An Anti-Aliasing Technique for Splatting

Transcription

1 An Anti-Aliasing Technique for Splatting J. Ewar Swan II 1,2,3 Klaus Mueller 2 Torsten Möller 1,2 Naeem Shareef 1,2 Roger Crawfis 1,2 Roni Yagel 1,2 1 Avance Computing Center for the Arts an Design 2 Department of Computer an Information Science The Ohio State University * 3 The Naval Research Laboratory Washington, DC ABSTRACT Splatting is a popular irect volume renering algorithm. However, the algorithm oes not correctly rener cases where the volume sampling rate is higher than the image sampling rate (e.g. more than one voxel maps into a pixel). This situation arises with orthographic projections of high-resolution volumes, as well as with perspective projections of volumes of any resolution. The result is potentially severe spatial an temporal aliasing artifacts. Some volume ray-casting algorithms avoi these artifacts by employing reconstruction ernels which vary in with as the rays iverge. Unlie ray-casting algorithms, existing splatting algorithms o not have an equivalent mechanism for avoiing these artifacts. In this paper we propose such a mechanism, which elivers high-quality splatte images an has the potential for a very efficient harware implementation. Keywors an Phrases: volume renering, splatting, irect volume renering, resampling, reconstruction, anti-aliasing, perspective projection. 1. INTRODUCTION In the past several years, irect volume renering has emerge as an important technology in the fiels of computer graphics an scientific visualization, an splatting is one of several popular techniques for irect volume renering. The majority of images prouce through irect volume renering have use orthographic projections, in part because such projections are useful in many of the application areas (such as biomeical an flui flow visualization) which have initially motivate wor in volume renering. Perspective projections offer a viewpoint which more naturally correlates to the way we perceive the physical worl, an perspective projections are essential when it is esirable to fly through the ata flight simulators are one example. A perspective projection of a volume ataset gives another perceptual cue which can be employe when comprehening spatial relationships. Any volume renering algorithm which supports perspective projections has to eal with the problem of non-uniform sampling prouce by iverging viewing rays. If not aresse this can result in potentially severe aliasing artifacts. Although other volume renering approaches have ealt with this problem (e.g. raycasting [15] an shear-warp [5][6]), to ate the problem has not been aresse in the context of splatting. In this paper, we present a moification to the splatting algorithm which prevents the aliasing that arises from this non-uniform sampling. The same type of resampling problems occur with an orthographic projection if the volume resolution is higher than the image resolution (e.g. if many voxels project into each pixel). Our moifie splatting algorithm also avois aliasing in this situation. * {mueller, moeller, shareef, crawfis, yagel}@cis.ohio-state.eu swan@acm.org In the next section we escribe the splatting algorithm an relate previous wor, an then we give some avantages an isavantages of splatting as compare to other volume renering techniques. In Section 4 we escribe our anti-aliasing technique an argue for its correctness. We follow this with implementation etails an example images. In Section 6 we iscuss our finings an inicate areas of future wor. 2. PREVIOUS WORK The splatting technique has been use to irectly rener volumes of various gri structures [11][21] an for both scalar [8][21][22] [23] an vector fiels [4]. The basic algorithm, first escribe by Westover [21], projects each voxel to the screen an composites it into an accumulating image. It solves the hien surface problem by using a painter s algorithm: it visits the voxels in either a bacto-front or front-to-bac orer, with closer voxels overwriting farther voxels. Splatting is an object-orer algorithm: the resulting image is built up voxel-by-voxel. This is in contrast to volume renering by ray-casting, which is an image-orer algorithm that buils up the resulting image pixel by pixel. As each voxel is projecte onto the image plane, the voxel s energy is sprea over the image raster using a reconstruction ernel centere at the voxel s projection point. This reconstruction ernel is calle a splat ; its name comes from the colorful analogy of throwing a snowball against a wall, with the spreaing energy analogous to the splatting snow. Conceptually, the splat is consiere a spherically symmetric 3D reconstruction ernel centere at a voxel. But because the splat is reconstructe into a 2D image raster, it can be implemente as a 2D reconstruction ernel. This 2D ernel, calle a footprint function, contains the integration of the 3D ernel along one imension. Because the 3D ernel is spherically symmetric, it oes not matter along which axis this integration is performe. The integration is usually pre-compute, an the footprint function is represente as a finely sample 2D looup table. The 2D table is centere at the projection point an sample by the pixels which lie within its extent. Each pixel composites the value it alreay contains with the new value from the footprint table. Uner certain conitions (regular volume gri spacing, orthographic view projection, raially symmetric splat ernel) the footprint table can be compute once an use unmoifie for all voxels. Uner ifferent conitions, the footprint function will vary, an consequently must be re-compute for each view (when there is a non-symmetric ernel) an possibly for each voxel (when there is a perspective projection). Recent wor has extene the original splatting algorithm to achieve higher quality as well as faster renering. To improve image quality, in later wor Westover [22][23] first accumulates splats onto a 2D sheet that is aligne with the volume axis most parallel to the, an then composites the sheets in epth orer into the image with a matting operation. Image quality is also affecte by the size, shape, an type of the reconstruction ernel use. Laur an Hanrahan [8] change the size of a splat base upon the cell it represents in an octree representation of the volume. Mao [11] uses spherical an ellipsoial ernels with varying sizes

2 to splat non-rectilinear gris. Mueller an Yagel [14] use an imageorer splatting approach which improves accuracy when using a perspective projection. An while to ate most splatting implementations have use a Gaussian reconstruction ernel, other ernel types can generate higher quality images. Max [12] an Crawfis an Max [4] propose quaratic spline functions, optimize for certain conitions, as splat ernels. To improve renering spee, Westover [22] maps view epenent footprints with a circular or elliptical shape to a generic footprint table which only nees to be compute once. Laur an Hanrahan [8] approximate splats with a triangle mesh an use graphics harware to quicly scan convert the footprint. Crawfis an Max [4] an Yagel et al. [26] also use texture mapping harware to quicly rener splats represente as textures mappe to polygons. Splatting can also be accelerate by preprocessing the volume an culling voxels which will not contribute to the final image. Laur an Hanrahan [8] cull with an octree structure, an Yagel et al. [26] extract an store only the most visually significant voxels. 3. ADVANTAGES AND DISADVANTAGES OF SPLATTING In this section we compare splatting to other renering algorithms. When listing the isavantages of splatting, we istinguish between inherent problems an those that are ue to inaccuracies in current splatting implementations. 3.1 Avantages of Splatting The main avantage of splatting over ray-casting is that splatting is inherently faster. In ray-casting, reconstruction is performe for each sample point along the ray. At each sample point a 3 convolution filter is applie. Even if, on the average, each of the n 3 voxels are sample only once, ray-casting has a complexity of at least 3 n 3. In splatting, on the other han, the convolution is precompute, an every voxel is splatte exactly once. Each splat requires 2 compositing operations. Therefore, one can expect a complexity of at most 2 n 3. This gives splatting an inherent spee avantage. An aitional benefit is that one can affor to employ larger reconstruction ernels an improve the accuracy of splatting, incurring an O ( 2 ) penalty instea of an O ( 3 ) penalty. Because splatting is an object-orer renering algorithm, it has a simple, static parallel ecomposition [9][23], where the volume raster is evenly ivie among the processors. It is more ifficult to istribute the ata with ray-riven approaches, because each ray might nee to access many ifferent parts of the volume raster. Splatting is trivially accelerate by ignoring empty voxels. It can be accelerate further by extracting an storing just those voxels which contribute to the final image [26], which prevents traversing the entire volume raster. This is equivalent to similar acceleration techniques for volume ray-casting, such as spaceleaping [25] or fitte extents [19], which accelerate ray-casting by quicly traversing empty space. Because splatting generates images in a strict front-to-bac or bac-to-front orer, observing the partially create images can give insight into the ata which is not available from image-orer techniques. In particular, with a bac-to-front orering, partial images reveal interior structures, while with a front-to-bac orering it is possible to terminate the renering early [14]. Finally, splatting is the preferre volume renering technique when the esire result is an X-ray projection image instea of the usual composite image [14]. This is because the summation of pre-integrate reconstruction ernels is both faster an more accurate than ray-casting approaches, which require the summation of many iscrete sums. Creating X-ray projection images from volumes is an important step in the reconstruction algorithms employe by tomographic meical imaging evices [14][3], such as CT an PET. 3.2 Inherent Disavantages of Splatting There are some isavantages inherent to the splatting metho. One is that while an ieal volume renerer first performs the process of reconstruction an then the process of integration (or composition) for the entire volume, splatting forces both reconstruction an integration to be performe on a per-splat basis. The result is incorrect where the splats overlap, an the splats must overlap to ensure a smooth image. This problem is particularly noticeable when the traversal orer of the volume raster changes uring an animation [22]. Another isavantage of splatting lies in the orering of the classification, shaing, an reconstruction steps. For efficiency reasons, in splatting both (transfer function-base) classification an shaing are usually applie to the ata prior to reconstruction. This is also commonly one in ray-casting [7]. However, this prouces correct results only if both classification an shaing are linear operators. The result of employing a non-linear classification or illumination moel may cause the appearance of pseuo-features that o not exist in the original ata. For example, if we want to fin the color an opacity at the center point between the two ata values a an b using linear interpolation, we woul compute ( Ca ( ) + Cb ( )) 2 performing classification first, but the correct value woul be C( ( a+ b) 2) performing interpolation first. Clearly, if C is a non-linear operator, these two results will be ifferent. Requiring C to be linear generally means that the shaing moel can only moel iffuse illumination. While methos exist for ray-casting that perform classification an shaing after reconstruction [2][13][17], this is not possible in splatting. 3.3 Implementation-Base Disavantages of Splatting With a ray-casting volume renering algorithm it is easy to terminate the rays early when using a front-to-bac compositing scheme, which can substantially accelerate renering. Although not reporte in the literature, early termination coul potentially be implemente for splatting by employing the ynamic screen mechanism [18] (also use by [6] for shear-warp volume renering). Also, the ray-riven splatting [14] implementation can support early ray termination. While ray-casting of volumes was originally implemente for both orthographic an perspective viewing, splatting was fully implemente only for orthographic viewing. Although ray-casting has to inclue some mechanism to eal with the non-uniform reconstruction that is necessary with iverging viewing rays, it seems splatting nees to aress several more inaccuracies. For the following iscussion, it is useful to aopt the efinitions given in [4], [14], an [26] which view the footprint table as a polygon in worl space centere at the voxel position with the pre-integrate filter ernel function texture-mappe onto it. As is escribe in [14], when mapping the footprint polygon onto the screen an accurate perspective splatting implementation must: (1) align the footprint polygon perpenicularly with the projector (sight ray) that goes through the polygon center; (2) perspectively project it to the screen to get its screen extent, an (3) ensure that the projector (sight ray) for every pixel that falls within this extent traverses the polygon at a perpenicular angle as well. All three conitions are violate in Westover s splatting algorithm [21]. Mueller an Yagel [14] give a voxel-riven splatting approach that taes care of con-

3 z p = 0 volume raster pixels volume raster s integration gri pixels p s +z p integration gri D p D p (a) Eye Space (b) Perspective Space FIGURE 1. Resampling the volume raster onto the integration gri. (a) In space. (b) In perspective space. ition (1) an (2), an a ray-riven approach that fulfills all three conitions. 4. AN ANTI-ALIASING TECHNIQUE FOR SPLATTING In this section we escribe our splatting-base anti-aliasing metho an argue for its correctness. In Section 4.1 we escribe why anti-aliasing is neee for volume renering algorithms. In Section 4.2 we evelop an expression (Equation 2) which, if satisfie by a given volume renering algorithm, inicates that the algorithm will not prouce the sample-rate aliasing artifacts that arise from the resampling phase of the renering process. In Section 4.3 we escribe our anti-aliasing metho, an in Section 4.4 we show that our metho satisfies the equation evelope in Section 4.2, which argues for the correctness of the metho. Section 4.5 escribes the anti-aliasing metho s frequency omain characteristics an iscusses the effects of using a non-ieal reconstruction ernel. Finally, Section 4.6 analyzes the error that results from the way the technique estimates the local sampling rate. 4.1 The Nee for Anti-Aliasing in Volume Renering The process of volume renering is base on the integration (or composition), along an integration gri, of the volume raster. This integration gri is compose of sight projectors (or rays) which pass from the point, through the, an into the volume raster. Before this integration can occur, the volume raster has to be reconstructe an then resample along the integration gri. This is illustrate in Figure 1 for a perspective view of the volume, where the volume raster is shown as a lattice of ots, an the integration gri is shown as a series of rays, cast through pixels, which traverse the volume raster. Figure 1a shows the scene in space, where the is locate at point ( 000,, ) an is looing own the positive z-axis (enote ). The perspective transformation means the integration gri iverges as it traverses the volume. Figure 1b shows the same scene in perspective space, after perspective transformation an perspective ivision. Here the volume raster is istorte accoring to the perspective transformation, an the integration gri lines are parallel. Because of this the is no longer locate at a point, but can be consiere the plane z p = 0. The reconstruction an resampling of the volume raster onto the integration gri has to be one properly to avoi aliasing artifacts. Ieally, aliasing is avoie by (1) sampling above the Nyquist limit, an (2) reconstructing with an ieal filter. The aliasing that results from an insufficient sampling rate (below the Nyquist limit) is calle prealiasing it is cause by energy from the alias spectra spilling over into the primary spectrum. The aliasing that results from a non-ieal reconstruction filter is calle postaliasing it is cause by the non-ieal filter picing up energy from the alias spectra beyon the Nyquist limit (see Figure 5 an [16][10][24]). In practice, it is not possible to implement an ieal reconstruction filter, an so criteria (2) cannot be achieve any realizable filter inevitably results in a traeoff among aliasing, blurring, an other artifacts [10]. However, reconstruction filters previously use for splatting, in particular Gaussian filters, contribute very little postaliasing at the cost of substantial blurring [10]. In current splatting implementations, the great majority of aliasing is prealiasing; it arises from not achieving criteria (1). Therefore, in the rest of this paper, when referring to the term aliasing we generally mean prealiasing, or aliasing that results from not sampling above the Nyquist limit. It may be possible to sample above the Nyquist limit, but if this is not possible then aliasing can also be avoie by low-pass filtering the volume to reuce its frequency content. For an orthographic view this low-pass filtering must be applie to the entire volume, but for a perspective view low-pass filtering may only be require for a portion of the volume. This can most easily be seen in perspective space (Figure 1b). Note that there is a istance along the +z p axis, enote p, where the sampling rate of the volume raster an the sampling rate of the integration gri are the same. Previous to this istance there is less than one voxel per pixel, an beyon this istance there is more than one voxel per pixel. When there is more than one voxel per pixel, the volume raster can contain frequency information which is higher than the integration gri can represent, an so aliasing is possible. In the next section this concept is evelope into an equation. Note that same concept hols in space (Figure 1a). Here there is an equivalent istance along the axis, enote, where the sampling rates of the volume raster an the integration gri are the same (note that in general p, but the two istances are relate by the perspective transformation). Aliasing artifacts can occur beyon, when the istance between ajacent rays is greater than one voxel. Volume ray-casting algorithms generally perform the reconstruction in space. Some avoi aliasing by employing recon-

4 struction ernels which become larger as the rays iverge [15][17]. This provies an amount of low-pass filtering which is proportional to the istance between the rays. Splatting algorithms generally perform the reconstruction in perspective space. Unlie raycasting algorithms, existing splatting algorithms o not have an equivalent mechanism to avoi aliasing. In this paper we propose such a mechanism. 4.2 Necessary Conitions to Avoi Aliasing In this section we give the conitions which are necessary for the volume renering resampling process to avoi introucing samplerate aliasing artifacts into the integration gri samples. Let s be the volume raster gri spacing (Figure 1a), an ρ = 1 s be the volume raster sampling rate. The volume raster contains aliasing when either (1) the sample function is not banlimite, or (2) the function is banlimite at frequency w but the sampling rate ρ is below the Nyquist limit: ρ < 2w. If the first conition hols then aliasing will be present no matter how large ρ becomes. However, if the function is banlimite at w, then as long as ρ 2w (1) there is no aliasing in the volume raster. Assuming that Equation 1 is true, our job is to resample the volume raster onto the integration gri in a manner that guarantees that no aliasing is introuce. Let φ represent the sampling frequency of the integration gri. For a perspective projection the integration gri iverges (Figure 1a) an therefore φ is a function of istance along the z e axis: φ = φ( ), where is the istance. For an orthographic projection we can still express φ( ) as a function, but it will have a constant value. As illustrate in Figure 1, at the sampling rates of the volume raster an the integration gri are the same: φ( ) = ρ. The istance means there are two cases to consier: Case 1: <. This is the case for the portion of the gri in Figure 1a previous to. Here φ( ) > ρ, an if Equation 1 hols then φ( ) > 2w an there is no aliasing. Case 2:. This is the case for the portion of the gri in Figure 1a beyon. Here φ( ) ρ, an so it may be that φ( ) < 2w. If this is the case, then the integration gri will contain an aliase signal once is large enough. This argument shows that, given Equation 1, volume renering algorithms o not have to perform anti-aliasing as long as Case 1 hols (previous to the istance ). However, once Case 2 hols (beyon ), it is necessary to low-pass filter the volume raster to avoi aliasing. Ieally the amount of this filtering is a function of *, an reuces the highest frequency in the volume raster from w to w ( ). To avoi aliasing there must be enough low-pass filtering so that φ( ) 2w ( ). (2) By showing that this equation hols for a particular volume renering technique, we can claim that the technique oes not introuce sample-rate aliasing artifacts when resampling from the volume raster onto the integration gri. * This is because a portion of the volume raster may not require any filtering (Case 1), an for the portion that oes require filtering (Case 2), if there is a perspective projection then the amount of filtering require is itself a function of. +y e ±x e -y e FIGURE 2. The geometry for attenuating the energy of splat 2. r An Anti-Aliasing Metho for Splatting As mentione in Section 4.1 above, volume ray-casting algorithms avoi aliasing by using reconstruction ernels which increase in size as the integration gri rays iverge, which satisfies Equation 2. In this section we give a similar anti-aliasing algorithm for splatting. As shown in Figure 1a, at istance the ratio of the volume raster sampling frequency ρ an the integration gri sampling frequency φ( ) is one-to-one. can be calculate from similar triangles: = s D ---, p where s is the sample spacing of the volume raster, p is the extent of a pixel, an D is the istance from the point to the screen. Figure 3 gives a sie view of splatting as implemente by Westover [21][22][23], as well as our anti-aliasing metho. In Figure 3 the y-axis is rawn vertically, the z-axis is rawn horizontally, the x-axis comes out of an goes into the page, an iagrams are shown in both space ( x e, y e, z e ) an perspective space ( x p, y p, z p ). The top row illustrates stanar splatting. As in Figure 1, D is the istance of the from the point, an is calculate from Equation 3. For this example we are renering a single row of splats, which are equally space along the z e axis (Figure 3a). Each splat is the same size in space. Figure 3b shows the same scene in perspective space. Here D p an p are D an expresse in z p coorinates. As expecte, because of the non-linear perspective transformation, the splat spacing is now non-uniform along the z p axis, an the size of the splats ecreases with increasing istance from the. The bottom row illustrates our anti-aliasing metho. Previous to we raw splats the same size in space (Figure 3c). Beginning at, we scale the splats so they become larger with increasing istance from the. This scaling is proportional to the viewing frustum, an is given in Equation 4 below. Figure 3 shows what happens in perspective space. Previous to p we raw the splats with ecreasing sizes accoring to the perspective transformation. Beginning at p, we raw all splats the same size, so splats with a z p coorinate greater than p are the same size as splats with a z p coorinate equal to p. Figure 2 gives the geometry for scaling splats rawn beyon. If a splat rawn at istance has the raius r 1, then the raius r 2 of a splat rawn at istance > is the projection of r 1 along the viewing frustum. This is calculate by similar triangles: r 2 splat 1 = r splat 2 r 2 (3) (4)

5 +y e Eye Space +y p Perspective Space splats ±x e ±x p D p +z p D p -y e (a) Stanar Splatting -y p (b) +y e +y p ±x e ±x p D p +z p D p -y e (c) Anti-Aliase Splatting -y p () FIGURE 3. A comparison of the stanar splatting metho (top) with our anti-aliase metho (bottom). (a) The stanar splatting metho in space. (b) The stanar splatting metho in perspective space. (c) The anti-aliase splatting metho in space. () The anti-aliase splatting metho in perspective space. Scaling the splats rawn beyon is not enough to provie anti-aliasing, however. In aition to scaling, the energy that these splats contribute to the image nees to be the same as it woul have been if they ha not been scale. As shown in Figure 2, both splat 1 an splat 2 project to the same size area on the. Because they are composite into the in the form of two-imensional footprint filter ernels [22], the amount of energy the splats contribute to the is proportional to their areas. We want the amount of energy per unit area contribute by the splats to be the same. We accomplish this by attenuating the energy of splat 2 accoring to the ratio of the areas of the splats: s q FIGURE 4. Calculating the integration gri sampling frequency. A 1 E 2 = -----E 1, A 2 (5) E 2 = -- 2 E. 1 (8) where A 1, A 2 are the areas of the splats an E 1, E 2 are some energy measure for the splats. Examples of energy measures inclue the volume uner the splat ernel or the alpha channel of the polygon efining the 2D splat footprint. Assume for now that the filter ernel is a circle for both 2 2 splats. Then the areas of the splats are A 1 = πr 1 an A 2 = πr 2. By Equation 4 we can express A 2 in terms of r 1 : Then A 2 which can be simplifie to = π r πr 1 E 2 = π r E 1 (6) (7) Although here we have erive Equation 8 using circular filter ernels, we can also use any other two-imensional shape for the ernel, such as an ellipse, square, rectangle, parallelogram, etc. an erive the same equation. We use Equation 8 to attenuate the energy of all splats rawn beyon. 4.4 Correctness of the Metho We now emonstrate that Equation 2 hols for our anti-aliasing technique. We begin by eriving expressions for the two functions in Equation 2 φ( ) (the integration gri sampling frequency at ) an w ( ) (the maximum volume raster frequency at ). We erive the integration gri sampling frequency φ( ) with a similar-triangles argument. Consier Figure 4, where q is the integration gri spacing at istance from the point. By similar triangles s - = q --, (9)

6 which can be written as 1 -- q 1 = s (10) ieal reconstruction ernel stanar splatting reconstruction ernel Now 1 s = ρ, an 1 q is simply φ( ), the integration gri sampling frequency at. Thus we have (11) The maximum volume raster frequency w ( ) can be erive from the scaling property of the Fourier transform [1]: (12) where inicates a Fourier transform pair. This shows that wiening a function by the factor a in the spatial omain is equivalent to narrowing the function in the frequency omain by the factor 1 a. In our anti-aliasing metho, the wiening for the splat ernels rawn beyon is given by Equation 4: Thus we have φ( ) = ρ --. fat ( ) 1 -- F 1 a ā -ω, a = --. (13) (a) (b) φ( ) φ ( ) --φ ( ) φ( ) alias spectrum 2 2 reconstructe spectrum primary spectrum ieal reconstruction ernel stanar splatting reconstruction ernel φ( ) φ ( ) --φ ( ) φ( ) alias spectrum 2 2 reconstructe spectrum primary spectrum ieal reconstruction ernel anti-aliase splatting reconstruction ernel (14) which shows that as the splat ernels are wiene by, the frequency components of the function they reconstruct are narrowe by. Since all the frequencies of the volume raster are attenuate by, the maximum frequency w is attenuate by the same amount, an we have: (15) Now we are reay to show that our technique satisfies Equation 2. We start with Equation 1: ρ 2w, which implies that the volume raster has sample the function above the Nyquist limit. Multiplying both sies by we have which we can write as: f --t -- F --ω, w ( ) = w --. ρ -- 2 w --, φ( ) 2w ( ). (16) (17) This erivation says that if the volume raster has sample the function above the Nyquist limit, our anti-aliasing technique provies enough low-pass filtering so that aliasing is not introuce when the volume raster is resample onto the integration gri. Note that this erivation only eals with the prealiasing that results from an inaequate sampling rate it oes not aress the aliasing or blurring effects which result from using a non-ieal reconstruction filter. 4.5 Frequency Domain Behavior This section escribes the frequency-omain behavior of the antialiasing metho, an it illustrates the effects of reconstructing with (c) φ( ) φ ( ) --φ ( ) φ( ) alias spectrum 2 2 reconstructe spectrum primary spectrum FIGURE 5. The behavior of the anti-aliase metho in the frequency omain. (a) Previous to ( < ). (b) Beyon ( ); stanar splatting metho. (c) Beyon ( ); anti-aliase splatting metho. a non-ieal reconstruction ernel. Figure 5 compares the frequency-omain behavior of the stanar splatting technique with the new anti-aliasing technique. For clarity, the iagrams are all rawn in one imension; the extension to three imensions is straightforwar. Figure 5a shows what happens in the frequency omain when resampling from the volume raster onto the integration gri previous to the istance (e.g. < ). There is a primary spectrum at the origin of frequency space, an alias spectra replicate at regular intervals of φ( ), the integration gri frequency for the istance. Assuming Equation 1 is true, then φ( ) > 2w an the primary an alias spectra o not overlap, an therefore there is no prealiasing [16][10]. As shown, an ieal reconstruction filter woul have the value 1 between ( 1 2)φ( ) an ( 1 2)φ( ), an 0 elsewhere, which woul yiel the primary spectrum. Of course, an ieal filter cannot be implemente, an a stanar, realizable splatting reconstruction ernel is also shown. This filter blurs the primary spectrum, yieling the reconstructe spectrum as shown. A realizable filter also typically pics up some energy beyon the Nyquist limit of ( 1 2)φ( ), an so it is susceptible to postaliasing as well [16][10]. Figure 5b shows what happens in stanar splatting beyon the istance (e.g. ). Here φ( ) has shrun, which pulls the alias spectra into the primary spectrum, resulting in prealiasing. An

7 pixels α FIGURE 6. Analysis of the integration gri sampling rate error. ieal reconstruction ernel woul shrin as well thus getting larger in the spatial omain which woul pic up some prealiasing but reject the postaliasing. A stanar splatting reconstruction ernel stays the same size, so it pics up both more of the prealiasing an the postaliasing. Figure 5c shows what happens with the anti-aliasing metho at the same location the splatting reconstruction ernel shrins in the frequency omain accoring to. This eliminates almost all of the postaliasing an most of the prealiasing, at the expense of blurring. 4.6 Integration Gri Sampling Rate Error As calculate from Figure 1 an Equation 3, the integration gri sampling rate at position is only exactly correct for splats centere on the axis. In this section we analyze the amount of error incurre for splats that o not lie along this axis. The geometry of the calculation is given in Figure 6. For the two rays shown, let r be the length, calculate from Equation 4, of the raius of the splat. A more accurate way to approximate the integration gri sampling rate is to measure the perpenicular istance between the two rays, given by r' (as shown r' is only perpenicular to the bottom ray; it is not quite perpenicular to the top ray). As calculate from the angle α, the error is given by r' rcosα, where we say approximately equal because the angle β is not exactly a right angle. This means that Equation 4 unerestimates w ( ) by a small amount, an thus voxels which o not lie on the axis receive a greater amount of low-pass filtering than they actually require. For example, at the view cone bounary with α = 30 Equation 4 chooses a ernel that is about = 1.15 times larger than it nees to be. It is important to note, however, that because this error results in more low-pass filtering than is actually require, Equation 2 is still satisfie an so this error oes not result in aliasing. 5. RESULTS volume raster β α r In our implementation of this algorithm we mae use of renering harware to quicly raw the splats, in a manner similar to [4], [14], an [26]. For each splat we raw a polygon in worl space centere at the voxel position. The polygon is rotate so it is perpenicular to the ray passing from the point through the voxel position. The splat ernel is pre-compute an store in a table which is texture mappe onto the polygon by the renering harware. We use the optimal cubic spline splat ernel reporte in [4]. We attenuate the alpha channel of the polygon as a measure of splat energy when evaluating Equation 8, an then composite the semi-transparent splat polygon into the screen buffer. r integration gri rays splat ernel +z voxel e Our renerer is a moifie version of the splat renerer reporte in [26]. For a given volume we extract an store a subset of the voxels. For each voxel we evaluate a transfer function t = F( ρ, ), where an ρ are the graient an ensity of the voxel, respectively; we inclue the voxel in the subset if t excees a user-efine threshol. We store this volume subset as a 2D array of splat rows, where each row contains only the extracte voxels. Each row is implemente as an array of voxels, but the voxels are not necessarily contiguous, an so we must store each voxel s location an normal vector. In general each row may contain a ifferent numbers of voxels. Despite not storing the empty voxels, we can still traverse this ata structure in either a bac-to-front or front-tobac orer. Figure 7 (see color plates) shows a collection of images obtaine from both a stanar splatting algorithm an the antialiase algorithm reporte in this paper. The images in the left column are renere without anti-aliasing, while the images in the right column are renere with anti-aliasing. Figures 7 (a) an (b) show a volume consisting of a single sheet, where alternate squares are colore either re or white to create a checerboar effect. The resulting ataset contains 198K splats which are renere into a image. In Figure 7 (a) a blac line is rawn at the istance ; beyon this line there is more than one voxel per pixel. As expecte, the left-han image shows strong aliasing effects, but these are smoothe out in the right-han image. Figures 7 (c) an () show a volume containing a terrain ataset acquire from a satellite photograph an a corresponing height fiel. The resulting ataset contains 386K splats (this is more than the expecte splats because extra splats are require to fill in the holes forme where ajacent splats iffer in height). Each column of splats is given the color of the corresponing pixel from the satellite photograph. The ataset is renere into a image. In Figure 7 (c) a blac line is rawn at the istance. The left-han image shows strong aliasing in the upper half of the image (containing about 90% of the ata); when animate, these regions show prominent flicering an strobing effects. In the right-han image these regions have been smoothe out; an although this technique oes not aress temporal aliasing effects, when animate these regions are free of flicering an strobing effects. Figures 7 (e) an (f) show a volume containing a microtubule stuy acquire from confocal microscopy. The resulting ataset contains 103K splats which are renere into a image. Unlie the previous images, where the viewpoint is set so that the atasets isappear into the horizon an thus the splats have a great range of sizes (covering several pixels to much less than a single pixel), in this image the entire ataset is visible, an the range of splat sizes is much smaller. However, because the volume has a higher resolution than the image it is still liable to aliasing effects (all of the splats are rawn beyon the istance ). This is shown in the left-han image, which contains jagge artifacts that shimmer when animate. In the right-han image these effects have been smoothe out; when animate this shimmering effect isappears. 6. FUTURE WORK An area of future wor is moifying our implementation to tae better avantage of the renering harware. As reporte in [26], epening on the machine an the number of extracte voxels, our renerer can give real-time performance. However, our current implementation oes not tae full avantage of the renering spees offere by the graphics harware. Since the anti-aliase splats efine by Equation 4 project to the same size on the screen, it woul be more efficient to raw the splats irectly on the screen

8 instea of rawing the scale splats in space. We are currently looing into harware-supporte bitblt operations to provie optimal renering spees. Another area of future wor is an effort in the opposite irection. In this project there is a mismatch between the type of renering algorithm we have written an the available programming tools: the Silicon Graphics renering harware we use is esigne to accelerate scenes using traitional surface graphics primitives such as polygons an spline surfaces; it oes not contain an optimize splat primitive. To this en, new harware architectures which better support operations which are common in volume renering are neee. Some possible avenues of exploration are: Extene bitblt-lie operators that can be sub-pixel centere an subsequently composite. Harware support for point renering using ifferent reconstruction ernels (cubic, Gaussian, etc.) with common footprints (circular, elliptical, etc.). Potentially this offers a far more efficient implementation of splatting than harware texture mapping. A higher resolution alpha channel to allow for the accurate accumulation of very transparent splats. Splat primitives with automatic size scaling base on their z- epth. A splat primitive has properties of both simple points (always lying in the projection plane) an texture maps (non-linear intensities across the primitive). We are currently woring on scanline algorithms to efficiently rener splat primitives, with the goal of future harware implementations. The implications of our technique for renering other types of objects nees to be explore. As shown in Figure 7 (a) (), one can employ our technique for traitional texture mapping. Our metho obviously provies an accurate solution to the texture sampling problem a solution which is much more accurate than either mip-maps or summe area tables [20]. We are exploring this point-base approach to viewing an renering as an alternative to the traitional scanline renering of iscrete objects, an we are exploring its applications for texture mapping, image-base renering, an volume renering. 7. ACKNOWLEDGMENTS We acnowlege Daniel Cohen-Or for the terrain ataset an Noran Instruments for the confocal ataset. We acnowlege The Ohio Visualization Lab at The Ohio Supercomputer Center for computer access. 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9 (a) (b) (c) () (e) (f) FIGURE 7. Renere images. The left column is renere with stanar splatting, the right column is renere with the anti-aliase splatting algorithm. (a) (b) Checerboar pattern. (c) () Terrain ataset from satellite an mapping ata. (e) (f) Microtubule ataset acquire from confocal microscopy.