Working Group « Pre-Filtering »

Why Pre-filtering?

● Complex light transports and complex materials extensively studied

● Today key issue = management of details● Very large scenes● Complex objects

● High quality, antialiased rendering requires costly integrals per pixel● Adaptive multisampling not a solution

● Pre-filtering = pre-integrate as much as possible

General equationScreen spaceIntegral over pixel

Change of variablesIntegral over pixel footprint

Introduction of aLocal Illumination Model

Change of variablesIntegral over surface maps

Uncorrelation hypothesisHypothesis

Consequence: average of product = product of averages

Application (when hypothesis valid – cf discussion):

Average color

Average BRDF

Average « visibility »

Average shadow

Parallax effects

Outline

● Using uncorrelation hypothesis & neglecting parallax:

– Color map filtering– Normal map filtering– Horizon map filtering– Shadow map filtering– Procedural map filtering– Summary

● Discussion of correlation & parallax effects● Conclusion

Color map filtering

● IF parallax effects neglected● THEN simple linear filtering of color map

→ Can use hardware anisotropic filtering

gives

Normal map filtering

Normal map filtering

● IF parallax effects neglected● THEN « simple » linear filtering of BRDF

● but BRDF = 4D function, not easy to store● Simplification hypothesis:

● BRDF represented with 2D distribution of normals

gives

normal map

Normal map filtering

● Direct methods● Represent fx with linear parameters, mipmap them

● Convolution methods● Further assume that● To use ergodicity relation● We get

● Represent px with linear parameters, mipmap them

Normal map filtering

● Representing 2D distributions:● Single Gaussian lobe represented with mean and

second moments (linear parameters)– 3D lobe, 2D lobe in tangent plane, isotropic or

anisotropic, etc– Second moments sometimes deduced from the mean

(Toksvig)● Multiple lobes● Spherical harmonics

Horizon map filtering

Horizon map filtering

● Fraction of S visible from viewer AND light

● Visibility represented with horizon map

● Reformulate with horizon distribution functions

● Or write V in a basis (e.g. Legendre Polynomials)

H = Heaviside function, non linear

V now linear in px!

Horizon map

(parallax effects neglected)

Horizon map filtering

● Representing horizon angle distributions:● Gaussian with mean and second moment

● Correlation between view and light directions: ● approximations

hotspot effect

[Ashikhmin et al.]

[Tan et al.]

Shadow map filtering

Shadow map filtering

● Average shadow (parallax neglected)

● Reformulate with depth distributions

● Or write S in a basis (e.g. use Fourier series)

H = Heaviside function, non linear

Shadow map (= depth map from light)

S now linear in ps!

gives

Other metod

Shadow map filtering

● Representing depth distributions● Gaussian with mean and second moment

Procedural map filtering

Procedural map filtering

● Global fading● Smooth transition from f(x) to its average● Drawback: attenuate ALL frequencies

● Analytic integration● Drawback: almost always impossible to compute

analytically● Workaround: apply on each term separately

● Frequency clamping● For methods using frequency synthesis (sum of

sinusoids, of trochoids, of noise functions, etc)

Summary

● To pre-filter a map M used in a non-linear function f(M,y)● Direct methods:

– use a « change of variable » to get f(M,y)=g(K,y) with g linear in the new variables K

– Pre-filter the new K variables● Convolution methods:

– Introduce the parameter distribution (= histogram) to get f(M,y)=∫f(m,y)px(m)dm, with px(m)=δ(m-Mx)

– Pre-filter linear parameters representing the px functions (e.g. mean and second moments, coefficients in a basis, etc). AND, at runtime, evaluate « convolution » f*pA

Summary

● Finding linear parameters characterizing a function:● Use moments● Use a basis of functions Bk

● Use a spanning set of functions

OR

→ spare representation

ki generally not linearly interpolable (counter-example: k = moments), requires alignment of « lobes »

Discussion

● Uncorrelation hypothesis● False for small footprints● Correlations between color and reflectance

– For each color channel, weight normals with reflectancel● Correlations between color and visibility● Correlations between reflectance and visibility● Correlations between view and light visibility

– Approximate solutions (cf above)

Discussion

● Parallax effects● Parallax offset

– Could be handled separately● Parallax Jacobian

– Not handled in existing works– Possible solution for normals

Discussion

● Silhouettes and curvature● Curvature and normal maps

– Curvature of base surface influences normals– Unless normals stored in global frame (instead of tangent

frame)● Curvature also influences horizon angles● Silhouettes

– Raw mesh resolution becomes visible● Silhouette clipping, shell mapping, relief mapping, etc

– Very anisotropic footprints

Discussion

● Mesh filtering and extreme filtering● At large distance, mesh itself must be pre-filtered● Removed mesh details must be incorporated in

surface maps (bump maps, normal maps, etc)● Topology changes can occur

– e.g. Tree foliage becomes a « volume » at large distance

Conclusion

● Many hard problems remain to be solved!