Hierarchical Least Squares Conformal Map

Nicolas RAY Bruno LEVY

Abstract

A texture atlas is an efficient way to represent informa-tion (like colors, normals, displacement maps ...) on trian-gulated surfaces. The LSCM method (Least Squares Con-formal Maps) automatically generates a texture atlas froma meshed model. For large charts (over 100k facets), theconvergence of the numerical solver may be slow. It is wellknown that the conformality criterion, minimized by LSCM,also corresponds to a harmonicity condition, meaning thatbarycentric coordinates are locally preserved through theparameterization. This has two different consequences :

• cascadic multigrid methods (coarse to fine) are welladapted to this criterion, and dramatically speed upthe convergence of the numerical solver ;

• the obtained parameterization naturally minimizes tex-ture swimming when used to texture-map a progressivemesh.

In this paper, we introduce HLSCM (Hierarchical LSCM),a cascadic multigrid version of LSCM. As an example ofpossible applications, the paper shows how normal mapsand simplified models can be automatically generated fromlarge scanned meshes. Using these normal maps, the visualappearance of the model can be preserved even when 90%of the vertices are removed from the initial model.

1 Introduction

Attributes such as colors, normal maps or displacementmaps, can be associated to a triangulated surface usingtextures. Such a texture is a 2D picture wrapped aroundthe object. More formally, the wrapping corresponds toa mapping functionX : R2 → R3, putting a surfacehomeomorphic to a disk in one-to-one correspondence witha 2D subset ofR2. It is natural to defineX to be piecewiselinear on each triangle. Then, the mapping function iscompletely defined by the texture coordinates at eachvertex. Constructing such a function is useful for severalapplications :

• Rendering : Thanks to vertex and pixel shaders, it isnow possible to program the way each individual pixelis rendered, by combining information stored in sev-eral textures. As a consequence, many special effects(bump mapping, light mapping, cell shading . . . ) canuse textures to store information. Constructing a map-ping function is of paramount importance for this typeof applications.

• Remeshing :Remeshing methods based on a parame-terization resample the 2D parameter space ofX , andtransform it back onto the original surface. This canbe used to create hierarchical meshes [7], or to repre-sent an object as a texture [10]. Alliezet al. [1] in-troduce another way to remesh a surface using a con-formal map. Their method allows controlling the meshdensity depending on the surface curvature.

• Surface fitting : A triangulated surface can be con-verted into a parametric surface (such as a Spline)by using a global parameterization. Several methodsbased on this idea have been proposed in the literature[8, 14, 9].

When the surface is not homeomorphic to a disk, it ispossible to partition it into a set of charts, and parameteriz-ing each chart individually, which results in a texture atlas[18, 20, 16]. This representation of an object has manypossible applications. Our goal in this paper is to providea practical method for constructing a texture atlas from alarge mesh, based on mathematical properties of conformalmaps.

Generating a texture atlas consists of three steps : theobject is first segmented into a set of charts, homeomorphicto disks, each chart is parameterized and all the chartsare packed in texture space. The most time-consumingpart of the method is the parameterization step, for whichwe present an efficient solution in this paper. Since ingeneral, a surface cannot be flattened without introducingdistortions, the goal of parameterization algorithms is tominimize those distortions. Most algorithms define anenergy function that estimates these distortions and opti-mizes texture coordinates in order to minimize it. These

A B C D

Figure 1. The decimation algorithm is of paramount importance for the efficiency of the multigridsolver. A: conformal parameterization of a surface with a bump; B: a rough approximation of thissurface; C: propagating the parameterization from the rough approximation to the finer level doesnot give a good initial solution; D: a better approximation, taking the geometry of the surface intoaccount, is a good initial solution for finer levels.

functions are defined to be the sum of local distortions foreach primitive (which can be vertices, edges or facets).The minimization of distortions is performed by numericalsolvers that cannot handle very large datasets.

This paper presents Hierarchical Least Squares Confor-mal Maps (HLSCM), an optimized version of LSCM [16].HLSCM combines a fast solver (conjugate gradient) with ahierarchical approach (cascadic multigrid [6], which makesit possible to efficiently generate texture atlases from largesurfaces. In addition, the so constructed parameterizationshows mathematical properties making it possible to texturemap level of details without introducing texture deviation(texture swimming).

The paper is organized as follows. Section 1 presentsprevious parameterization approaches. The hierarchical al-gorithm and the cascadic multigrid numerical solvers arethen introduced (Section 2 and 3). The paper concludeswith some results and application to the automatic gener-ation of normal-mapped level of details.

2 Previous Work

In most parameterization algorithms, one of the goal isto minimize the deformations. To reach this goal, severalauthors have developedgreedyapproaches [22] [2] [27],where the mesh is cut and parameterized simultaneously.The main limitation of this method is that it only takes intoaccountlocal information.

Other approaches are based on aglobal optimization.Most of them consist in locking the border and minimiz-ing the distortion inside the mesh. The distortion can bedefined to respect either shapes [21] or scales [24]. Basedon Tutte’s barycentric mapping theorem, Floateret al.’smethod [9] ensures that a one-to-one parameterization isgenerated. Another way to estimate distortion consists inminimizing local shape deformations, which means con-structing a conformal (i.e. locally isotropic) map. To con-

struct such a parameterization, Pinkallet al. [21] lock themesh border and minimize the conformal energy inside themesh.

Other methods [26, 16, 5] can directly extrapolate theborder and benefit from these new degrees of freedom tominimize the conformal energy. Schefferet. al[26] opti-mize the angles at the corners of the triangles in parameterspace and guarantees a constant sign of the Jacobian (notriangle flipping). However, this latter approach requiresa time-consuming constrained solver, and may suffer fromnumerical problems when texture coordinates are retrievedfrom the angles. The LSCM method [16] and DCP method[5] minimize the same functional, measuring local angle de-formations (LSCM considers theconformalpoint of view,and DCP theDirichlet energypoint of view).

Minimizers of the Dirichlet energy are known to behar-monicfunctions, i.e. maps preserving local barycentric co-ordinates [19]. When applied to meshed models in theComputer Graphics domains, this remark has two conse-quences:

1. If a vertex is added into a triangle of the surface, theparameterization does not change, and the barycen-tric coordinates of the vertex in parameter space areequal to the barycentric coordinates in the triangle in3D space (see [16] for a proof). This means that if afine mesh represents a surface with mostly low spatialfrequencies, a conformal parameterization of a coarseversion of the surface will be a good approximation.

2. When considering a textured surface where texture co-ordinates correspond to a conformal parameterization,the surface can be simplified without introducingtex-ture swimming. In other words, conformal param-eterizations naturally minimize texture deviation be-tween LODs (level of details), without needing the ex-tra terms introduced by Sanderet. al[24].

The first remark suggests that a hierarchical solver(multigrid) will perform well when minimizing a criterionfor which the minimum is a harmonic map. As such,

LSCM and DCP are good candidates. Since LSCM resultsin a symmetric matrix with good conditioning, fastersolvers (conjugate gradients) can be used to optimizethe conformality. The second remark suggests that thehierarchical structure used by the solver may be also usedfor LOD-based visualization. The remainder of this sectionpresents previous hierarchical parameterization approaches.

Lee et al. [15], Hormannet al. [12], Sanderet al.[23] and Guet al. [10] have presented multiresolutionalgorithms to parameterize triangulated surfaces. Leeet al. coarsest chart representation is a triangle, so theyjust propagate texture coordinates to finer levels. Sanderet al. and Guet al. optimize the map for a particularsignal (information associated to the surface). Theirmultiresolution framework enables the algorithm to finda map that will evenly distribute signal frequencies in thetexture. However, the minimum may not be reached sincetheir criterion is non-linear and since only a small numberof relaxation iterations is performed.

The algorithm proposed by Hormannet al.[12] pa-rameterizes the coarsest mesh of a PM, inserts a smallnumber of vertices (25% of the actual mesh), relaunches theparameterization step and repeats these operations until thefinest level is reached. This strategy enables a non-linearobjective function to be optimized on surfaces of averagesize. Since the PM creation process they use does not takeinto account the geometry of the surface, coarse levelsmay not approximate well the original surface. As shownin figure 1, some geometric details may not be correctlyhandled in coarse levels, which reduces the efficiency ofthe multigrid scheme.

The next section introduces a hierarchical solver forLSCM, based on a cascadic multigrid approach, then, it willbe shown how to construct the hierarchical structure used bythe multigrid algorithm.

3 A hierarchical solver for LSCM

3.1 Least Squares Conformal Maps

The LSCM parameterization algorithm[16] generates adiscrete approximant of a conformal map. A conformal mappreserves the local isotropy (see Figure 2), which can beuseful for many applications, such as texture mapping[16,5] or remeshing[1].

LSCM minimizes theconformal energyEC of the map-pingU , defined by :

EC(S) =∫

S

∣∣∣∣∂U(s)∂x

+ i∂U(s)

∂y

∣∣∣∣2 ds

u

v

(u,v)

conformal

X(u,v)

N

iso−v

iso−u

Figure 2. In a conformal map, the tangents toan iso-u and an iso-v curve are orthogonaland have the same norm

Figure 3. Evolution of LSCM with 100, 200,300 and 2500 iterations. Top : iso-u and iso-v. Bottom : parametric space

whereU : R3 → C ; (x, y, z) 7→ u + i.v is the inverseof the parameterizationX (the parameter space is assimi-lated to a subset of the complex plane).

For a piecewise linear parameterization, the conformalenergyEC(.) expressed in function of the parameter-space(uk, vk) coordinates of the vertices is a positive definitequadratic form. Therefore, the(uk, vk) coordinates can befound thanks to the Conjugate Gradient method.

3.2 LSCM performance analysis

When using the Conjugate Gradient method, even witha good preconditioner, convergence is still slow for largedatasets. This paragraph analyzes the behavior of LSCMfor large models. The conformal energy is defined over eachtriangle, which means only relations between two neigh-bors vertices are explicitly represented. In other words, therelative positions of two distant vertices is mainly a con-sequence of all the triangle chains connecting them. Theimmediate consequence is that the relative positions of twoneighbors will be fast to determine. For larger neighbor-hoods, the energy propagation on the mesh may take a longtime. The conclusion is that the system is able to efficiently

solve high frequencies, but convergence is slower for lowfrequencies. Figure 3 presents the evolution of the mini-mization process in 3D and in parameter space. After a fewiterations, the smoothness of the map shows that high fre-quencies are almost solved. On the other hand, we can seein the parametric space that low frequencies take many iter-ations to converge. The multiresolution idea is to solve lowfrequency using a coarse representation of the minimizationproblem.

3.3 Cascadic Multigrid

Classic multigrid algorithms solve each frequency usingthe most appropriate level of representation of the problem.This idea applied to linear system gives the followingalgorithm :

Algorithm 1 function Multigrid (level,Ah, uh, fh)1: if level is the coarser meshthen2: Solve the problem using classic resolution3: else4: SmoothAhuh = fh (m1 times)5: Compute the residualrh ← fh −Ahuh

6: Restrictfh−1 ← Rrh

7: Multigrid(level − 1,Ah−1, vh−1 = 0, fh−1) (mc

times)8: Interpolatevh ← Pvh−1

9: Correctuhnew ← uh + vh

10: SmoothAhuh = fh (m2 times)11: end if

WhereAhuh = fh is the linear system to solve at res-olution h andR andP are respectively the restriction andextrapolation operators used to propagate results betweenresolutions.

High frequencies are estimated using a few passesof a smoother. At this step, the unsolved part of thesolution is only composed of low frequencies. A coarserrepresentation is then used to correct the solution. The laststeps solves once again high frequencies, without beinglimited by low frequencies problems. Using a recursivealgorithm makes it possible to separate more than twofrequencies. Splitting the spectrum enables the algorithmto solve problems with both high and low frequencies. Themc parameter in the algorithm enables to schedule thesequence of frequency bands resolution.

Classic multigrid algorithm such as V-cycle, W-cycle,or full-multigrid are defined by different values ofm1,m2 andmc. The simplest multigrid algorithm (m1 = 0andmc = 1) is called cascadic multigrid. Bornemannetal. [3] show that it can be very efficient, provided that

Figure 4. Level of details (top row) can be ef-ficiently represented by a progressive mesh(bottom row)

the coarse solution is a good approximation of the finersolution. In this case, cascadic multigrid is fast and simpleto implement. The idea is to solve the coarser problem andto use it as initial solution to solve higher frequencies. Eachfrequency is just visited once.

As mentioned in the introduction, conformal param-eterization are alsoharmonic mappings, which meansthat the conformal energy does not depend on the meshresolution[19, 16]. As a consequence, the LSCM functionalmeets the requirement of cascadic multigrid if the geometrydoes not vary too much between two consecutive levels ofthe hierarchy. This latter point depends on the multiresolu-tion analysis, evoked in the next section.

4 Progressive Meshes

The most common multiresolution representation of sur-faces is a set of level of details (LODs), shown in Figure4. It consists of a set of triangulated surfaces, approxi-mating the same surface with different resolutions. Thisrepresentation is very efficient for rendering. However, fornumerical computations, information propagation betweenlevels is critical. The Progressive Mesh [11] representation(PM), shown in figure 4 facilitates the implementation ofthe multigrid algorithm. In a PM, the surface is representedas a coarse mesh and a list of operations to refine / coarsenit. In the original paper, the used basic operations areedgecollapse andvertex split . Kobbelt [13] proposesto usehcoll (halfedge collapse), which makes it easierto store the list of operations and to propagate informationbetween the different levels.

Figure 5. Estimating the geometric impor-tance.

A Progressive Mesh can be generated from the initialmesh by applying successive decimations [11, 17]. Thealgorithm reduces the number of triangles by iteratively re-moving the least important vertex relative to a well-chosenmetric. This approach is fast and can efficiently construct aseries of coarse meshes approximating the original surface.As in Kobbelt’s approach[13], we use thehcoll opera-tion to coarsen the mesh, to facilitate the representation ofthe operation list and to propagate attributes between levels.

To decimate the original mesh, Hormann [12] applieshcoll randomly and locks the neighborhoods of the re-moved vertices until all the vertices are locked. The mainlimitation of this approach is that the coarse mesh does nottake into account the geometry of the fine mesh, which mayslow down the convergence of solver (see Figure 1). As in[25, 11, 17], our goal is to remove in priority the halfedgesintroducing the smallest geometric information (see Algo-rithm 2). The halfedges are sorted according to the follow-ing two criteria:

• Validity of hcoll : A hcoll may create non-manifolds mesh. For this reason, the topology needsto be checked. For our particular application, we addanother constraint: the angular defect of the vertex toremove has to be smaller than a given threshold (70degrees in the examples shown here). This constraintensures that the decimated mesh will preserve the ge-ometric details that influence the parameterization.

• Halfedge geometric importance : The halfedge tocollapse is determined using an estimate of its geomet-ric importance. In our approach, this function is thesum of three criteria, obtained by combining a volume-based approach [17] with global error management[4]:

– Volume difference[17] between the original meshand the simplified one;

– Area difference[17] (when the halfedge is on theborder);

– Accumulated loss of quality on mergedvertices[4].

Note that to have compatible dimensions, these param-eters are normalized (cube root for volume and squareroot for area).

This algorithm is fast and provides good results. In thenext section, the influence of the quality of the coarse lev-els on the hierarchical parameterization algorithms will bediscussed.

Algorithm 2 Hierarchical triangulated surfaces generation1: List¡hcoll ¿operations list2: while Somehcoll can be performeddo3: Halfedgeh← the less geometrically

important halfedge that can be collapsed4: Add the correspondinghcoll to the operations list5: Add h to operations list6: Collapseh7: end while

Figure 6. Hierarchical LSCM solution shownat coarse, middle and fine resolution. Eachlevel is a good initial solution for subsequentones.

4.1 Hierarchical LSCM

As discussed in the multigrid section, we use a cascadicapproach. As shown in Figure 6, the coarsest mesh is pa-rameterized, the result is propagated to a finer one, and theprocess loops until the finest level is reached. This algo-rithm requires a way to propagate results from one level tothe next one, to fix the number of levels and the number ofconjugate gradient iterations at each level:

• Results are propagated during thevsplit operationsused to refine the mesh. A new vertex is created ateachvsplit , its textures coordinates are estimatedusing a barycentric interpolation in the nearest trian-gle. This method does not guarantee the abscence oftriangle flips in the initial solution. However, LSCMremoves them during the next parameterization step.

Figure 7. A: The ‘hand’ data set (650K faces); B: normal map; C: simplified model (29K faces); D:normal-mapped simplified model (rendered with pixel shaders). Note how the fine-scale geometry isreproduced (A and D closeup).

Venus Horse Scanned

] triangles 37K 72K 1.12MPM construction 10s 20s 285smultigrid 8s 11s 415sHLSCM (total) 18s 31s 704s

LSCM (monogrid) 50s 390s ??

Table 1. Results: single chart (1.2 GHz PIV)

• At each level, the convergence of the solver is fast, andthe time required to create the linear system is negli-gible. Note that monogrid LSCM is very efficient tosolve high and average frequencies, which means it isunnecessary to create too many levels. In our imple-mentation, LSCM is launched each time there are 10times as many vertices as in the previous level.

• The number of iterations needed at each level dependsof the quality of the coarse approximation. In the bestcase, the parameterization of the coarsest mesh just hasto be slightly relaxed during subsequent steps, which isthe case when using the decimation method presentedin the previous section.

Using these settings, the cascadic multigrid strategy ap-plied to LSCM results in a highly efficient parameterizationalgorithm. The next section shows timings and presents apossible application to the automatic generation of normal-mapped simplified models from large meshes.

5 Results

Table 1 shows the timings obtained with mono-chartmodels of various sizes, parameterized by LSCM (mono-

Figure 8. A: the ‘hand’ data set, with iso-parametric curves displayed; B: associatedparameter space.

grid version) and HLSCM. As can be seen, the hierarchicalversion results in dramatic speed improvements, making itpossible to parameterize very large charts. All hierarchi-cal approaches ([12, 24, 23] and HLSCM) are able to fastlygenerate acceptable results if only a few smoothing steps areperformed on finer levels. However, HLSCM minimizes aquadratic harmonicity energy. These two characteristics ofthe objective function make it possible to reach theglobalminimum, even for large meshes.

Our method can be integrated in an automatic atlas gen-eration method (see Figure 8), which can be used to gener-ate a simplified model and a normal map [24] (see Figure7). Table 2 gives the statistics and timings of the methodused in this context.

The hierarchical solver makes it possible to parameter-ize large charts in one go, which facilitates the generationof a texture atlas and a normal map. The models can bedramatically simplified, more than 90 % of the faces can be

Figure 9. A: Parameterized Stanford Bunny (70K faces); B: simplified model (4K faces); C: normal-mapped simplified model; D: normal map, texture and per-pixel specular lighting.

Bunny Horse Hand BuddhaFig 9 Fig 10 Fig 7 Fig 11

] trgl (orig.) 70K 96K 650K 1.97M] trgl (result) 4K 8K 29K 46K] charts 14 58 102 627atlas constr. time 52s 112s 534s 1184s

Table 2. Results: normal-mapped simplifiedmodel generation (1.2 GHz PIV)

removed without introducing any visual artifact. Note thatit is also possible to dynamically change the resolution ofthe PM during the visualization process, as done by Sanderet. al[24].

Figure 10. A: the ‘horse’ data set (97K faces)with a texture atlas; B: normal-mapped sim-plified model (8K faces); C: normal map.

Conclusion

In this paper, we have introduced HLSCM, an efficientparameterization method for large meshes. As a possibleapplication, we have demonstrated how the method can ef-ficiently generate normal maps. The hierarchical numeri-cal strategy (cascadic multigrid) is justified by mathemati-cal foundations (a minimizer of the conformal energy is aharmonic map), also meaning that LSCM can texture-mapprogressive meshes without introducing any texture devia-tion between the levels. The method is simple and easy toimplement, and can use the progressive mesh algorithm forboth solving and display purposes.

To parameterize large-scale models (such as full bodyscans or statues), we will study parallel and out-of-core ver-sions of the solver. Finding numerically stable expressionsof stretch[24] and signal-adapted[23] metrics for which aglobal minimum can efficiently be found is another possi-ble direction of future research.

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