Parametric Master Element for Global Illuminationvegas.loria.fr/papers/jcp/parametric.pdf ·...

Online Submission ID: papers0381

Parametric Master Element for Global IlluminationCategory: Research

Left: PME global illumination applied to “David” (500K∆). Center: PME decouples the representation of the lighting from the geometry.Sharp shadows are accurately captured by adaptive wavelets, represented in parameter-space. The image shows the geometric triangulatedmesh with the wavelets FEM control mesh superimposed; Right: PME Stanford Bunny with area lights (70K∆). Both are displayed at 40 fps

Abstract

Complete numerical simulation of global illumination is a compu-tationally intensive process. The complexity of the problem has twomain aspects:� geometric complexity: Highly tesselated models dramatically

increase computation times;� lighting complexity: Sharp shadows and small-scale lighting

variations are difficult to represent.For these two reasons, finite element methods perform poorly

when applied to highly tesselated models, due to the tight couplingthey introduce between the geometry and the lighting. In contrast,Monte-Carlo methods completely decouples the lighting from thegeometry, and the underlying integration is less sensitive to the dis-continuities of the functions. However, it is difficult to use the com-puted solution for real-time rendering. Moreover, In our case, ourgoal is to compute acompletesolution.

In this paper, we introduce the Parametric Master Element(PME), a new structure that makes finite-element methods appli-cable to large tesselated meshes, by using a parameterization to de-couple the representation of the energy from the geometry of thescene. Both can beindependentlyandcontinuouslyconsidered atdifferent scales, which optimizes large-scale transfers and enablescapturing sub-facet lighting details. PME may be used by variousfinite-element global illumination methods. The view-independentsolution, represented in parameter-space, can be directly exploitedby graphics hardware for interactive walkthroughs.

CR Categories: I.3.3 [Computer Graphics]: Picture / Image Gen-eration I.3.7 [Computer Graphics]: Three-Dimensional Graphicsand Realism

Keywords: global illumination, parameterization, tesselated mod-els, multiscale methods, wavelets, Galerkin

1 Introduction

Computing global illumination in general environments is achallenging task. Global illumination may be considered in termsof computing the interactions between thelighting signal andthe geometric signal(i.e. the scene). These interactions occurat variousscales. To deal with this issue, thesignal processingfamily of approaches is a well-suited formalism. As such, themulti-scale approach is a natural choice, which dramaticallyimproves performances. When computing an energy transfer, themain idea consists in considering the problem at a suitable scale.For large-scale variations of lighting, grouping primitives makescomputations more efficient in uniform zones. For small-scalevariations, adaptively subdividing primitives enables capturinglighting variations of high frequency. The same approach is usedby ray-tracing methods, with the difference that the adaptive dis-cretization is performed in ray-space rather than in primitive-space,based on the theory of sampling.

Environments composed of a large number of primitives, suchas highly tesselated models, show a high variability of these scales,which makes classic finite-element methods unapplicable. The fol-lowing four classes of interactions can be considered (see e.g. [Ka-jiya 1985]):� large scale→ large scale: uniform lighting conditions applied

to simple geometry;� large scale→ small scale: uniform lighting conditions applied

to complex geometry;� small scale→ large scale: irregular lighting conditions applied

to simple geometry. For instance, when a complex object castsshadows onto a simple one;

� small scale→ small scale: irregular lighting conditions appliedto complex geometry. For instance, when a complex object castsshadows onto itself.



In classic finite element approaches, the representation of the ge-ometry and the lighting signal are tightly coupled. For instance,if the illumination is represented at the vertices of a triangulation,interactions between large-scale lighting and small-scale geome-try requires to oversample the lighting signal, which dramaticallyincreases the total computational cost. Conversely, representingsmall-scale lighting details applied to large-scale geometry requiresrefining the geometric representation.

To deal with this problem, we introduce the Parametric MasterElement (PME), a new representation of tesselated models that de-couples the lighting and the geometric signals, by using a parame-terization of the mesh. We demonstrate how this structure can over-come the main limitations of finite element methods (scale problem,mesh artifacts and performances). The global light simulation algo-rithm can choose the appropriate discretization scale for the lightingsignal, stored in parameter space using wavelets.

The paper is organized as follows. The remainder of this in-troduction presents the previous work, the advantages and limita-tions of our approach. Section 2 gives the definition of a ParametricMaster Element and an algorithm to construct it from a tesselatedmodel. Section 3 shows how PME can be used to optimize Galerkinfinite elements methods and proposes an implementation for pre-computed lighting transfer. The conclusion gives some suggestionsfor future work.

1.1 Previous Work

FEM Radiance Computing

Numerical solutions to the light transport equation introduce sam-pling or discretization errors. Monte-Carlo methods distribute lightpaths in space, and FEM require to recast the global illuminationproblem in terms of finite-dimensional quantities (i.e. basis func-tions), and finite processes by which the approximation of the func-tion is selected from this space. On the one hand, FEM is veryeffective to solve for the complete solution of the radiance equa-tion, on the other hand, it is strongly dependent on the geometriccomplexity of the scene. Our goal in this paper is to address thislimitation of FEM by decoupling the light representation from thegeometric support.

One of the main improvement to FEM radiosity and radiance isthe introduction of wavelets bases [Gortler et al. 1993],[Schroeder1994], [Christensen 1995]. These functions combine the power ofadaptative subdivision with functional analysis. Often, a smooth-ing heuristic is applied to a piecewise-constant approximation fordisplays purpose, although this does not necessarily improve theaccuracy of the approximation. When non-constant elements areemployed, the degrees of freedom of the elements adds to the di-mension of the approximation space. The introduced coupling be-tween the light and the geometric support makes wavelet radiosityand radiance unpractical for highly tesselated objects. Importancedriven refinement has been applied to solve for one variable (i.e.the viewpoint) of the radiosity equation [Smits et al. 1992] and ra-diance equation [Christensen et al. 1996]. This approach accuratelycomputes a subset of the scene, but the solution becomes view de-pendent and consequently cannot be used for interactive viewing.In that case, Monte-Carlo methods are more appropriate since theyare very efficient in solving for one variable of the integral equation.

In this paper, we show how our mesh representation makes FEMglobal illumination scaling well with the geometry, and accuratelycaptures small-scale lighting details.

Clustering

Clustering approaches have appeared as a promising avenue to sim-plify the geometry of space. Clustering approaches were motivatedby the FFM approach [Greengard. 1988], initially proposed as an

efficient way of solving the N-Body physical problem. FFM uses ahierarchical decomposition of the computational domain as well ashigher-order series of expansions to group sets of sufficiently dis-tant particles into “super-particles”.Volume clusteringapproaches[Smits et al. 1994], [Sillion et al. 1995], [Sillion 1995], [Gibsonand Hubbold 1996], [Christensen et al. 1997] exploit this idea, bygrouping primitives into volumes (see Figure 1-A). However, sinceFFM volume cluster are meant to approximate “well separated”bodies, they fail to account for the topology of surfaces in a ge-ometric 3D model. Consequently, volume clustering approachesrequire a costly final gather pass.

The face clusterapproach [Wilmott et al. 1999], [Garland et al.2001] attempts to preserve more geometric information, by group-ing facets having approximatively the same normal (see Figure 1-B). However, as shown in [Hasenfratz et al. 1999], the variabilityof the normals within the cluster still causes shading artifacts wheninterpolating the scalar irradiances accross the volumes. For thisreason, it is difficult to obtain a continuous lighting (a ad-hoc solu-tion to this problem uses post-processing smoothing heuristics). In-stead of pushing irradiances to the leaves as done in [Sillion 1995],[Smits et al. 1994], [Christensen et al. 1997], the approach pro-posed in [Gobbetti et al. 2003] partially solves the problem by us-ing a shooting-based linkless method and higher order functions,improving the smoothness of the result. However, face clusters arestill a rough approximation of the geometry, for which error controlis difficult.

All the approaches above usecoupledrepresentations for the ge-ometry and the lighting. To overcome this limitation, this paperintroduces theParametric Master Element, a higher-level structurethat preserves and structures all the geometrical information (seeFigure 1-C). Higher order bases can be used, which means the so-lution is smooth, without requiring any post-processing. In contrastwith other clustering approaches, PME uses a parameterization todecouple theses function bases from the geometry. This facilitatesboth macro-scale computations and sub-facet adaptive refinement.

Pre-computed Radiance Transfer (PRT):

PRT [Sloan et al. 2002] computes the light transfer function of anobject, i.e. the function that maps source illumination into trans-ferred incident illumination at each point of the object. SphericalHarmonics are used to represent this function, and their coefficientsare estimated at the vertices using Monte-Carlo integration. Morerecent work [Sloan et al. 2003a] uses clustered principal compo-nents to decrease both storage requirements and computation time.A bi-scale extension is proposed in [Sloan et al. 2003b], which sep-arates the mesocopic, i.e. at Bidirectional Texture Function (BTF)scale [Dana et al. 1999], from the geometric details. Mesoscopicdetails are mapped onto the object using a parameterization. Pro-grammable graphics hardware can interactively display theseRadi-ance Transfer Textures (RTT).

The approach we propose here is also based on a parameteriza-tion of the meshed model. In our case, since our goal is to handlelarge scanned meshes, fine-scale details areexplicitelyrepresentedin the geometry, and cannot be captured by a repeatedly mappedBTF or RTT. Rather than using twofixed scales, our hierarchicalrepresentation isadaptivelyrefined to capture small lighting de-tails. In the results section, we show an example of precomputedillumination based on our PME representation.

1.2 Overview

� PME uses a parameterization to decouple the lighting from thegeometric representation. The relevant discretization scales canbe used to represent the lighting signal;



Figure 1:A: volume clustering is best suited to unstructured objects, such as plants. B: facet clustering keep more geometric information, bygrouping triangles that approximatively share the same normal. C: PME may be thought of as an evolution of these clustering methods, thatpreserves all the geometrical information. Each cluster (a) is provided with a parameterization (b), and illumination can be represented inparameter-space by high order wavelets (c). This decouples the lighting representation from the geometric support.

� soft and hard shadows are naturally represented, the lighting sig-nal stored as wavelets in parameter space dynamically adapts tothe details to represent;

� the linear and quadratic wavelets yield a smooth function space.Therefore, no smoothing post-processing heuristic is required;

� quadrature methods can be applied to compute integrals over thegeometric primitives, which helps formalizing and quantifiingthe approximation error;

� the view-independent computed solution, stored in parameterspace, can be easily transformed into light maps, displayed inreal-time by the hardware on a consumer’s PC.

� Limitations: Our algorithm is slower than fast interactive ap-proaches (e.g. [Keller 1997],[Benthin et al. 2003]). However,note that our goal here is to provide aview independentandcom-pletesimulation rather than aninteractiveone.

2 Parametric Master Element

Our Parametric Master Element representation is a natural exten-sion of theMaster Elementapproach (also calledParent Element)[Zinkiewicz and Taylor 1989], used in FEM theory. The Master El-ement is a general, powerful concept, that facilitates adaptive sub-division. In this approach, 1D, 2D or 3D elements are “mapped”into distorted shapes, using parameterizations. For instance, 2Delements can be mapped onto 3D surfaces. This correspondencemakes it possible to simplify the computations, by transformingthem into an undistortedcanonicalspace. The term “master ele-ment” refers to the canonical undistorted element, seen as a “tem-plate”, instanciated multiple times to create the FEM mesh.

To our knowledge, the Master Element concept was first men-tioned in the context of global illumination in [Cohen and Wallace1993]. This approach was generalized to higher order surfaces in[Alonso et al. 2001], which uses a single finite element to representthe illumination of a quadratic surface.

This is similar to our approach, with the difference that we con-sider tesselated surfaces. In this context, instead of using a finiteelement per facet, we decompose the model into topological discs(or charts), and associate a PME with each chart (see Figure 1-C). Since each chart comprises a large number of facets, this rep-resentation dramatically improves performances. In addition, allthe theoretical background from the Master Element method can bereused. However, when considering a chart, in contrast with a para-metric surface, no parameterization is knowna priori. As shown inthe next section, recent advances in mesh parameterization enableto deal with this issue.

Note that the curvilinearu,v coordinates yielded by the param-eterizations need to satisfy some differential properties, depend-ing on the PDE to solve. In our context, to accurately manipulatethe energy in canonical space, the Jacobian of the parameterizationneeds to be constant over each element. Moreover, it is well knownthat numerical computations are more stable if the parameteriza-tion approximates an isometric mapping. Ensuring this propertyhas the following advantages. First, it gives a natural way of ex-tending traditional FEM techniques to complex primitives; second,it simplifies the computation of push-pull coefficients on parametricsurfaces. Using an energy-preserving parameterization, they onlyinvolve a double sum of coefficients in the general case; and third,most importantly, the push-pull coefficients do not depend on thehierarchy level. They can thus be computed and stored at the toplevel of the hierarchy.

Using this parameterization, it is possible to apply the MasterElement approach to our problem, and to store the illuminationin parameter-space, as wavelets. Therefore, as shown in Figure 2,PME is a decoupled geometric / lighting multi-resolution represen-tation. Lighting can be adaptively represented by wavelet bases,without any constraint yielded by the initial mesh discretization.

The next section shows how to construct a PME from a tesselatedmodel and how to implement FEM global illumination based on aPME representation.

Figure 2:The mesh of the “David” dataset and the decoupledwavelet control mesh stored in parameter-space.



Figure 3:PME Construction. A: scanned meshes often have small holes and irregularities. They can be repaired using simple heuristics; B:the object is segmented into charts, using a discrete counterpart of centroidal Voronoi diagrams, driven by an estimate of the curvature; C:each cluster is parameterized and extrapolated, which puts it in correspondence with a square domain, where high-order wavelets are stored.

2.1 Construction

This section presents afully automaticmethod to convert a tesse-lated model into a set of PMEs. The model is decomposed intocharts, i.e. parts that are homeomorphic to discs. Each of this chartis then transformed into a master element. The method consistsof three distinct steps, involving recent results in Digital GeometryProcessing. Robust implementations of the required basic tools arenow available on the web [Fabri et al. 2000],[Graphite 2003].

1. Model repairing (Figure 3-A): It is well known that finite el-ement methods require a clean geometry. However, scannedmeshes are often “punctured” by tiny holes, which can ar-tificially increase the number of clusters created by our al-gorithm. For this reason, we propose here a simple methodto “clean” the geometry before applying subsequent steps.To fill a hole, edges are iteratively inserted. Each new edgeconnects the pair of vertices(pi ,p j ) that maximizes the ratio(si − sj )/d(pi ,p j ), whered(pi ,p j ) denotes the 3d Euclidiandistance betweenpi andp j , and wheresdenotes a curvilinearcoordinate measured along the border. The constructed trian-gles are then regularized by applying Laplacian smoothing;

2. Segmentation(Figure 3-B): we use an approach inspired by[Sander et al. 2003] to decompose the model into charts. Tofacilitate the parameterization step, our goal is to improve thecurvature balance in the charts. The approach proposed in[Sander et al. 2003] may be thought of as a discrete version ofCentroidal Voronoi Diagrams [Lloyd 1982]. In this context,balancing the curvature in the charts can be achieved by us-ing a curvature estimator to drive the algorithm. We use thecurvature estimator proposed in [Cohen-Steiner and Morvan2003].

3. Parameterization and Extrapolation (Figure 3-C): the hier-archical finite-element method requires a squareu,v parame-ter space (the used Master Element corresponds to the canon-ical [0,1]× [0,1] square parametric domain). It is possible toconstruct the corresponding parameterization by constraining

the chart boundaries on a square in parameter space, and ap-plying the method proposed in [Sander et al. 2001]. However,in our case, it is better to use anatural boundariesparame-terization, such as [Sheffer and de Sturler 2001],[Levy et al.2002], [Desbrun et al. 2002] and use Dual Domain Extrapo-lation [Levy 2003] to obtain a square border. Applying thismethod to the segmentation obtained at step 2 creates a nearlyisometric parameterization, suitable for the subsequent nu-merical computations (as explained in the previous section).At this step, the charts having a Jacobian deviation greaterthan a certain threshold (1 percent) are subdivided and recur-sively re-parameterized.

Once the model is converted, it can be considered as a set ofparametric surface on which thevirtual meshmethod proposed in[Alonso et al. 2001] can be directly applied. In other words, the al-gorithm proposed here extends the feasability of the virtual meshmethod beyond tesselated model with several hundred thousandfacets. Using our PME representation, it is now possible to group alarge number of triangles into a single master element that keeps aone-to-one mapping with the geometric information. To operate ona PME, thevirtual meshalgorithm only requires the two followingqueries, implemented by classic geometric search data structures:

� 3D space query:find the parameter-spaceu,v coordinates asso-ciated with a given pointP in 3D. A BSP is used to efficientlyfind the 3D triangle that contains the pointP;

� 2D space query:find the 3D pointP corresponding to a givenu,v location in parameter space. A regular 2D grid is used to findthe 2D triangle that containsu,v in parameter-space.

The wavelets stored in the parameter spaces of the PMEs maybe thought of as Mip-Maps. In this context, the extrapolation pro-cess is similar to the push-pull operations done when constructinga texture atlas (see e.g. [Sander et al. 2001]). As in Mip-Mappingapproaches, and as explained in [Alonso et al. 2001], the extrapola-tion makes it possible to meaningfully compute integrals at variousscales, even when the integration domain crosses a chart boundary.



FEM PME]∆ time time

simplified bunny 1.5K 18 min. 1 min. 13 s.bunny 70K 23 h. 4 min. 28 min.

Table 1:Compared timings for classic FEM radiosity and PME(Origin 3000, 20 processors).

]∆ timesimplified bunny 1.5K 1 min. 13 s.bunny 70K 28 min.David head 85K 54 min.David 500K 2 h. 46 min.Museum 1100K 8 h. 12 min.

Table 2:Timings of PME global illumination applied to variousdata sets (Origin 3000, 20 processors).

3 Results

This section presents results obtained by applying our method tofinite-element global illumination and pre-computed illumination.

Wavelet radiosity

In Table 1, we compare the timings obtained with classic FEMwavelet radiosity (one finite element per facet) and our PMEmethod. The test scene comprises a mesh located in a Cornell box(see the 1st page), which generates many interactions. As can beseen, grouping many facets in a single element results in dramaticperformance increases.

Table 2 shows the timings obtained with various classic data sets(put in a cornell box). To test how the method scales up, we havecreated a “Museum” data set (shown in Figure 4). It comprises thefollowing meshes: Feline (100K), Isis (380K), kiss (48K), David(500K), Bunny (70K), room (10K) and 14 spot lights. This makesa total of 1.1 million facets with complex interactions (see Figures4 and 5). The solution was computed in 8 hours, on an Origin3000 with 20 processors. Interactive walkthrough is done with22.4 frames per seconds (average), on a 3 GHz Pentium IV witha GeForce FX 5900, using textures, visibility culling and LODs.

Figure 4:Our “Museum” data set contains a total of 1.1 millionfacets. This view-independent solution was computed in 8 hours onan Origin 3000. Interactive walkthrough is achieved with 22.4 fpson a consumer’s PC.

Precomputed lighting transfer

In [Sloan et al. 2002], a new method is proposed to display in realtime dynamic radiance transfers. This method precomputes the ra-diance incoming from an infinite sphere in many pointsO of themodels. Then the radiance in a point of the environment can beinterpolated from the pre-computed data using the nearest points.

More precisely, SH (Spherical Harmonics) are used to code theincoming light on the sphere and to store the radiance going out ev-ery pre-computed pointO. This yields for each point a matrix (ofdimension up to 25) that codes for each incoming function the out-going radiance for each direction. To display the model, the influ-ence of the environment on the ingoing SH functions is computed.For each point, the nearest points ofO are found and the associatedSH functions are combined. For diffuse surfaces, the transfer ma-trix can be reduced to a vector, that codes for each input lightingfunction the outgoing radiosity.

Recently, using a local basis, [Lehtinen and Kautz 2003] pro-poses a method to reduce the size of the matrices and improvethe speed of GPU-assisted display. Another method proposed by[Sloan et al. 2003a] consists in clustering the triangles, so that oneach cluster, each matrix ofO can be replaced by an linear inter-polation of a limited number of matrices. Knowing a parametrisa-tion on a object, this latter approach can be seen as finding a goodcompresion of the given matrices (functions) in the parametrisationspace. Wavelets are known to give such a good scheme. Based onour PME representation, a new algorithm for PRT can be designedas follows :

1. compute PRT of the models in the parameter-space of thePME;

2. convert the precomputated matrices in the local basis;

3. for each PME, compress the data using multi-wavelets (e.g.M2 multi-wavelets or M3 multi-wavelets);

4. prune the wavelets trees in order to keep only the part thatcorresponds to the real meshes;

5. display the result as done in [Sloan et al. 2003a] (if M2 multi-wavelets are used) or modify the algorithm to do quadratictransformations (if M3 multi-wavelets are used).

The interests of this approach is first that we can use the waveletstheory in order to control the approximation errors, and second,wavelets being hierarchical structures, this can be easily used todefine LODs.

In order to test this approach, we have used a radiosity algorithmto precompute radiosity coming from 25 spherical harmonics, thecompression is done using M2 multi-wavelets (see Figure 6).

Figure 6:PME-Based Precomputed Lighting Transfer; “Bunny”shown under three different lighting conditions.



Figure 5:Closeups in the “Museum” data set. Note the global lighting effects and the precision of the shadows.

4 Conclusions

Parametric Master Element is a natural extension of the MasterElement formalism to tesselated models. By decoupling the lightrepresentation from the geometric support, hierarchical FEM meth-ods fully show their strength. A complete view-independent solu-tion can be computed for scenes comprising millions of primitives.FEM theory can be fully reused to control the error.

To visualize the solution, since lighting is represented in param-eter space, generating light maps from a PME is trivial. Moreover,to speed-up the display, the decoupled geometric - lighting repre-sentations facilitates using LODs. These techniques combined withvisibility culling make it possible to implement interactive walk-throughs on consumers PCs.

A natural idea would be to use the LOD not only for the visual-ization, but also to accelerate the lighting computations. However,since the bottleneck of the simulation process comes from the visi-bility queries, since the algorithm already uses a BSP to acceleratethem, and since the lighting is already represented by hierarchicalfunctions, this approach does not improve the efficiency, which wasconfirmed by the experiments we did.

In future works, we will experiment PME-based radiance simu-lations and precomputed radiance transfer.

Aknowledgements

The aknowledgements will be given in the final version.

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