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Computers & Graphics 31 (2007) 730–736
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Technical Section
A classification-based rendering method for point models
Haitao Zhang�, Arie Kaufman
Center for Visual Computing (CVC) and Computer Science Department, Stony Brook University, Stony Brook, NY 11794-4400, USA
Received 3 December 2006; received in revised form 2 May 2007; accepted 14 July 2007
Abstract
We present a classification-based high quality rendering method for large scenes with point-based models. Each model is represented
by a uniformly sized point hierarchy. All the points at the same resolution in the hierarchy share the same splat radius, and the splat
radius ratio between any two neighboring resolutions are the same. We use this data structure to minimize the number of rendering
points with no compromise in the image quality. Compared to the standard OpenGL point primitive, it is quite expensive to rasterize the
perspectively correct splat projection even when using the advanced features of modern GPUs. We propose a classification-based
rendering algorithm to reduce the number of points rendered by the complicated splatting. The potentially visible points are collected and
classified into two types: a pixel point, whose image projection size is within one pixel; and a multi-pixel splat, whose image projection size
is larger than one pixel. Only the multi-pixel splats are rendered by a perspectively correct splatting algorithm. The point classification is
integrated with the hardware accelerated rendering. By adopting the uniformly sized point hierarchy and the classification-based
rendering, we can achieve effective rendering of large scenes.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Point-based graphics; Point-based rendering; Classification-based rendering; Uniformly sized point model
1. Introduction
With the advent of graphics hardware and the 3Dscanning techniques, point has become an importantmodeling and rendering primitive. The main difficulty inpoint-based rendering is how to generate a hole-free visiblesurface on the screen without any coherence information ofthe points. Surface splatting is a commonly used method,which defines a disk-shape splat on the tangent plane of themodel at every point. During rendering, these splats areprojected onto the image plane and overlapping visiblesplats are blended together to generate a smooth hole-freesurface.
It is an expensive operation to compute the perspectivelycorrect projection of a splat even when taking advantage ofthe programmability of modern GPUs. For a complexscene consisting of a large number of point-based models,surface splatting would be relatively slow, especiallycompared to the use of the standard OpenGL point
e front matter r 2007 Elsevier Ltd. All rights reserved.
g.2007.07.004
ing author.
ess: [email protected] (H. Zhang).
primitive with one pixel size. One advantage of the surfacesplatting is that a high quality image can be generateddirectly from the point-based models from any viewpoint,which is not guaranteed by the use of the standardOpenGL point primitive.In this paper, we present a classification-based method
for efficient rendering of large scenes of point-basedmodels. A uniformly sized point hierarchy is used formodeling, which reduces the number of points in therendering. The points in the same resolution shares thesame radius value, and these radius values in a pointhierarchy form a geometric sequence. An effective resolu-tion is chosen for each model for fast rendering withoutany compromise in image quality. A classification-basedrendering method classifies the potentially visible pointsinto two types: a pixel point and a multi-pixel splat. Theimage plane projection size for the pixel point is within onepixel, while the multi-pixel splat projects to more than onepixel. Only those points classified as multi-pixel splats needperspectively correct splatting for high quality rendering.The classification process is incorporated in the hardwareaccelerated rendering for effective rendering.
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From the point of view of modeling, our uniformly sizedpoint hierarchy is a hybrid model hierarchy composed ofpoints and splats. The coarser resolutions are all points,and the finest resolution can be either splats or points,which is determined by the classification-based rendering.
2. Related works
2.1. Point-based Rendering
After Levoy and Whitted [1] proposed the use of point asa modeling and rendering primitive, many point-basedrendering methods have been introduced, especially in therecent years. Grossman and Dally [2] use an image spacesurface reconstruction method for hole-free rendering ofthe points. Surfels [3] and QSplat [4] use multi-resolutiondata structures for the point-based modeling and render-ing. Zwicker et al. [5] use EWA surface splatting for highquality rendering. Botsch et al. [6] design a hierarchicalstructure for efficient storing and rendering of point-basedmodels.
Using the programmability of modern GPUs, hardwareaccelerated methods [7–10] have been introduced forefficient point-based rendering. For the GPU-based ren-dering algorithms, the main difficulty is how to blend onlyvisible splats together and discard the invisible ones. Mostof the GPU-based methods use a three-pass splattingmethod: a visibility splatting pass to get the depth, followedby a splatting pass for rendering those splats that pass thedepth test, and finally a normalization or shading pass. Byshifting the depth away along the view direction in thevisibility splatting, overlapping visible splats are blendedtogether in the splatting pass. In the fragment shaders, theproper splat projection shape and depth are computed,together with the weight for blending. The final image isgenerated by a normalization of the color buffer, or by aper-pixel shading pass, such as Botsch et al.’s [11] method,which splats point attributes into multiple frame buffers forper-pixel Phong shading. Guennebaud et al. [12] add anadditional pass for rendering point indices to limit thenumber of points rendered by splatting. Dachsbacher et al.[13] use a sequential list to store the point tree hierarchy.Each point in the sequential point tree has an error value,and the point is rendered only when its error is within anerror range computed from the view distance. Ourrendering method is also based on the GPU-based three-pass splatting method. We collect the potentially visiblepoints, and classify them according to the projected imagesize for better performance. Furthermore, we minimize thenumber of rendered points by a uniformly sized pointhierarchy.
There are indirect point rendering systems [14–16] whichfit a piecewise quadratic function in a local region forsurface reconstruction. In the rendering, these quadraticfunctions are resampled with image resolution, or con-verted into other rendering primitives such as meshes. Ray
tracing also can be used by finding the intersectionsbetween the rays and the implicit surface [17].
2.2. Hybrid model hierarchy
Hybrid model hierarchy consisting of points and otherprimitives can be used for rendering large models. POP [18] isa model hierarchy that uses polygons as leaf nodes and pointsfor inner nodes. Cohen et al. [19] adaptively replace the nodesby points in a triangle hierarchy construction. Dey andHudson [20] use both triangle hierarchy and the correspond-ing point hierarchy in the rendering. Proper resolution in themodel hierarchy and the corresponding primitive type arechosen according to the viewing or some user defined errortolerance. Our uniformly sized point hierarchy consists ofpure points. In the rendering, these points are rendered eitherby splats or one-pixel size points. Thus, our model can bethought of as a hybrid model with splats and points.
3. Uniformly sized point hierarchy
3.1. Motivation
Given a point-based multi-resolution model, we want toselect one resolution for the best image quality with thelowest rendering time possible. Fig. 1 shows threerenderings at three different resolutions of a Santa modelwhen the splat projection is larger than one pixel for all thepoints. The finest resolution will always generate the bestimage, which should be used to guarantee the renderingquality. Fig. 2 shows the results of the same threeresolutions when the splat projection is within one pixel.From the enlarged view (Fig. 2), we can see that there is nonoticeable difference in the image quality among the threeresolutions. If a limited number of points are blended foreach pixel, such as Guennebaud et al.’s [12] renderingmethod, using the coarsest resolution whose splat projec-tion is still within one pixel may even result in betterfiltering. This is because the attributes of a point in acoarser resolution are the averages of the correspondingpoints in the finest resolution.To sum up, if the splat projection is larger than one pixel
for the finest resolution of the model, this finest resolution(Fig. 2) should be rendered for the best image qualitywithout much increase with the rendering time comparedto any coarser resolution. If the splat projection is withinone pixel, the coarsest resolution whose splat projection isstill within one pixel (Fig. 2) is used to minimize thenumber of rendering primitives and get the best filtering. Inorder to efficiently determine the splat projection size for apoint-based model, we use a uniformly sized pointhierarchy.
3.2. Point hierarchy
For a uniformly sized point hierarchy, all the points atthe same resolution share a same splat radius. Thus, it is
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Fig. 1. The Santa model at three resolutions with splat projection larger than one pixel for every resolution.
Fig. 2. The Santa model at three resolutions same as Fig. 1. (a) splat projection within one pixel and (b)–(d) the enlarged view of (a).
H. Zhang, A. Kaufman / Computers & Graphics 31 (2007) 730–736732
efficient to determine the size of a splat projection on themodel level. Furthermore, the splat radii at differentresolutions form a geometric sequence, with the splatradius at each coarser resolution is K times the value at itsneighboring finer resolution. After computing the splatprojection size for the finest resolution, if it is within onepixel, it is easy to find the coarsest resolution whose splatprojection is still within one pixel.
Given a uniformly sized point-based model M0 and aratio K, a point hierarchy is constructed so that the splatradius ratio between two neighboring resolutions is K.There are many methods for simplifying point-basedmodels directly [21]. Wu and Kobbelt [22] select a subsetof the original points as a hole-free approximation of themodel within a prescribed error tolerance. Progressivesplatting [23] uses two error metrics defined on the splatdistance and normal difference in the clustering. We use aclustering method based on splat growing.
The original uniformly sized model M0 is used as thefinest resolution. Assume the splat radius for M0 is r, forany coarser resolution Mm with splat radius Kmr, it isgenerated by simplifying M0. All the points in M0 areclustered, and each cluster forms a point in Mm, whoseposition is the center of the cluster. The cluster is generatedby splat growing. A cluster C is initialized by randomlychoosing an un-clustered point p0 in M0, and thecorresponding initial splat is centered at p0 with radius r.Nearby un-clustered points are added into the cluster one
by one to grow the splat. An un-clustered point pk can beadded to the cluster if it satisfies:
(i)
Radius requirement: After adding pk, the splat centerck is recomputed as the average of all points in thecluster, and the splat radius rk is the maximum distancebetween ck and the points in the cluster plus r forconservatively covering all the points:rk ¼MAXpi2Cðjckpij þ rÞ. (1)
The new splat radius rk should not exceed Kmr.
(ii) Normal requirement: The normal difference between pkand any point already in the cluster should be within athreshold to prevent clustering points on disconnectedopposite surfaces. This threshold increases with m
value to alleviate this requirement for coarser resolu-tions.
If there are more than one point satisfying the tworequirements, the point with the minimum total distanceto all the points in the cluster is chosen. The splat growthstops when no more qualifying points can be added. Theattributes of the point from the cluster for Mm, such asnormal and color, are the averages of the original points inthe cluster.Compared with Wu and Kobbelt’s splat growing method
[22], our method dynamically changes the splat center in
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Fig. 3. Point distribution in a portion of the Santa model at five resolutions with splat radius ratio 2.
Fig. 4. Uniformly sized point hierarchy of the Dinosaur model with five resolutions. The splat radius ratio is set as 2.
H. Zhang, A. Kaufman / Computers & Graphics 31 (2007) 730–736 733
the growing while theirs fixes it, and our splat growing isbounded by the splat radius and theirs stops when reachingan error tolerance. This is because the two simplificationmethods work for different purposes. Wu and Kobbelt tryto find a good approximate of the original model in termsof surface error, and our method is used for ourclassification-based rendering. We only care that noartifacts would appear in our classification-based render-ing. While our simplified model may not be the bestsimplification in term of surface error, it is sufficient for ourclassification-based rendering. In our classification-basedrendering, discussed in the next section, the simplifiedmodels are only rendered when the splat projection on theimage plane is within one pixel size. This prevents anynoticeable artifacts from the simplified models. Fig. 3shows the point distribution in a portion of the Santamodel at five resolutions with splat radius ratio K ¼ 2.Fig. 4 is the Dinosaur model hierarchy with five resolutionsgenerated by our simplification method.
When a model has a large number of points, or themodel shape is irregular in its three dimensions, it wouldnot be efficient in our rendering when creating the pointhierarchy on the whole model. A finer resolution has to bechosen even if only a few points in the next coarserresolution project to multiple pixel size, which increases thenumber of rendered points dramatically. In order to dealwith this problem, we segment such models into severalclusters using k-means clustering, and create a uniformlysized point hierarchy for each cluster.
3.3. View-dependent resolution selection
Each model M (or each cluster in the large models) isassociated with several attributes: (1) a bounding sphere S,with CS and RS as its center and radius; (2) a splat radius r
for the finest resolution M0; and (3) the splat radius ratio K
between any neighboring resolutions Mm and Mmþ1. In therendering, an effective resolution is selected for M if itpasses a view frustum culling.Assuming the aspect ratio of the field of view is the same
as the image aspect ratio, the maximum possible splatprojection size for M0 on the image plane is defined as
dðM0Þ ¼ �h= tanðfovy=2Þ � r=ðZðCSÞ þ RSÞ, (2)
with h as the image height, fovy as the field of view angle inthe vertical direction, and ZðCSÞ as the depth of CS. IfdðM0Þ41, this finest resolution is selected. Otherwise, thecoarsest possible resolution Mn that still satisfiesKndðM0Þp1 is selected. Points in the selected resolutionare sent to the GPU for rendering.
4. Classification-based rendering
The points in the selected resolutions of the pointhierarchies are rendered by a classification-based methodimplemented totally on the GPU. Our rendering methodshares some similarities with Guennebaud et al.’s [12]deferred splatting method. Their method renders the pointindices for visible point selection, and performs EWAsurface splatting on the selected points. In our method, wecollect and classify the potentially visible points and storetheir attributes in textures using the multiple draw buffersextension (MRT). There are two types of classification:a pixel point and a multi-pixel splat, determined by thesplat projection size, with the former within one pixel, andthe latter larger than one pixel. EWA surface splattingis only applied to the multi-pixel splats, whose attributesare collected in the textures. The pixel points can be effici-ently rendered by texture mapping, because their imagepositions are already computed in the point collection and
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classification step. The visible point attributes are blendedtogether, and per-pixel Phong shading is used for the finalimage.
There are six steps in our rendering pipeline, which aredescribed below:
Step 1: Point collection and classification. Points arerendered using the standard OpenGL point primitive withpointsize 1. By enabling depth test, the potentially visiblepoints are collected, one per pixel. The point classificationis determined by the splat projection size. The pointposition and normal under the eye coordinates, the splatradius, the color, and the classification are stored in two 32-bit floating-point RGBA textures (TEX PR and TEX NC)using the MRT extension. The RGB channels of TEX PR
store the point position, and the A channel stores the splatradius (which is greater than 0) if the classification is multi-pixel splat, �1 for pixel point, 0 for the background. Thepoint normal is stored in the RGB channels of TEX NC , andthe remaining A channel holds the color, which is packedinto a 32-bit float using the Cg function pack_4ubyte. Thispass is also a part of visibility splatting to get the depthfrom the pixel points and the splat centers of the multi-pixel splats.
Step 2: Vertex array creation. TEX PR and TEX NC areconverted into a vertex array for rendering using the vertexbuffer and pixel buffer object extensions (VBO and PBO).The image size is the length of this vertex array, whichconsists of position (4 floats), normal (3 floats) and color(3 ubytes) attributes.
Step 3: Multi-pixel splat visibility splatting. To finish thevisibility splatting, the multi-pixel splats need to berendered by a perspectively correct surface splattingmethod. The vertex array generated in Step 2 is rendered,and those points with a non-positive 4th value of theposition attributes are culled away in the vertex shader torender the multi-pixel splats only. Only the A channel ofTEX PR is rendered, which is set to 0 to erase possibleinvisible pixel points behind the splats.
Step 4: Pixel point rendering. The color and normalattributes of the visible points are to be blended togetherand stored in two textures for deferred shading. These twotextures are initialized with the pixel points rendering inthis pass.
For each pixel point, its attributes are stored in TEX PR
and TEX NC at a position the same as its image projection.Thus, pixel point rendering becomes very simple, similar toa texture mapping of an image size rectangle: TEX PR andTEX NC are binded, the color and normal attributes arerendered if the 4th component of TEX PR is smaller thanzero; otherwise, the corresponding pixels are cleared to 0.
Step 5: Multi-pixel splat rendering. The multi-pixel splatsare rendered with blend enabled to accumulate the colorand normal attributes, using a perspectively correctsplatting similar to Step 3.
Step 6: Shading pass. Per-pixel Phong shading is per-formed using the color and normal values generated in theprevious steps.
Step 3 and 5 can be skipped if no finest resolution isselected in the view-dependent resolution selection, becauseall the points would be classified as pixel points. We use theray-casting method [9], which finds the ray-splat intersec-tions to get the perspectively correct splat shape and depth.For each fragment, the ray goes through the center of thecorresponding pixel. A perspectively correct splat projec-tion on the image plane is a near ellipse shape. When thesplat is nearly parallel to the view direction, the short axisof this near-ellipse will become too short to cover any pixelcenter, resulting in holes in this ray-casting method. Inorder to avoid the possible holes, we extend this method toguarantee that all the pixels that the long axis of the near-ellipse goes through will be rendered. The correspondinglong axis direction Alo on the splat plane and the short axisdirection Asi on the image are computed for the splats. Ifthe distance between the ray-splat intersection and the splatcenter long Alo is within the splat radius, and the distancealong Asi is smaller than
ffiffiffi
2p
=2, this fragment should berendered regardless of whether the intersection and splatcenter distance exceeds splat radius or not. Since thisperspectively correct splatting is only performed on themulti-pixel splats, there is only a tiny effect on therendering speed from this extra computation.
5. Results
We use a 3.59GHz PC with an NVIDIA Quadro FX4500 graphics card to conduct the experiments for ourclassification-based rendering for uniformly sized pointhierarchies. All the vertex/fragment shaders are written inthe Cg language.We have created two large scenes SA and SB, which
consist of 9604 and 18; 480 uniformly sized point hierar-chies, respectively. The total number of points is835; 670; 451 for SA, and 1; 608; 071; 219 for SB, countingat the finest resolutions. Although we use only four uniquemodels (Dinosaur, Horse, Rabbit and Santa) and repeatthem to create the scenes, in the rendering, the models areregarded as different models and rendered separately fortesting purpose. A uniformly sized point hierarchy with fiveresolutions is created for each of the four test models,and the number of points at every resolution is shown inTable 1. The computational costs for building these 4 pointhierarchies range from 1 to 3min.Figs. 5(a)–(c) show the renderings for SA, and Figs. 5(d)–
(f) for SB. The image sizes are 800� 800. Table 2 is thecorresponding frame rate, the time spent on each of therendering steps, and the number of points sent to the GPUfrom the selected resolutions of the point hierarchies. Therunning time for Steps 2–6 is relatively fixed by the imagesize. The number of classified multi-pixel splats will affectthe time for Steps 3 and 5, with an upper bound also set bythe image size. The running time for Step 1 is approxi-mately linear with the number of points sent to the GPU.Since the time for Steps 2–6 is relatively fixed with a sameimage size, Step 1 takes most of the time in the rendering
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when the number of points is large. Once the image size ischanged, the time for Steps 2–6 will be affected accord-ingly. The image plane splat projection size will also bealtered, resulting in a possible change of the selected modelresolutions, which determine the number of points handledby Step 1. The last column of Table 2 shows thecorresponding frame rate when the image size is reducedto 512� 512.
Table 1
Point count of the models used in the tests
The splat radius relative to the finest resolution is used to represent the
resolution level.
Fig. 5. Renderings of the large scenes with an image size 800� 800: (a)–(c) ren
consisting of 18; 480 models.
6. Conclusions and future work
We have implemented a classification-based high qualityrendering method for large scenes composed of point-basedmodels. A uniformly sized point hierarchy is used for pointminimization in the rendering. In the GPU-based render-ing, the potentially visible points are classified according totheir image plane projection sizes, and effective renderingmethods are applied to the different classifications.For a large scene, the rendering speed is mainly
determined by the number of points sent to the GPU.Therefore, it is essential to minimize the number of pointsusing different culling techniques. In our renderingmethod, view frustum culling and level-of-detail method(the uniformly sized point hierarchy) are used. It is possibleto integrate other culling methods such as occlusion cullingand backface culling to further reduce the number ofpoints.We use 2 as the splat radius ratio for the uniformly sized
point hierarchy, and five resolutions are used for eachmodel. It is important to study how to determine the best
dering of a scene consisting of 9604 models and (d)–(f) rendering of a scene
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Table 2
The performance for large scenes with an image size 800� 800, including the number of points sent to the GPU, the frame rate, and the time for the six
steps in the rendering
The last column shows the corresponding frame rate when the image size is reduced to 512� 512.
H. Zhang, A. Kaufman / Computers & Graphics 31 (2007) 730–736736
splat radius ratio and the number of resolution for a givenmodel to optimize the rendering. Since only one splatradius is used in the finest resolution, a small radius shouldbe used if there are detailed features in some parts of themodel. This is not an efficient modeling when the modelhas both flat regions and regions with high curvatures. Weplan to extend our multi-resolution data structure to dealwith these models. In the finer resolutions, more than onesplat radius should be allowed to preserve the shapefeatures while not increasing the point density in therelatively flat regions.
Acknowledgments
This work has been partially supported by NSF GrantCCR-0306438. The Dinosaur, Horse, Rabbit, and Santamodels are courtesy of Cyberware, Inc.
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