptive shadow maps [Fernando et al. 2001; Lefohn et al. 2006]

Plural sunlight buffers [Tadamura et al. 1999,2001]

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Warping + partitioning

4x4 matrixLixel for every pixel[Chong and Gortler 2004]

PSM with cube maps[Kozlov 2004]

Warping + various frustum partitioning schemes [Lloyd et al. 2006]

Dual paraboloid [Brabec et al. 2002]

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Specialized representations

Silhouette shadow maps[Sen et al. 2004]

Irregular shadow maps[Johnson et al. 2004,2005; Aila and Laine 2004]

Solves the aliasing problemGPU support requires major changes

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Outline

Related workAliasing errorLogarithmic parameterizationHardware architectureResults

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Aliasing error

shadow plane

viewfrustum

eye

light

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light beam

'wl

'wi

'wi 'wi 'wi

Aliasing error

'wl

image beam

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wi

wl

light beam

image beam

θl

θi

'wi

Aliasing error

'wl

m='wl

'wicos θl

cos θi

wl

wi

Perspectivealiasing

cos θl

cos θi Projectionaliasing

wl

wi≈

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wi

wl

θl

θi

'wi

Aliasing error

'wl

m='wl

'wi

wl

wi

Perspectivealiasing

wl

wi≈

light beam

image beam

cos θl

cos θi

cos θl

cos θi Projectionaliasing

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wi

wl

light beam

image beam

Controlling aliasing error

To eliminate perspective aliasing:

wl ≤ wi

Light beam width depends on:

Shadow map resolutionParameterization

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Outline

Related workAliasing errorLogarithmic parameterization

Directional lightPoint light

Hardware architectureResults

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Parameterization

Standard Perspective warped

light imageplane

shadow map shadow map

light light

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Directional light

imagebeam

light imageplane

t

wl

light

z

1

wi

1. Light beam widths

2. Texel spacing function

3. Parameterization

[Wimmer et al. 04]

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Quadratic

z

beam

wid

th

Uniform

z

beam

wid

th

Linear

zbe

am w

idth

Directional lightWith 4x4 matrix With logarithm

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Comparison

Standard

Perspective

Logarithmic

101

102

103

104

100

102

104

106

108

f / n

Sh

ad

ow

map

texe

ls /

imag

e te

xels

frustum fov = 60○

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Point lights

light

view frustum

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Point lights

light

light imageplane

view frustum facez

y wl ~ z

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Point light spacing function - z

facey

zzylight

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Point light spacing function - z

z lighty

faceylight

y

zz

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Point light spacing function - z

faceylight

y

zz

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Point lights – parameterization

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Fitting the spacing function - z

Parameterization for exact function is complicatedFit two parabolas:

Parameterization:0

z

Point light spacing function

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4x4 matrix + logarithm

vc : [xc zc yc wc]T - clip coordinate

vn : [xn zn yn 1]T - normalized device coordinate

v : [x z y 1]T - world coordinate

P :perspective or orthographic projection matrix

a0-a3 : constants

b0-b3 : constants

To screen coordinates

s = a0ln( a1xn + a2 ) + a3

t = b0ln( b1zn + b2 ) + b3

To NDC coords.

vc = P vvn= vc /wc

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Parameterization summary

Directional Point

x:

z:

without log with log

ort

ho

.p

ersp

.

Actual spacing functions Spacing functions from4x4 matrix + log

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Outline

Related workAliasing errorLogarithmic parameterizationHardware architectureResults

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Logarithmic parameterization

Unwarped

viewfrustum

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Vertex program

Transform vertices on the GPURequires finely tesselated model

Increases burden on vertex processor

Adaptive tesselation complicated

Easier with DX10 geometry shaders

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Fragment program

Brute force rasterization

Render bounding primitiveTransform fragments to original triangleDiscard if not inside

10x slow downComputing bounding primitive is complicated

Easier with DX10 geometry shaders

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Graphics pipeline

fragment processor

vertex processor

rasterizer

alpha, stencil, & depth tests

blending

clip & back-face cull

memoryinterface

depthcompression

colorcompression

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Rasterizing equations

Coverage determination

use sign of edge distance equations

Attribute interpolation

depth, color, texture coordinates, etc.

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Rasterizing equations

Parameterization:

Edge distance and attribute interpolation:

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Incremental computation

1 MULT + 2 ADDper step

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Depth compression

First order Second order

Compressed tiles shown in red

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Feasibility

Current hardware trendComputational power increasing rapidlyBandwidth lags behind

Log shadow mapsIncrease computationReduce memory/bandwidth consumption

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Nonlinear rasterization

Configurable rasterizerOther rasterization functions are possibleMight be useful for other effects

Reflections, refractions, caustics, general multi-view perspective [Hou et al. 06]

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Outline

Previous workAliasing errorLogarithmic parameterizationHardware implementationResults

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Results

Power plant13 Mtris

Oil tanker82 Mtris

Town scene59 Ktris

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Results

Standard Perspective warping Logarithmic

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Results

Perspective warping Logarithmic

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Results

Perspective warping Logarithmic

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Comparison to z-partitioning

101

102

103

104

100

102

104

106

108

k=1

k=2

k=4

k=8

f / n

Sh

ad

ow

ma

p t

ex

els

/ i

ma

ge

te

xe

ls Perspective with kz-partitions

Logarithmic

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Results

z-partitioning (k=4) Logarithmic

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Results

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Results

Perspective warping Logarithmic

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Results

Perspective warping Logarithmic

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Results – Point lights

Standard Perspective warping Logarithmic

high error

higherror

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Advantages

Logarithmic shadow maps require less bandwidth and storageSmoother parameterization than z-partitioning

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Disadvantages

Does not handle projection aliasingRequires multiple shadow maps

Up to 5 directional lightUp to 4 per frustum face for point light

Warped depth values

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Conclusion

Logarithmic parameterizationLower error than previous methods

Hardware architecture for log rasterization

Requires only small enhancements to current graphics hardwareExploits current hardware trends

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Future work

More details for hardware implementation

precision requirementsfilteringshadow map bias

Other applications for nonlinear rasterization

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Acknowledgements

Aaron Lefohn for the town modelSupported in part by:

NSF Graduate FellowshipARO Contracts DAAD19-02-1-0390 and W911NF-04-1-0088 NSF awards 0400134 and 0118743ONR Contract N00014-01-1-0496DARPA/RDECOM ContractN61339-04-C-0043Intel

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Questions?

http://gamma.cs.unc.edu/logsm

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Coordinate systems

view

y

zx

warping frustum

viewfrustum

light

shadow map

t

s

Light space [Wimmer et al. 04]

z

x

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Directional light - general case

Highest error (wl /wi) occurs at the facesPartition frustum to handle each face separately

wl

light

shadow map t

z