A Heightfield on an Isometric GridMorgan McGuire and Peter SibleyBrown University

A Heightfield on an Isometric Grid

The Isometric HeightfieldThe Isometric Heightfield

• 10% more accurate, subjectively smoother

• Practical for real applications

• GPU accelerated

• Drawback: requires resampling

MotivationMotivation

Traditional Ortho-HeightfieldTraditional Ortho-Heightfield

y-displacement

3D View

Grid

Heightfield BenefitsHeightfield Benefits

• Represent scientific data at highest fidelity

• Practical

• Great for dynamic data

• Storage efficient for high-frequency data

• Simple!

Worst Case Ortho FitWorst Case Ortho Fit

vs. Best

Musgrave’s HeightfieldMusgrave’s Heightfield

• Shear axes to produce a parallelogram• Equilateral tessellation

rrcSrc ,, r

c x

z

Grid Space World Space

New Isometric HeightfieldNew Isometric Heightfield• New Mapping

– Shift odd rows ½ edge length– Slide end vertices to make square tiles– Border padding for computing vertex normals

• Drop-in replacement for ortho-heightfield

c

rGrid Space World Space

1,1&2, rrcSrc

x

z

Related WorkRelated Work

• Musgrave 88– Grid Tracing: Fast Ray Tracing for Height Fields

• Duchaineau et al 97– ROAMing terrain: real-time optimally adapting meshes

• de Boer 00– Fast Terrain Rendering Using Geometrical MipMapping

• Shankel 02– Fast Heightfield Normal Calculation

• Middleton 01, 02– Edge Detection in a Hexagonal-image Processing Framework– Markov Random Fields for Square Hexagonal Textures

AnalysisAnalysis

Signal Processing AnalysisSignal Processing Analysis

• Underlying “true” function h(x, z)• Heightfield approximation:

– Low-pass filter and sample– y(x, z) at integer x, z– Barycentric interpolation for non-integer x, z

• Reasons for poor fit:– Aliasing from inadequate sampling– Inappropriate reconstruction filter (i.e. triangles)

ExperimentExperiment

• Approximate a 2D Sinusoid

h(x, z) = sin(f[x cos + z sin ] +)• Vary phase, angle, and frequency• Error Metrics

– Elevation– Shading– Normal deflection– Curvature

• Any function is a weighted sum of sinusoids• Draw generalizations about arbitrary functions

Cleaner High Freq.Cleaner High Freq.

Ortho Iso

Smoother Low Freq.Smoother Low Freq.

Ortho Iso

Shaded

Smoother Low Freq.Smoother Low Freq.

Ortho Iso

Shaded

Curvature

Shading Error

Isometric: 25% more accurate shading.

(lower is better)

AlgorithmsAlgorithms

ResamplingResampling

• Existing ortho-grid sampled data– Reconstruct true continuous surface– Filter and downsample on hex pattern– Lossless if original was correctly sampled

• Artist created data– Iso-heightfield as modeling primitive

Moving on the SurfaceMoving on the Surface

• We know elevation at vertices

• We need elevation between vertices– Must exactly match rendered elevation!

• 4 cases, easily derived from Barycentric interpolation

H

G I

L=(1-)H+IJ=(1-)H + G

P=(1-)J+L

Fast Vertex NormalsFast Vertex Normals

• Indices of neighbors:

• Grid space edge vectors:

• Vertex normal contains 6 cross-products:

D E

IP

F

H G

Cps

pisCphsCpg

pfsCpesCpd

11

11

0,,1sin,,

sin,,0,,1

sin,,sin,,

321

321

321

321

PyIyPIPyHyPH

PyGyPGPyFyPF

PyEyPEPyDyPD

PIPDPHPIPGPH

PFPGPEPFPDPEN

Fast Vertex NormalsFast Vertex Normals

• Algebra dramatically simplifies:

• Operation count: 6 add, 1 mul

– Cheaper than 1 cross-product!

yyyyy EHGDFIEHGDN ,4,2sin 364

D E

IP

F

H G

Normals in Vertex ShaderNormals in Vertex Shader

const uniform mat4 K =

vec3 A(tex2D(d), tex2D(h), tex2D(i));vec3 B(tex2D(g), tex2D(e), tex2D(f));vec4 C(A – B, 1);vec4 N = mul(K, C);

0000

0011

4000

03

23

13

1

1TMV

Precomputed outside vertex shader

Actual per-vertex work

• 6 texture lookups (NV50?)

• Neighbor indices are the same for each tile– Precompute and store in tex coord stream

Fast Shadow VolumesFast Shadow Volumes

Heightfield TilesHeightfield Tiles Page from disk Unit of LOD, texturing

Wake Island from Battlefield 1942 courtesy of Digital Illusions CE

Triangle StripsTriangle Strips

Start

End

Sub-sampled LODSub-sampled LOD

• Geo-mipmapping would be ideal– Cannot afford to touch vertices on CPU

• Use subsampling– Keep edges at maximum resolution – Carefully alpha-blend transitions

LOD MovieLOD Movie

Efficient Storage (NV20+)Efficient Storage (NV20+)

• Pack x, z in int16 vertex stream – Same for every tile

• Pack y in int16 texture coordinate stream

• Combine streams in vertex shader

• Modify y on CPU, upload a relatively small vertex array– 2 bytes/vertex vs. 12 bytes/vertex

Efficient Storage (NV40)Efficient Storage (NV40)

• Put y in a texture– SM 3 allows texture lookup in vertex shader!

• Modify y using render-to-texture– Navier-Stokes– Render deformations– Tire tracks & footprints– [insert your favorite GPU hack here]

ApplicationsApplications

VisualizationVisualization

GamesGames

AcknowledgementsAcknowledgements

• Advisor John F. Hughes

• Battlefield 1942 data courtesy of Digital Illusions CE• Mars data from Dr. James Head III, Brown University

• Additional coding:– Hari Khalsa– Nick Musurca

Hardware and data made available by:

Elevation Error

Isometric: Same quality with 10% fewer polygons.

(lower is better)

What about ROAM?What about ROAM?

• Irregular tessellation:– y(x, z) for a data-dependent set of samples– Barycentric interpolation between samples– Can produce a better fit

• Less aliasing: put samples where needed• e.g. ROAM, Triangulated Irregular Networks

• Heightfields are still preferred for many applications…

Our Mapping (2)Our Mapping (2)• Grid to world space:

• Grid space to memory:

• Row and column sizes:

• Also tweak end-vertices on even rows• Drop-in replacement for orthogonal heightfield

– Square tiles– Convenient texture & normal mapping

odd ,33

sin2 ;13 ;, RCRCCR

1,1&2, rrcSrc

rCcrc ,

Need Indices and WeightsNeed Indices and Weights

H

G I

P

Indices: g = ?, h = ?, i = ?

Weights: wG = , wH = ?, wI =

Barycentric InterpolationBarycentric Interpolation

• Simple equations exist for g, h, i, , • 4 triangle orientations: {even row, odd row} x {up, down}

H

G I

L=(1-)H+IJ=(1-)H + G

P=(1-)J+L

wG = 1 –, wH = 1 , wI =

ExampleExample

• Given P = (c, r)• Let• Example: odd row, up triangle

• 3 other cases

Crcprrvccu ,,

)( and )12mod( vur

g = p + C, h = p, i = g +1

= v, = u

Self-Shadowing is HardSelf-Shadowing is Hard

Problems:

• Shadow maps are expensive for terrain– High fill rate

• Shadow volumes expensive for terrain– High fill and vertex rate

• Pre-computation, horizon map no good for dynamic data

A Trick for ShadowingA Trick for Shadowing

• Constrain the sun orbit the z-axis – i.e. moves strictly East-West

• (x0, z0) only be shadowed by (x0 – k, z0)

• Start at the sun end of the x-axis, iterate over vertices

• Incrementally compute the max shadowed elevation

• Works on ortho-heightfield too

Fast Shadow VolumesFast Shadow Volumes

• Terrain shadows other objects• Other objects do not shadow terrain (use projective

shadows, shadow maps, volumes, etc. for that)• Iterates along rows, which are memory-aligned• Almost free

– Compute shadowing inside the loop that uploads elevation to the graphics card

Future WorkFuture Work

• Compute true Geo-mipmapped LOD in hardware?

• LOD w/ normal maps

• Update shadow elevation in hardware?

• Experiments with deformation and destruction

Irregular SamplingIrregular Sampling

Regular

Irregular spacing captures shape better:

Irregular

General Ortho ProblemGeneral Ortho Problem