Eurographics Symposium on Point-Based Graphics (2004)M. Alexa, S. Rusinkiewicz, (Editors)

Phong Splatting

Mario Botsch Michael Spernat Leif Kobbelt

Computer Graphics GroupRWTH Aachen University

Abstract

Surface splatting has developed into a valuable alternative to triangle meshes when it comes to rendering of highlydetailed massive datasets. However, even highly accurate splat approximations of the given geometry may some-times not provide a sufficient rendering quality since surface lighting mostly depends on normal vectors whosedeviation is not bounded by the Hausdorff approximation error. Moreover, current point-based rendering systemsusually associate a constant normal vector with each splat, leading to rendering results which are comparable toflat or Gouraud shading for polygon meshes.In contrast, we propose to base the lighting of a splat on a linearly varying normal field associated with it, andwe show that the resulting Phong Splats provide a visual quality which is far superior to existing approaches. Wepresent a simple and effective way to construct a Phong splat representation for a given set of input samples.Our surface splatting system is implemented completely based on vertex and pixel shaders of current GPUs andachieves a splat rate of up to 4M Phong shaded, filtered, and blended splats per second. In contrast to previouswork, our scan conversion is projectively correct per pixel, leading to more accurate visualization and clipping atsharp features.

1. Introduction

Surface splatting is a well-established technique to renderhigh quality images of geometric objects that are given bya sufficiently dense set of sample points. The idea is to ap-proximate local regions of the surface by planar ellipses inobject space and then render the surface by accumulatingand blending these ellipses in image space. From the geo-metric point of view, the set of ellipses defines a piecewiselinear approximation of the given geometry, and the size andaspect ratio of the ellipses depend on the local principal cur-vatures (and the approximation tolerance prescribed by theuser).

Usually, each splat is associated with a normal vector inorder to compute local lighting, which leads to a piecewiseconstant shading — similar to flat shading for polygonalmeshes. Blending the splats’ color contributions by Gaus-sian filtering improves the rendering from flat shading toGouraud shading. For polygons, much better visual qualityis achieved by Phong shading, where normal vectors are lin-early interpolated across triangles, leading to a continuouslyvarying lighting [Bui75]. However, normal interpolation re-

quires connectivity information which is usually not avail-able for surface splats.

The basic idea of Phong splatting is to associate a linearlyvarying normal field with each splat instead of keeping thenormal constant. By this, we achieve the same visual qualityas Phong shaded polygons, but we preserve all the importantadvantages of point-based surface representations, e.g., wedo not have to construct a globally consistent connectivity.

The first to propose the use of a varying normal field forsurface splatting were [KV01], generating normal mappedsplats by sampling position and curvature information ofNURBS surfaces. In this paper we present an algorithm forthe generation of Phong splats that provides a simpler formu-lation as well as a more robust and accurate approximationof the geometry as well as of its normal field.

In order to generate Phong splat representations we as-sume that the input data consists of a set of sample pointswith associated normal vectors. Hence, each point can beconsidered as a 5-dimensional sample. The first three entriesare the spatial point coordinates and the last two entries rep-resent the (normalized) normal vector.

c© The Eurographics Association 2004.

Botsch, Spernat, Kobbelt / Phong Splatting

To approximate a certain surface region by an ellipticsplat, we fit a least squares plane to the corresponding setof sample points that belong to this region. Then we projectthe samples to this plane and fit a minimum enclosing el-lipse. To derive the linear normal field, we do exactly thesame, i.e. we fit a linear function to the normal vectors in theleast squares sense.

The rendering of the resulting Phong splat representa-tions can be implemented very efficiently by exploiting thefeatures of modern programmable graphic processors. Oursystem delegates all rendering tasks to the vertex and pixelshader units of the GPU and achieves a rendering speed of upto 4M high-quality filtered Phong splats per second. Com-pared to standard point-based rendering algorithms, Phongsplatting needs significantly fewer splat primitives to pro-duce the same visual quality. Taking this into account, ourmethod is also more efficient than previous high-qualitysplatting approaches.

Besides higher performance, another advantage of map-ping the complete rendering process to the GPU is that theCPU is free to handle the actual processing of the geometrydata. Hence, the geometry processing algorithms also bene-fit from Phong splats, as less primitives have to be stored andprocessed.

2. Related Work

Points have first been proposed as rendering primitives byLevoy and Whitted [LW85], followed by point-based render-ing approaches based on image-space reconstruction tech-niques [GD98, PZvBG00, SD01] or object-space resam-pling [ABCO∗01, FCOAS03].

In contrast to this, surface splatting [ZPvBG01] avoidsholes in the rendered image by associating a normal vec-tor and a radius with each point, thereby consideringthem as small discs or ellipses in objects space. Build-ing on this work, several point-based rendering approacheshave been developed, the early ones being implementedin software and therefore putting a high load on the CPU[ZPvBG01, BWK02].

The increasing efficiency and programmability of moderngraphic cards triggered the development of hardware-basedsplatting methods, starting with [RL00]. The approach of[RPZ02] renders splats using object space quads, causing amultiplication of the number of vertices to be processed bya factor of four. More efficient approaches manage to ren-der just one vertex per splat and use vertex and pixel shadersfor the splat rasterization. But also these methods representa trade-off between high visual quality and efficient render-ing, differing in the rendering primitives and filtering meth-ods they use. Splat shapes range from simple image spacesquares [DVS03] over circular splats [CH02, BK03] to el-liptical splats that nicely adapt to the local surface curvature[ZRB∗04].

High quality anisotropic anti-aliasing can be achievedby assigning a Gaussian filter kernel to splats. Combin-ing these object-space reconstruction kernels with a band-limiting image-space filter results in the high quality EWAsplatting technique [ZPvBG01, RPZ02, ZRB∗04].

However, all methods mentioned above use a constantnormal vector for lighting splats, leading to results similar toflat shading in the polygon rendering case. Blurring the shad-ing discontinuities by Gaussian filtering achieves render-ings comparable to Gouraud shading. Better results can beachieved by a per-pixel EWA averaging of normals in com-bination with deferred shading, as proposed by [ZPvBG01].However, non-constant normal fields yield an even highervisual quality, as first proposed by Kalaiah and Varshney[KV01, KV03]. Since their approach is the one most simi-lar to ours, we will compare to it in more detail in Sec. 5. Acomparison to [ZPvBG01] is given in Sec. 6.

Being piecewise linear primitives, surface splats alsogained increasing attention as an alternative geometry repre-sentation, since they exhibit the same approximation poweras triangle meshes. As a consequence, several geometry pro-cessing algorithms have been developed for point-based ge-ometries [PG01, ZPKG02, PKKG03, PKG03]. The runningtime of these algorithms is mostly dominated by the num-ber of splat primitives. To enable efficient processing andrendering of these datasets, simplification and resamplingtechniques for point-based geometry have been developed[PGK02, Paj03, WK04]. We build upon this kind of ap-proaches by adding a linearly varying normal field to thesplat primitives.

We simplify and improve the splat generation phase of[KV03] in Sec. 3 and propose a Phong splat rendering sys-tem that combines per-pixel lighting and high-quality fil-tering and correctly computes the projection per pixel inSec. 4. Finally Sec. 6 shows that the resulting Phong splat-ting framework provides a superior ratio of quality to speedcompared to existing splatting approaches.

3. Phong Splat Generation

Given a dense set of (input) surface samples pi we use a tech-nique like the one described in [WK04] to generate an op-timized set of elliptical surface splats that approximates theinput data within a prescribed error tolerance ε. The idea ofthis technique is to locally estimate the principal curvaturedirections at every input sample and to align the ellipses’axes accordingly. An initial selection of splats that cover thecomplete set of samples is then optimized by a global relax-ation procedure.

The output of this procedure is a set of elliptical surfacesplats S j that provide an approximation of the input pointcloud. A splat is defined by its center c j and two orthogo-nal principal tangent directions u j and v j . The tangents arescaled according to the corresponding ellipse radii such that

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Botsch, Spernat, Kobbelt / Phong Splatting

(0,0)

(0,0,1) (x,y,1)

n

Figure 1: Since lengths are not important for the normalfitting procedure, normals are represented as homogeneouspoints on an offset tangent plane.

an arbitrary point q in the plane spanned by u j and v j lies inthe interior of the splat S j if its local parameter values u andv satisfy the condition

u2 + v2 =(

uTj (q− c j)

)2+

(

vTj (q− c j)

)2≤ 1. (1)

The inside test can be applied even to points that do not lie inthe supporting plane of the splat. In this case, the two param-eter values u and v correspond to the orthogonal projectionof q into that plane.

Each splat S j is associated with a sub-set Pj of the samplespi which are covered by it, i.e., which are contained in theelliptical cylinder generated by offsetting the splat in normaldirection by ε and by −ε respectively. Notice that usually weexpect Pj ∩Pk 6= ∅ for neighboring splats due to the mutualoverlap.

The basic idea of Phong shading is to assign explicit nor-mal vectors to the vertices of a polygon mesh and then to in-terpolate these normals in a piecewise linear fashion. Phongsplatting does the same, except that we cannot interpolatenormal vectors of neighboring splats, since in general noconnectivity information is available. Hence, for each splatwe derive a linear normal field from the associated set ofsamples Pj .

For the geometric fitting the splat’s center and tangentialdirections are usually derived by fitting a plane to the samplepoints pi ∈ Pj in the least squares sense. We do exactly thesame in order to fit a linear normal field to the given normalsni associated with the sample points pi. This normal field N jis specified by a center normal n j and two scalar values α jand β j , such that the (unnormalized) normal of a point q onthe splat S j with parameter values (u,v) is

N j(u,v) = n j +u α j u j + v β j v j, (2)

i.e. we tilt the center normal along the tangential directions.

For the normal fitting we represent a given normal vectorni w.r.t. the local frame spanned by the splat’s tangent direc-tions (u j , v j) and its normal (u j × v j). Note that the centernormal n j generally differs from u j × v j in order to mini-mize the normal error over the whole splat.

For later lighting computations the length of the inter-polated normal vector is not relevant, since it can be re-normalized or is used for accessing a cube-map (Sec. 4). Asa consequence, we can set the third local-frame coordinateof ni to 1 such that each normal vector is actually repre-sented by a point (x,y) on an offset tangent plane, similar tohomogeneous coordinates (cf. Fig. 1).

If the center normal n is represented by (x, y), and if(ui,vi) denote the parameter values of the sample pi and(xi,yi) its local-frame normal vector, the normal fitting canbe written as a set of linear equations

(

xy

)

+

(

ui αvi β

)

!=

(

xiyi

)

∀ pi ∈ Pj,

which are solved for (x, y), α and β in the least squares sense,e.g. using the normal equations (AT Ax = AT b). Since in theabove equation the x and y components are uncoupled, it canbe further simplified to the solution of two 2× 2 linear sys-tems:

(∣

∣Pj∣

∣ ∑ui

∑ui ∑u2i

)(

xα

)

=

(

∑xi∑xi ui

)

(3)

(∣

∣Pj∣

∣ ∑vi

∑vi ∑v2i

)(

yβ

)

=

(

∑yi∑yi vi

)

, (4)

where the summation is done over all pi ∈ Pj . If these sys-tems happen to be underdetermined, e.g. if Pj contains lessthan 2 samples, we compute the least norm solution usingthe pseudo-inverse [GL89].

Note that the overall method works best if the tangentialdirections are roughly aligned to the directions of minimaland maximum normal deviation of the sample normals ni,i.e. to the principal curvature directions. If this is not alreadyprovided by the geometry fitting scheme, like e.g. [WK04],the directions u j and v j can easily be estimated by the eigen-vectors corresponding to the two smaller eigenvalues of thecovariance matrix of sample normals ∑i ninT

i (see [Gar99]for details).

The result of this normal fitting process is a center normaln j and two scalars α j and β j associated with each splat S j ,representing an optimal fit to the given input normals.

4. Phong Splat Rendering

In this section we show how to render the Phong splat repre-sentation based on the features of modern GPUs. The Phongsplatting system we propose is targeting at superior visualquality, therefore we combine per-pixel lighting using linearnormal fields with elliptical splat shapes and antialiasing byGaussian filtering and blending.

For efficiency reasons we want to render each splat bysending only one vertex through the graphics pipeline. Likein previous approaches, we determine the projected size ofthe splat and adjust the OpenGL point size, such that a suffi-ciently large square will be generated and rasterized. In order

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Botsch, Spernat, Kobbelt / Phong Splatting

t c

b

−n

q

−f −z

q

n

n

Figure 2: We compute the point q corresponding to a givenwindow pixel by casting a ray through the respective pointqn on the near plane and intersecting it with the splat plane.

to avoid the complicated computation of the exact projectedsize [ZRB∗04], we conservatively estimate it by perspec-tively foreshortening the larger of the ellipse radii r usingthe depth value zeye of the splat center c like in [BK03]

size = 2r ·n

zeye·

hvp

t −b, (5)

where n, t and b are the parameters of the viewing frustum(cf. Fig. 2) and hvp is the height of the viewport.

This will rasterize a size× size image-space square suchthat we have to determine the local splat parameter valuesfor each pixel generated by it. Based on these coordinateswe decide whether the pixel is within the splat or shouldbe discarded, leading to the correct elliptical splat shape. Incontrast to other approaches we base this on the computa-tion of the exact 3D point corresponding to the current pixelposition by inverting the viewing and projection transforma-tions. The first step is to compute the point qn on the nearplane that is being projected to the current window pixel po-sition (x,y), which is basically an inversion of the viewporttransformation:

qn =

x · wnwvp

− wn2

y · hnhvp

− hn2

−n

,

where w{n,vp} and h{n,vp} denote the width and height of thenear plane and viewport, respectively. Casting a ray from theorigin (the eye) through this point and intersecting it with thesplat’s supporting plane yields the corresponding point q onthe splat (cf. Fig. 2):

q = qn ·cT nqT

n n, (6)

where c and n denote the splat’s center and normal vectorin eye coordinates. After computing q, the parameter values(u,v) can easily be determined by Eq. 1. If the pixel is ac-cepted, i.e. u2 + v2 ≤ 1, we compute its normal vector usingEq. 2 and derive a color by lighting it.

If technical datasets are to be rendered by surface splat-ting, sharp edges or corners can be represented by clippedsplats, as first proposed by [PKKG03]. Based on the object-space parameter values (u,v), this method can easily be in-tegrated into our framework. We represent a clip line in thesplat’s tangent space by three scalars (a,b,c), such that thepoint q is to be clipped if au + bv + c < 0, i.e. at the costof one dot product. Notice that this is much easier comparedto [ZRB∗04], since their approach requires the projection oftwo points representing the clip line to image space in orderto do the half-space test there.

What remains to be done is correcting the pixel’s depthvalue, that would otherwise equal the depth value of the cen-ter point all over the splat, causing blending artifacts as wasfirst pointed out by [BK03]. Since we know the exact depthin eye-coordinates qz, we can derive the fragment’s depthvalue zvp by

zvp =1qz

·f n

f −n+

ff −n

.

Note that this actually results in per-pixel correct depth val-ues, in contrast to previous approaches using affine approxi-mations to the projective mapping [BK03, ZRB∗04].

4.1. Implementation Details

In order to render an elliptical Phong splat we send one(colored) vertex located at the center position c through theOpenGL rendering pipeline and pass the (scaled) tangentialvectors (u, v) and the normal field (n, α, β) as texture coor-dinates. Hence, in comparison to standard elliptical splattingapproaches we need 17 instead of 12 floating point valuesper splat. However, since we need one order of magnitudefewer splat primitives to achieve the same visual quality, theoverall memory consumption is lower for Phong splats.

A vertex program transforms the splat center to windowcoordinates and the tangents and center normal to eye-space.It also computes the size of the image-space square to be ras-terized (Eq. 5) and optionally (for closed models) a conser-vative backface test based on the transformed normal field.In addition, the vertex program precomputes the scaled nor-mal n · (cT n)−1 for the later ray intersection (Eq. 6).

In the fragment program we then need only 6 instructionsto compute the local parameter values. Based on them, weeither discard the fragment or compute its normal vector.For efficient per-pixel lighting, we use a cube map to storeprecomputed lighting terms as a function of normal vectors.This precomputation derives the lighting components thatare independent from the surface color, based on lightingconditions and achromatic surface reflectance. Colored ortextured models can then be rendered by multiplying theseintensity values by the color of the splat or pixel, respec-tively. In the case of achromatic light sources (r=g=b), wecan even store separate diffuse and specular intensity valuesin one luminance-alpha cube map.

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Botsch, Spernat, Kobbelt / Phong Splatting

Figure 3: A least squares plane (dashed) generally providesa better approximation to a set of sample points than a tan-gent plane (solid).

When rendering models containing clipped splats, we or-der them such that first all non-clipped splats are renderedand then all splats with one or two clipping lines. This min-imizes the overhead caused by additional splat attributes byswitching the respective shader programs.

For high-quality filtering and blending we exactly followthe approach of [BK03]. We assign a radially decreasingweight to splat pixels and sum up their weighted contribu-tions by alpha blending. In order to blend only overlappingpixels with slightly different depths, we have to use two ren-dering passes, the first one doing so-called visibility splat-ting. A final normalization, i.e. a per-pixel division by thesum of weights, is done in a pixel shader by rendering aviewport-sized quad that is textured by the outcome of thetwo rendering passes.

5. Discussion

Using varying normal fields for splat rendering was first pro-posed by [KV01]. They generate point datasets by samplingcontinuous smooth NURBS surfaces or triangle meshes[KV03], using a local Taylor expansion to estimate differ-ential properties at the splat center. The geometry is approx-imated by the tangent plane at the splat center and normalvectors are constructed based on the principal curvatures atthe center point.

In contrast, the input data we consider are just samplepoints with associated normal vectors, i.e. we do not requireexplicit surface geometry or topology information. In thissense, our approach more closely follows the spirit of point-based graphics. In addition, our splat generation phase doesnot even require the sample normals to be consistently ori-ented, as the orientation is canceled out by the homogeneousnormal representation. To get consistently oriented normalfields for the rendering process, only the splats S j (definingthe offset tangent planes) should be consistently oriented,but their number is significantly smaller than the number ofthe input samples.

Comparing the two approximation methods, least squaresplanes can approximate the local geometry better than a tan-gent plane (cf. Fig. 3). In strictly convex/concave configura-

tions, the approximation error of the least squares solutionis roughly half of the error of the tangent plane. In the samesense our least squares linear normal field approximates bet-ter than the normal field of a quadratic surface approxima-tion used in [KV01].

From a signal processing point of view their approach is asubsampling process, while we approximate both geometryand normals by least squares methods, which can be consid-ered as a low-pass filtering process. Hence, our approxima-tion effectively avoids alias artifacts, that would otherwiseresult in shading discontinuities.

The normal approximation of [KV01] can be consideredto be the normal field of a quadratic patch parameterizedover the tangent plane f (u,v) = au2 +bv2 +cuv+du+ev+f . Since at the splat center (0,0) they interpolate positionand normal vector, i.e. f (0,0) = fu(0,0) = fv(0,0) = 0, andsince the axes u and v are aligned to principal curvature di-rections, i.e. fuv(0,0) = 0, the quadratic patch actually hasthe form f (u,v) = au2 + bv2. In contrast, we use a linearfunction n(u,v) = au + bv + c for the normal fitting, whichcorresponds to using the normal field of a general quadraticfunction f (u,v) = au2 + bv2 + du + ev + f . Again, the termcuv vanishes since the tangent coordinate system is alignedto the principal axes. The additional degrees of freedom dand e correspond to the adjustment of the center normal n inorder to improve the normal fitting. The offset f representsthe fact that our least squares plane is not fixed at the splatcenter pi.

More important from a practical point of view is that ourdecoupling of the normal fitting from the geometry fittingleads to the simple solution of two 2 × 2 linear systems(Eq. 3). Our method is also more robust in the presence ofimperfect real-world datasets, since the least squares methodsmoothly distributes the error over the whole splat, whilethe estimated curvature information at the splat center willget unreliable. The slight shading discontinuities caused byinevitable approximation errors of the linear normal field re-quire the use of high-quality filtering and blending — thiswas not necessary for the clean datasets used in [KV01].

An important difference to other GPU based surface splat-ting methods is that our approach is projectively correctsince it does not approximate the projection by an affinemapping. The resulting exact parameter values are impor-tant for splat shape, normal computation, and splat clipping,and they enable a simpler implementation. Using exact depthvalues for each pixel leads to more robust blending andavoids artifacts even for large splats which are almost per-pendicular to the image plane.

On the other hand, not using an affine approximation tothe projective mapping disables us to use EWA splatting,i.e. to combine object-space reconstruction filter and image-space band-limiting filter into one Gaussian. However, usingthe object-space reconstruction filter alone turned out to besufficient in practice, as also mentioned in [BK03].

c© The Eurographics Association 2004.

Botsch, Spernat, Kobbelt / Phong Splatting

Figure 4: The same torso dataset containing 3k splats (left) rendered using flat shading without blending (center left), flatshading + Gaussian blending (center right) and Phong splatting (right).

Figure 5: To achieve comparable visual quality the model complexity has been adjusted for each rendering mode. From left toright: flat shading (170k splats), flat shading + Gaussian blending (33k splats), Phong splatting (3k splats).

6. Results

We compare our Phong splatting system to related work bothin terms of rendering quality and rendering efficiency. Forthe models we present, a Phong splat representation has beengenerated using the algorithm described in Sec. 3.

Fig. 4 shows a torso model of 3k splats using different ren-dering techniques. Using standard surface splatting withoutany filtering clearly shows the flat shading discontinuities.Gaussian blending (using two rendering passes) manages todecrease these artifacts, but also blurs the image noticeably.Phong splatting uses per-pixel lighting and clearly achievesthe highest rendering quality. Note that these three resultsare comparable to flat shading, Gouraud shading (smoothshading but no sharp highlights) and Phong shading in thetriangle mesh rendering case.

The same model is shown in Fig. 5, but the number ofpoints has been adjusted such that each rendering modeachieves about the same visual quality. Although using avery high splat count (170k splats), the flat shading artifacts

can hardly be removed without blending. Even with blend-ing enabled, the sampling has to be quite dense (33k splats)in order to result in high visual quality and sharp high-lights. Using Phong splatting, we need one order of mag-nitude fewer splat primitives (3k splats) in order to achievethe same visual quality. This is confirmed by Fig. 8, thatshows the decrease in visual quality for several simplifica-tions of a scanned statue. It is clearly noticeable that the ren-dering quality is not as tightly related to geometric complex-ity for Phong splatting as for standard splatting. An exampleof Phong shading for textured models is shown in Fig. 6.

In order to avoid shading discontinuities, [ZPvBG01] pro-posed to splat normal vectors in addition to colors and tocompute the final pixel colors by a deferred shading of theseaveraged per-pixel values. However, when blending the nor-mal vectors, the gradient of the normal field depends on thedistance of neighboring splats while in the case of Phongsplats we derive the normal gradient directly from the highlydetailed input data (cf. Fig. 7). Additionally, this method isnot suitable for an implementation on current GPUs, since

c© The Eurographics Association 2004.

Botsch, Spernat, Kobbelt / Phong Splatting

Figure 6: A comparison of standard splatting (top closeup)to Phong splatting (bottom closeup) for the coloredchameleon model consisting of 100k splats.

the accumulated (alpha blended) normals suffer from dis-cretization artifacts.

Finally, the combination of splat clipping and Phongsplatting is shown in Fig. 9, where the well-known fandiskdataset is rendered as point-based representation using thedifferent shading modes.

Besides considerably higher visual quality, our Phongshading system also provides splat rates comparable to re-cent approaches. On a 3GHz Pentium4, GeForceFX 5950,we achieve splat rates between 3.5M and 4M splats/sec fora window size of 512 × 512. Taking into account that weneed significantly fewer splats for the same visual quality,our method can even be considered to be more efficient thanprevious approaches.

7. Conclusion

In this paper we proposed to use a linearly varying normalfield for per-pixel lighting of surface splats. The resultingPhong splats have been shown to result in a much higher ren-dering quality compared to flat shaded splats. Although non-constant normal fields have been used before by [KV01], ourPhong splat generation provides an easier formulation andalso offers a more robust fitting of both the geometry and thenormal field, which is especially important for noisy real-world datasets.

The presented Phong splatting system is implementedcompletely on the GPU, leading to a rendering speed ofup to 4M splats/sec. Additionally, our approach results ina projectively correct rasterization with pixel-exact depthvalues, leading to more precise results. The example im-plementations of the vertex and pixels shader programs in-cluded in the supplementary material will also be availableat http://www.rwth-graphics.de.

As general geometry processing algorithms also benefitfrom the lower splat counts, we believe that Phong splats canevolve to an interesting alternative to standard “flat” splats.

Figure 7: A coarse model (9k splats) rendered by de-ferred per-pixel shading of averaged splat normals (left)and Phong splatting (right). Since the normals of the for-mer method are not based on the detailed input data (pi,ni), highlights are not represented sufficiently. The highestfrequency perturbations in the left image are due to GPUdiscretizations and not related to the approach in general.

Figure 8: The decrease in visual quality related to modelcomplexities of 350k, 110k and 35k splats, shown as close-ups for standard splatting (left) and Phong splatting (right).

c© The Eurographics Association 2004.

Botsch, Spernat, Kobbelt / Phong Splatting

Figure 9: Phong splatting can easily be combined with splat clipping, resulting in sharp features. A splat representation of thefandisk dataset (left) is rendered using flat shading (center left), flat shading + Gaussian blending (center right), and Phongsplatting (right).

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[PKKG03] PAULY M., KEISER R., KOBBELT L., GROSS M.:Shape Modeling with Point-Sampled Geometry. InProc. of Siggraph 03 (2003).

[PZvBG00] PFISTER H., ZWICKER M., VAN BAAR J., GROSS

M.: Surfels: Surface elements as rendering primitives.In Proc. of Siggraph 00 (2000), pp. 335–342.

[RL00] RUSINKIEWICZ S., LEVOY M.: QSplat: a multires-olution point rendering system for large meshes. InProc. of Siggraph 00 (2000), pp. 343–352.

[RPZ02] REN L., PFISTER H., ZWICKER M.: Object spaceewa surface splatting: A hardware accelerated ap-proach to high quality point rendering. In Proc. ofEurographics 02 (2002), pp. 461–470.

[SD01] STAMMINGER M., DRETTAKIS G.: Interactive sam-pling and rendering for complex and procedural geom-etry. In Proc. of Eurographics Workshop on Rendering01 (2001), pp. 151–162.

[WK04] WU J., KOBBELT L.: Optimized subsampling of pointsets for surface splatting. In Proc. of Eurographics 04(2004).

[ZPKG02] ZWICKER M., PAULY M., KNOLL O., GROSS M.:PointShop 3D: An Interactive System for Point-BasedSurface Editing. In Proc. of Siggraph 02 (2002).

[ZPvBG01] ZWICKER M., PFISTER H., VAN BAAR J., GROSS

M.: Surface splatting. In Proc. of Siggraph 01 (2001),pp. 371–378.

[ZRB∗04] ZWICKER M., RÄSÄNEN J., BOTSCH M., DACHS-BACHER C., PAULY M.: Perspective accurate splat-ting. In Proc. of Graphics Interface 04 (2004).

c© The Eurographics Association 2004.