Fast Soft Self-Shadowing on Dynamic Height Fields

John Snyder

Microsoft Research

Derek Nowrouzezahrai

University of Toronto

Related Work

• horizon mapping [Max88; …]

– hard shadows– precomputed for static geometry

[Sloan&Cohen00]

Related Work

• shadow map filtering [Reeves87; …]

– light “bleeding” artifacts– small light sources– no complex environmental lighting

[Donnelly06]

Related Work

• ambient occlusion [Bunnell05; Kontkanen05; …]

– AO in screen-space [Shanmugam07;…]

– “cone” blocker model [Heidrich00;…]

[Oat07]

[Dimitrov08]

Related Work

• static relighting [Sloan02; Ng04; …]

• dynamic relighting [Bunnell05, Ren06, Sloan07]

Goals

• strong response to lighting direction (cast shadows)

low-frequency SH [Ren06]ambient occlusion

Goals

• strong response to lighting direction (cast shadows)• environmental + directional lighting

Goals

• strong response to lighting direction (cast shadows)• environmental + directional lighting• dynamic geometry (not precomputed)• real-time performance

• limitation: geometry is height field• applications:

– terrain rendering (flight simulators, games, mapping/navigation)– data visualization

1. approximate horizon map via multi-resolution- create height field pyramid [Burt81]- sample height differences from each pyramid level- use coarser levels as distance to receiver increases- reduces sampling

2. convert horizon map to SH visibility for soft shadowing- use visibility wedges [Dimitrov08]- get good directional lighting response- sharpen shadows by restricting wedges azimuthally- fast: {2D lookup + SH z rotation + Σ over wedges}

Summary of Main Ideas

Horizon Map [Max88]

height field: z=f(x)=f(x,y), point: p(x) = (x,f(x))

horizon angle: (x,) max angle horizon makes at p

in azimuthal direction

Sample at all points x along set of directions i.

Calculating the Horizon Map

t

yxftytxft

),()sin,cos(tanmax),( 1

),0(

x

Calculating the Horizon Map

t

yxftytxft

),()sin,cos(tanmax),( 1

),0(

x

t

Calculating the Horizon Map

t

yxftytxft

),()sin,cos(tanmax),( 1

),0(

x

t

Calculating the Horizon Map

t

yxftytxft

),()sin,cos(tanmax),( 1

),0(

x

t

Calculating the Horizon Map

t

yxftytxft

),()sin,cos(tanmax),( 1

),0(

x

t

Calculating the Horizon Map

t

yxftytxft

),()sin,cos(tanmax),( 1

),0(

x

t

Calculating the Horizon Map

t

yxftytxft

),()sin,cos(tanmax),( 1

),0(

x

t

Calculating the Horizon Map

t

yxftytxft

),()sin,cos(tanmax),( 1

),0(

x

t

Calculating the Horizon Map

t

yxftytxft

),()sin,cos(tanmax),( 1

),0(

x

t

Calculating the Horizon Map

t

yxftytxft

),()sin,cos(tanmax),( 1

),0(

x

Problem: aliasing – need many samples in t.Solution: prefilter height field, apply multi-scale derivative.

t

max

As t ↑, i ↑ : sample coarser levels further from x.

Multi-Resolution Approximation

i

iiiiii d

yxfdydxfdfD

),()sin,cos(),,,(

x

kii ld 2

multi-scale derivative

sampling distance for level i

if pyramid level i

fi fi-1 fi-2 fi-3

di di-1

di-2di-3

i i-1 i-2 i-3

Multi-Resolution Horizon Angle

horizon angle sample i

di di-1

di-2di-3

),,,(tan 1iii dfD x

Up-sample coarser levels with 2D B-splines:

Reconstruct blocking angle function with 1D B-splines:

)),,(),...,,,(,(spline-b),,( 1100 NN dxdxx

i i-1 i-2 i-3

Multi-Resolution Horizon Angle

knot

mid

Take max over sample knots and midpoints:),,(max),(

}1,0{

xx

N

Pyramid Level Step

• k = 1 (standard power-of-2 pyramid) abrupt transitions

• k = 4 smoothes transitions

k = 1 k = 2 k = 3 k = 4

Pyramid Level Offset

• pyramid level bias when sampling height differences• increasing bias increases shadow sharpness• increases sampling requirements

o = 0 o = 1 o= 2 o = 3 o = 4

1. approximate horizon map via multi-resolution- create height field pyramid [Burt81]- sample height differences from each pyramid level- use coarser levels as distance to receiver increases- reduces sampling

2. convert horizon map to SH visibility for soft shadowing- use visibility wedges [Dimitrov08]- get good directional lighting response- sharpen shadows by restricting wedges azimuthally- fast: {2D lookup + SH z rotation + Σ over wedges}

Summary of Main Ideas

Reconstructing Visibility

• so far discussed sampling in single azimuthal direction• large lights sample multiple azimuthal directions• linearly interpolate horizon angle as function of

• sequential pairs of i determine visibility wedges

Projecting Visibility to SH

• Project visibility wedge to SH (order 4)

• fix i = 0 and Δφ. Store as 2D table

• Visibility for a single wedge

otherwise. 1,

)( and ] ,[ if ,0),( 1

iiv

S

ibabai dsv )(),,(),( sysv

Reconstruct Full Visibility

• (N – 1) table lookups + (N – 1) SH Z-rotations

rotate wedge from i = 0 to azimuthal direction

• Sum over all wedges

• N azimuthal sampling directions N horizon angles• (N – 1) adjacent horizon angle pairs (wedge boundaries)

All operations (including horizon angle calculation) in a single

GPGPU shader. See [Snyder08] for full source code.

Azimuthal Swaths

• azimuthal swaths contain many visibility wedges– key lights: restrict swath

– env lights: use complete swath

get sharper shadowsacts as a geometric maskonly sample where necessary

Soft Shadowed Shading with SH

xV visibility vector at x

diffuse reflectance: clamped

cosine around normal Nx

L lighting environment

)( xH N

xN

wedges

izrot ) ,(

Soft Shadowed Shading with SH

LHV xx ),(, N SH triple product

xV )( xH N

or

BRDF x Visibility SH Product

and dot with lighting

xT

LTx ,

Comparison with Ground Truth

ground truth k = 1, o = 1 k = 2, o = 2 k = 3, o = 3 k = 4, o = 4

Measured PerformanceHF resolution pyramid levels FPS (key only) FPS (env + key)

256x256(130k triangles)

33 165 57.2

512x512(522k triangles)

37 34.5 11.5

1024x1024(2.1M triangles)

41 7.05 2.48

Image Results

Video Results

Clip 1

Clip 2

Conclusions

• multi-resolution approximation for horizon map• soft shadowing via fast SH projection• key + env lighting decomposition• simple GPU implementation• real-time up to 512x512 dynamic height fields• performance independent of geometric content

Thanks! Any questions?