*"Light Makes Right"*

June 25, 1999

Volume 12, Number 1

Compiled by

All contents are copyright (c) 1998,1999, all rights reserved by the individual authors

Archive locations: text version at
http://www.acm.org/tog/resources/RTNews/text/

HTML at
http://www.acm.org/tog/resources/RTNews/html/

You may also want to check out the Ray Tracing News issue guide and the ray tracing FAQ.

- Introduction
- Ray Tracing Roundup
- Comparison of Hierarchical Grids , by Vlastimil Havran and Filip Sixta
- Great Shadow Hack , summarized by Eric Haines
- Octree Traversal and the Best Efficiency Scheme , by Ben Hutchison, Eric Haines, Hanan Samet, and Erik Jansen
- Quicker Grid Generation via Memory Allocation , by Eric Haines

Actually, real-time ray tracing rendering techniques (RTRTRT) is a hot topic now. Remember SIGGRAPH 98? The Ray Tracing Roundtable was more than just a shmooze-fest last year, with the idea of using ray tracing in a real-time setting discussed as a serious possibility. Well, real-time ray tracing has been possible for years on a PC, but only through demo programs - see http://www.acm.org/tog/resources/RTNews/demos/overview.htm for a page put together by Piero Foscari. That a 4K program can do anything, let alone rapidly ray trace, is pretty great. For an online Java interactive ray tracer in action, see http://www.eyeone.no/raytracing/index.html.

Parker et al. at the University of Utah just presented a paper, "Interactive Ray Tracing", at the 1999 Symposium on Interactive 3D Graphics. 35 million spheres at 15 frames per second on 60 processors. Admittedly, a lot of those spheres were occluded by being in a dense array (hey, even if spread out, there are only so many pixels on the screen), but the point is that a hidden surface algorithm which simply threw the entire database at the screen currently cannot compete. The announced Playstation 2 (which I've read one reviewer say is about 4.5x as fast as an InfiniteReality 2) can do 20 million polygons a second. This is pretty good (no, actually, it's incredible), but a far cry from 525 million spheres a second. With Moore's Law it would take pure Z-buffer based machines almost a decade to catch up to Utah's feat. See their paper at http://www2.cs.utah.edu/~bes/. The one word summary of their paper: optimization. For every hour spent programming algorithms they spent four hours optimizing the code.

It is interesting to see that, as architectures change, the importance of avoiding cache misses and branches and whatnot has become much more important than worrying about algorithm operation counts (i.e. how many adds and multiplies there are). See "Efficiency Issues for Ray Tracing" by Brian Smits, published in the journal of graphics tools. It is also available online at http://www2.cs.utah.edu/~bes/

The Utah paper has one of the best hacks I've seen in years for ray tracing, a great way of faking soft shadows. It's explained in the article in this issue, and is one of the main reasons I felt the need to get out an issue. It was going to be the lead article, but then I received Havran and Sixta's article on hierarchical grid efficiency scheme testing. Their results are surprising, plus they give a nice grid traversal speed-up, so they're now in the lead spot.

To finish up, here's an interesting puzzle, from an MAA question book: given an axis-aligned cube with corners at (0,0,0) and (1,1,1), cut it with the three planes x=y, y=z, and x=z. How many pieces is the cube cut into?

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________

Mark VandeWettering contributed a much better hashing function by Bob Jenkins for the fractal mountain generator in the Standard Procedural Databases code. If you make fractal mountains using the SPD and generate a huge number of polygons you'll notice patterning artifacts. The new hashing function gets rid of these. http://www.acm.org/tog/resources/SPD/overview.html

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Teddy is an amazing modeler which will be presented at SIGGRAPH 99 - easy to learn, use, and it's fun. Play with the Java version at http://www.mtl.t.u-tokyo.ac.jp/~takeo/teddy/teddy.htm. For links to other SIGGRAPH 99 papers, see Fredo Durand's site at http://w3imagis.imag.fr/Membres/Fredo.Durand/Book/sig99.html.

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Xfrog creates extremely impressive plant forms and more, exporting to POV-Ray, Rayshade, DXF, OBJ, RIB, VRML, and Maya formats. Currently on SGIs, they're seeking beta sites for the Windows and Linux versions. http://www.greenworks.de/.

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3DWin, a shareware 3D file converter, is available at http://www.stmuc.com/thbaier/

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The paper

Image-Based BRDF Acquisition Including Human Skin, by Steve Marschner, myself, Eric Lafortune, Ken Torrance, and Don Greenberg,

presented at the 10th Eurographics Workshop on Rendering, is now available online at

http://www.graphics.cornell.edu/~westin/pubs/egwr99-marschner.pdf

It describes how we measured the BRDF of various surfaces, including living human skin, using only a digital camera, an electronic flash, and a Cyberware scanner. The measurements have been verified against measurements on our gonioreflectometer; in the best case, they appear to be accurate to within the limits of the gonioreflectometer.

Stephen H. Westin <westin@graphics.cornell.edu>

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For mathematical definitions and tidbits (along with collections on other topics), try Eric Weisstein's treasure trove: http://www.treasure-troves.com/. The book made from his math collection, "CRC Concise Encyclopedia of Mathematics", is an amazing achievement - no great depth, but an incredible range of material and fun to just page through.

Various geometric formulae can be found at http://freeabel.geom.umn.edu/docs/reference/CRC-formulas/. Linear algebra definitions can be found at http://maths.uwa.edu.au/~keady/M25n/M25nS1/laglossary.html. Finally, for a quick reference to mathematical formulae in general, try Dave's Math Tables at http://www.sisweb.com/math/tables.htm.

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In RTNv10n3 we observed an interesting discussion concerning hierarchical grids. The problem with the algorithms published was that they were developed more or less independently on different platforms and hardware, and they could not be compared against one another. We decided to take up the gauntlet and make some comparisons. There have been about three methods published; let us recall the basic properties of these approaches:

a) Jevans and Wyvill's article (GI'89) - we call this approach the recursive grid here. The principle is simple: construct a uniform grid over the set of objects, assign the objects to all voxels where they belong. Then recursively descend; that is, for each voxel that contains more objects than a given threshold construct a grid again. The construction of grids is terminated when the number of primitives in the voxel is smaller than a threshold or some maximum depth for grids is hit (usually 2 or 3).

b) Cazals, Drettakis, and Puech articles (EG'95 and SCG'97) - we call this approach the hierarchy of uniform grids (HUG) here. The principle is more complex; first, we compute the histogram according to object size. Using the histogram we split the objects into groups, we use three groups here for explanation. The first group contains large objects, the second group middle-size objects, and the last group small-sized objects. Within the middle-size and small-size groups we perform clustering according to the distance between the objects. Thus each group contains several clusters. For large-size group objects we construct an initial global grid. For all clusters having a large enough number of objects we also construct the grid. These grids are inserted into the global grid recursively, so the smaller grids are contained in bigger ones. The author's statement in the paper's introduction that the method is fully automatic is true as far as it goes; however, it does require at least two parameters to be set initially (the number of groups and delta-connectivity).

c) Klimaszewski and Sedeberg (Klimaszewski PhD thesis and IEEE'97 article) - we call this approach the adaptive grid here. The algorithm principally differs from HUG. The first step of algorithm is clusterizing of objects according to some criteria based on the distance between two candidates. A candidate can be the current cluster or an object. The criteria consider candidates' and the resulting cluster's surface areas. Moreover, the resulting cluster must be small enough in comparison with the scene bounding box. In the second phase of the algorithm, bounding volume hierarchy (BVH) is built up over the clusters. The grids are then constructed for the clusters (leaves of BVH). We have found that the number of children in the interior nodes of BVH is small, so it is unsuitable to create grids for these nodes. In the last phase, so called sub-voxel grids are constructed for voxels, where the number of objects is greater than a given threshold.

The hierarchical grids (b) and (c) differ from (a), since voxels in recursive grids cannot overlap, so each nested grid along the ray path is tested exactly once. The HUG approach requires having mail boxes in the traversal algorithm, where the grids are taken as an object. The ray traversal for hierarchical grids is somewhat similar for all the approaches, but it differs between (b) and (c), since the BVH is traversed in (c).

In forming the list of objects for a grid cell, we use the simple scheme of comparing only the object's bounding box to the grid and adding the object to overlapped cells. This can result in a cell having objects in its list which in fact do not overlap the grid cell, thereby causing superfluous ray intersection tests to occur. However, we also implemented pruning (exactly testing the primitives' surfaces against the cells). The results do not improve very much for SPD scenes (by 10% at most).

Ray Traversal:

- If this grid was already tested, results are already known, return.
- Prepare the constants for 3DDDA algorithm for the current grid instance.
- Within current voxel, test all its children for intersection; if there is an intersection and it lies in the current voxel, return.
- Otherwise, proceed to the next voxel in the current grid along ray path.
- If there is not another voxel in the current grid, return - no intersection. Otherwise, go to step (3).

There are several issues concerning the hierarchical grids construction. How many voxels should be in the grid of a given size containing N objects? The same question arises for uniform grids as well. The answer is usually the pragmatic rule often called the (N^1/3) rule: the axis-aligned bounding box is split on each axis into (N/1^3) cells. The construction of adaptive grids also provides another approach called the heterogeneous grid, where the number of voxels is proportional to the number of objects, but the voxel is made to be cubic as much as possible (that is, the number of cells along an axis is proportional to the relative length of the axis). We found out this setting is generally more efficient than the regular one. It can lead to better performance since it assumes the direction of rays is uniform, even if this assumption is usually violated. The second important issue is the parameter settings for grid hierarchy construction. What should be the threshold and the number of objects to construct a grid over its child and maximum hierarchy depth? What should be the distance criteria (delta-connectivity) and number of groups for the HUG approach, or the two parameters for merging in adaptive grids and the threshold for construction of subvoxel grids?

Suitable settings should be found experimentally, even if the authors recommend some values to use. The problem of parameter setting is complex and the best settings can differ from scene to scene.

We implemented the uniform grid and hierarchical grids schemes in the GOLEM rendering system (http://www.cgg.cvut.cz/GOLEM). Due to space restriction and readability we present here only the final timings for the SPD scenes. Several other parameters reflecting the finer points of hierarchical grids behavior obtained from testing can be found in supplementary statistics at http://www.acm.org/tog/resources/SPD/overview.html. The interested reader will find much more information in this statistics document.

The testing was conducted on an Intel Pentium II, 350 MHz, 128 MBytes, running the Linux operating system, compiler version egcs-1.1.2.

T_B[s] .. time for construction of grid hierarchy T_TR[s] .. time for ray tracing a scene of resolution 513x513, depth of recursion 4, according to the SPD description. Scenes Timings --------------------------------------------------------- -------------- balls gears lattice mount rings teapot tetra tree Uniform Grid - D = 1.0 T_B[s] 0.19 0.38 0.38 0.26 0.35 0.3 0.13 0.22 T_TR[s] 244.7 201.0 54.68 28.99 129.8 28.68 5.54 1517.0 Uniform Grid - Customary ( D = 20.0 ) T_B[s] 0.39 1.13 1.27 0.4 0.98 0.65 0.34 0.33 T_TR[s] 38.52 192.3 34.21 25.15 83.7 18.6 3.86 781.3 Recursive grid T_B[s] 0.39 5.06 0.16 1.98 0.39 1.55 0.47 0.28 T_TR[s] 36.73 214.9 82.05 30.28 113.9 22.67 7.23 33.91 HUG T_B[s] 0.4 1.04 0.3 0.16 0.45 0.53 0.24 0.48 T_TR[s] 34.0 242.1 71.62 62.31 116.3 25.61 7.22 33.48 Adaptive grid T_B[s] 2.79 5.88 0.6 2.54 1.82 3.16 1.19 1.46 T_TR[s] 30.51 330.0 129.6 59.05 167.7 43.04 8.71 18.38It is apparent that the time for construction is always less than 10% of the time for ray tracing itself. We devoted special care to the efficiency algorithm for grid traversal. It requires four comparisons per traversal step when the voxel is empty. We studied the Rayshade grid traversal code. The result of our studies is our traversal code is much more efficient than Rayshade's. One trick is to compute in advance the coordinates of the voxel in the grid where to terminate the traversal (i.e. the voxel where the ray leaves the grid), as it saves comparisons during the traversal.

The uniform grid (Uniform Grid Customary) constructed with a high number of voxels can be very efficient for SPD scenes (excluding scenes which are extremely sparse). With custom setting of the grid resolution, it outperforms hierarchical grids for some SPD scenes. The full results show that the rule of thumb of having the number of voxels equal to the number of objects is not very good. We call the ratio between the number of voxels and number of objects the voxel density. Uniform Grid Customary has the density set to D = 20.0 and uses a resolution setting according to Woo's GI'92 article: "Ray Tracing Polygons Using Spatial Subdivision" (i.e. strive to create cubical voxels). It always outperforms density D=1.0, and in many cases outperforms any hierarchy of grids, regardless the construction algorithm. The disadvantage of setting a high voxel density is the total memory used; the memory consumed for a hierarchical grid is typically much smaller, with similar performance.

Recursive grids is the simplest hierarchical grid scheme to implement and in most cases shows reasonable performance in comparison with methods (b) and (c). In our opinion, implementation of HUG and adaptive grids is rather difficult and brings interesting efficiency improvements only for the scene tree and, we suspect, for other very sparse scenes as well. You should consider carefully if the implementations of HUG and adaptive grids pays off for your application.

The three methods for construction of the grid hierarchy are not the only ones which can be used, someone can develop his own method with different performance characteristics. In order to compute ray shooting quickly, the methods try to organize the free space and the objects in the scene in some reasonable, adaptive way according to the local scene properties. We would like to stress that even if the traversal of the grid is a very efficient algorithm, there are unavoidable drawbacks of hierarchical grids.

First, the initialization of the variables for ray traversal in the grid is expensive; that is why, for some scenes, the uniform grid with a high number of voxels can outperform the hierarchical grid. Second, we need to set the grid resolution somehow, and how to compute the resolution setting for real scenes is a difficult and still rather unsolved problem. Moreover, the cells' boundary positioning does not take into account the objects boundaries, which seems to be important. That is, a slightly different resolution along an axis can noticeably change how many cells the objects overlap and the tightness of the fit. To traverse the grid efficiently, the voxels along the axis have to be the same size, so local changes in cell size seem unwise.

To conclude our observations from testing, we consider the approaches of hierarchical grids appealing and promising, but their best/optimal/suboptimal construction algorithms (clustering, resolution setting, etc.) is still relatively unsolved and is a difficult problem needing more research. We agree with the opinion that the best setting of parameters is scene dependent. Generally speaking, performance also depends on the viewer position and global illumination algorithm, since these affect the distribution of rays. It should be also interesting to test efficiency of the hierarchical grids for scenes with huge number of objects, say hundreds of thousands, but we do not have such models available (The SPD models can be made to produce more primitives, but we have found the computation time to grow very slowly with increasing size, perhaps due to the fractal nature of these models. We would like to try different large scale models). It would also be worth researching how grid resolution is dependent upon the object distribution in space, not just upon the number of objects. We have done some research along these lines for BSP trees (see "Rectilinear BSP Trees for Preferred Ray Sets", http://sgi.felk.cvut.cz/~bittner/abstracts.html).

The future work in this area should also include the optimization of hierarchical grids construction that will be based on the known or predicted rays distribution for given global illumination task. It is interesting this topic of the non-uniformity of ray distribution (what is the distribution? can we predict and utilize this distribution knowledge somehow?) has not yet been researched.

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Start with a triangle casting a shadow on a ground plane. One way to get a soft shadow is to shoot a number of rays from a point on the ground plane at the area of a light source. Count up the times the light was hit vs. missed and you have a rough guess of the coverage of the triangle and can attenuate (dim) the light's contribution accordingly. Do this for all pixels and you've got a soft shadow cast on the ground plane. However, the cost is excessive: just to get 16 gray levels you need to cast 16 shadow rays from each test point. Even with this the soft shadows are noisy: at one pixel 14 rays hit the light, the next has 10 hit, the next 13 hit, due to changing sampling patterns (and you do want to change the light's sampling pattern, otherwise visible patterns such as banding can arise; noise is better than banding). There are various acceleration schemes which can trim away at having to fire 16 shadow rays for each sample (e.g. by identifying areas on the plane that must be fully illuminated or fully obscured). But in general this classic stochastic sampling of area lights is costly and noisy, sparkling when you animate it.

So, wouldn't it be nicer if the penumbra of the shadow wasn't noisy? What if I said that such a technique exists, and that it costs little more than sending one shadow ray per sample? Now how much would you pay? But wait, this technique can also be added to any ray tracer in just a few hours. Well... there is a little fine print, the shadows are a bit overstated and some banding is also possible. Let's dispense immediately with the criticism that this method is not realistic. The authors present it in the context of making an interactive ray tracer, where the computing time is short. I believe the method is useful in a number of other areas, such as previewing and in artistic or even technical rendering. Many techniques in computer graphics, such as environment maps and bump maps, Phong lighting, etc. are attempts at realism with simplifications. Perceived realism often does not have to match physical realism (that said, physical phenomena and theory are great sources of ideas and algorithms for computer graphics; hacks do have their limits). A soft shadow, improperly drawn, is almost always better than a hard shadow, which can be misinterpreted as a crease or a material change in the occluded object instead of as a shadow.

The idea developed at Utah is that the penumbra grows linearly with the distance of the occluder from the shadowed surface. This isn't quite true in reality, but accept it for now. In a classic, point light ray test the ray is shot from the plane (or whatever) to the triangle and the triangle either blocks it or not. In the Utah test, the ray is first tested against the triangle's plane. If the plane is hit, the length of the ray (the distance from the ground plane to the triangle's plane) is used to extend the original triangle outwards. You may wish to look at their figure at http://www.cs.utah.edu/~bes/papers/coneShadow/shadow-node5.html. The idea is that the triangle itself fully blocks the shadow test ray. As we get farther away from the triangle's edges, the amount of light stopped by the triangle falls off, until finally at some maximum distance the light is not blocked at all by the triangle.

Because the width of this "extended area" of the extended triangle is proportional to the distance of the triangle from the ground plane, the penumbra works about right. When the triangle is close to the ground plane, the extended area, and hence the penumbra, is narrow; as it moves further away, the extended area increases and so does the penumbra. The same idea can be easily done with occluding spheres (other primitives are trickier - you want to grow their extents in a smoothly changing and easily calculable way).

Less obvious, but also good, is to notice what happens with a triangle which is, say, perpendicular to the ground plane and touches at one vertex. Near this vertex the shadow cast will be sharp, because the distance from the ground plane to the triangle will be small. At the other vertices, further away from the ground plane, the shadow generated will be fuzzy.

One question is how this system works with, say, a mesh of triangles. Say three extended triangles are hit by a shadow test ray, letting through 0.4, 0.8, and 0.3 of the light. How much light really gets through? The authors tried a number of methods (multiplication and addition of various sorts); the most effective method, called thresholding, is simply to take the lowest value found, e.g. 0.3. This gives physically unrealistic results, but does give a reasonable looking soft shadow. The umbra is over- or understated: the object casts a hard shadow, but is extended outwards with a false penumbra. In reality an object smaller than the light's area can cast a shadow with no umbra; an object larger than the light's area can cast a larger umbra. With the Utah method the umbra is always the size of the object (though I guess you could simply change the interpolation function in the extended, fringe areas to be an umbra, then drop off as a penumbra, so this understated umbra is less of a problem).

One idea is to make the overstated umbra smaller by having the penumbra area not only extended beyond the triangle, but also extended inwards towards the center of the triangle. This then would give a triangle an umbra that would decrease in size as it moved away from the ground plane. Nice in theory, but it has a fatal flaw. Imagine a checkerboard, and imagine each checker square pulling in as the checkerboard moves away from the ground plane. With each checker shrinking, light can now leak in between neighboring checkers, and a checkered shadow produced. Obviously this is not what should happen with a solid checkerboard.

Note that the scheme is somewhat more expensive than classic shadow testing, but not much more. The ray/triangle test is more complex and returns a percentage instead of a hit/miss bit. Hitting these extended triangles does not necessarily end testing; only a full occlusion by the initial triangle does. The efficiency scheme holding the shadowing objects also has to take into account the maximum increased size of the triangles.

How much to move out the edges of the triangle is an interesting question. As mentioned, the distance extended is proportional to the distance the ray travels from the test location to the intersected plane. This in fact is how Steve Parker first implemented it, and it looks pretty good. However, the angle between the ray and the occluding plane's normal should also affect the amount to extend the edges. Think of the edge of a square shadowing polygon. Now rotate the square around one of its edges. As it turns on this edge, compared to the light this edge will need to be extended further to cast the same sized penumbra. The square's opposite, parallel edge will also have to be extended more. Notice that the other two edges do not have to be extended, even though the square is being rotated. Each edge has to be extended a variable amount, something like 1/DotProduct(d,en), where "d" is the ray direction and "en" is the edge's normal in the triangle's plane. However, even this is not quite right, as these extended edges could go out quite far and so the intersection distance can increase considerably, causing a larger penumbra. Also, you don't even want to think about how to join variable sized edges' endpoints to give curved corners. Finally, such huge triangles will cost extra processing time, as the largest possible triangle has to be used in forming the efficiency structure storing the scene.

Happily, just recall this is a hack and then live with the simple solution of extending edges based only on distance. While one particular triangle may have its edges understated due to its orientation, recall that there are normally two triangles touching any silhouette edge and that the odds are with you that one of these will be fairly well aligned with the ray direction and so cast a reasonable penumbra.

However you do it, these extended triangles are larger than their original primitives. This means that the efficiency structure put around them must take their larger size into account. Either a separate structure must be created for shadow computations, or the shared efficiency structure must be made less efficient for eye/reflection/refraction rays (since these do not need to have extended triangles).

As far as actually testing the ray against this extended triangle, here are a few approaches. If you're not interested in implementing this method, skip to the last paragraph or two. I'm making some of this up as I go, so feel free to correct me if you try something out and it fails miserably. The original paper says you extend the edges and round the corners. In practice, Steve Parker (first author) said they extend the edges of the triangle out, also clamping the corners by this extension distance, and do not bother with rounding.

One way to implement extending the edges and corners is to use barycentric coordinates (see RTNv5n3) and extend the edges out by loosening the range 0 to 1 requirement. In barycentric coordinates, the point is in the basic triangle if all three coordinates are between 0 and 1. However, giving different ranges for these coordinates results in different primitives. For example, if the intersection point's barycentric coordinate is -0.1 for a given vertex, this means the point is 1/10th of the way outside the triangle edge opposite from this vertex. Values above 1 means the point is beyond the vertex compared to its opposite edge. Say you allow the triangle to extend by +/- 0.5 for a given coordinate, i.e. from -0.5 to 1.5. From 0 to 1 is then the umbra, and something like 1.3 would dim the light by 0.4 (the light would be multiplied by 0.6), since it is 60 percent of the way from 1.0 to 1.5. Compute the attenuation for each coordinate and choose the one that dims the light the least (i.e. is highest). Varying the allowed range of these coordinates allows you to limit how far the triangle extends, vary the penumbra, and also compute the attenuation.

However, just using barycentric coordinates will give a somewhat chopped off extended triangle, with sharp corners. Allowing the values to be below 0 pushes the edges out, and values above 1 move the amount each corner extends out (essentially, controls the bevel formed at the corner). The corners are worth limiting because a corner with a sharp tip, when extended, will have a penumbra which extends quite a distance as a sharp extension.

The amount each barycentric coordinate is extended is relative to the width of the triangle for that coordinate. For example, say corner A is across from the long side of a right triangle (a triangle where corner A is 90 degrees and the other two corners B and C are each 45 degrees). Say the triangle is 4 units across from corner A to the opposite edge. To extend the opposite edge and corner A out 1 unit, change the range from [0,1] to [-0.25,1.25]. However, for corner B (and C), the opposite edge is 4*sqrt(2) far away, so to extend it 1 unit the range is only [-0.1767,1.1767].

I've tried to figure out a way to add rounding to the corners while using barycentric coordinates but can't quite figure out a foolproof way to do it. Enter method two, which is less efficient but higher quality: simply build the extra geometry needed on the fly and test against it. First test against the triangle, of course - if it's hit, you're done. Else, each edge is extended perpendicularly outwards to form a rectangle, each of which is tested. If any of these are hit, immediately return the amount of attenuation from it, since only one rectangle can be hit. You might test the longest edge's rectangle first, since this is most likely to be hit. If none are hit, then test against a circle around each vertex. If inside one or more circles, return the value which dims the light the most (i.e. is closest) - note you have to test against all circles, not just stop when you first get a hit [well, this isn't quite true - you have to test against all circles only if the radius of these circles is larger than the length of the smallest edge of the triangle].

Up to this point I've assumed a linear falloff in the penumbra region. This is in fact the sort of shadow you would get if a shadow edge was aligned with a light source which was rectangular. However, this gives sharp slope changes where the penumbra starts. To avoid the discontinuities at both edges of the penumbra, the falloff could be determined instead by a form of the Bernstein polynomial (essentially, an S curve):

f = (3 a^2 - 2 a^3)

with a=0 at the umbra's edge, a=1 at the point entering full illumination, and f being the resulting falloff.

That's about it. Now someone find some time to code it up into POV-Ray or Rayshade or somesuch! Should be easy.

I do not believe this idea has yet been used in Z-buffer projective shadows, i.e. extend each triangle with an alpha-blended fringe. This sort of thing should work, though as mentioned, thresholding seems to yield the best results, which is not normally supported in graphics accelerators. Still, such soft shadows would be of higher quality than typical methods used, such as accumulation buffers (jitter the projection of the shadow and average the results) or post-process blurring of the projection.

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I'm trying to find out about fast methods for traversing octrees. It strikes me that you could use some kind of integer math method like a 3D Bresenham algorithm to get a list of cells a ray intersects. I assume I'm not the first to think of this, and that somewhere there is some C source code for such a method, or at least a detailed description of how to do it. I found a fragment in Graphic Gems IV, by Daniel Cohen. Unfortunately, it's complex, uncommented code, and the email address listed is no longer in use.

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Eric Haines <erich@acm.org> replies:

People do essentially use a 3D Bresenham to traverse a grid efficiency structure, getting the cells as they go. There are many ways of looking at octree traversal, Erik Jansen wrote an article discussing some of these, in the book _Data Structures for Raster Graphics_, Springer-Verlag. In the RT News, Jim Arvo's "Linear-Time Voxel Walking for Octrees" in RTNews2 is an interesting method of examining the octree and minimizing octree traversal. Graphics Gems III has an article by Sung and Shirley on walking through octrees (BSP trees, but essentially the same thing) and some skeletal code. Personally I tend to just find the next three intersections (one for X, Y, and Z) and take the closest one and walk to the next neighbor. But I've definitely become a nested grids convert over the years so have lost track of what the best octree traversal methods are.

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Hanan Samet <hjs@umiacs.umd.edu> replies:

With respect to the octree Bresenham, etc. I saw the Ray Tracing News issue of sometime back that discussed this problem. There are fast methods of doing this and in fact am close to finishing up a paper on doing it on the sphere as well. So, the bottom line is that there are very fast ways of doing the neighbor finding. I wrote it up in the style of my book for a rewrite of the section. However I found some errors while writing the paper on the sphere. I will try to fix it next week (it is on my stack). If I get somewhere I will let you know.

By the way, the ray tracing code in my Applications book is rewritten and published in the newer printings of the book. Look for the latest printing which came out in 1995.

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Erik Jansen <F.W.Jansen@twi.tudelft.nl> writes:

As far as I know there is no dominant "fastest" method yet/still. The "nested grids" is indeed a good choice. Easy, robust, and fast. The BSP traversal is also fast and if you have a bintree spatial subdivision a good choice. It should be very easy to adapt the BSP-traversal to octrees (do some extra recursions for the x,y,z direction). You can also apply some intelligent coding like in the HERO-algorithm (I have no copy any more of the paper). The SMART algorithm is also a variant in the same style. Then there is the DDA-Octree traversal, which is the grid traversal but mapped to an octree (or more the other way around: the octree mapped to a grid).

The last comparison we did is with the grid, our BSP-traversal, and the DDA-Octree. It is reported in RTNv7n2, Comparison of Ray Traversal Methods.

That was the last thing we did. The (nested) grid (with some tweaking) was the winner. But with other data models, the results may be slightly different, but not dramatically.

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Ben later writes:

One easy way to convert a grid traversal to octree traversal is to pretend the octree is actually a grid of resolution equal to the octree depth. In many data sets I've tried, the average depth of an octree is close to the max depth, so this simplifying assumption isn't too wasteful. Then you must cull out grid cells which aren't actual octree cells, by doing some lookup into the real octree.

I have developed a 2 pass algorithm over the last few days to do this. I've coded it up but it isn't debugged, so I've no runtime figures yet.

Pass 1 traverses a grid and fill and array with grid coordinate for the cells pierced by the ray. Essentially a normal grid traversal. In my implementation, I convert these (i,j,k) co-ordinate into a 64bit key which uniquely specifies an octree cell location. Each depth level is encoded into three bits in the key, with a depth value occupying the last few bits of the key. The conversion process is all bit manipulation should it should execute fast.

Pass 2 culls out the missing cells. I map ID values to Cell pointers via a hash table. If they table returns null for an ID, I derive that Cell's parent's ID (bit manipulation again) and try the hash table again, repeat until found. The found cell will have 0 or more subsequent cells in the octree which are also missing, depending on how high in the tree it was. The ID of the cell can be quickly bit-manipulated into a template which can be matched against subsequent IDs in the list, enabling further culling without a hash table lookup.

I believe it will run fast because its basically integer traversal, then bit manipulation and hash table lookups. The actual octree never needs to be touched. Result - little bus activity, most calculation stays in the registers or the L1 cache. it would work well in SMP enviroments and on ultra fast CPUs with (relatively) slow buses/main memory.

Anyway, hopefully I can get the thing to work soon and I'll know. The bit-manipulation stuff is pretty heavy and its tedious to debug. Once again, I hate to be doing this if someone else has (I try never to reinvent a wheel) already, so if what Im saying sounds all too familiar, please send me the right way.

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Erik Jansen <F.W.Jansen@twi.tudelft.nl> replies:

Ben,

The thing you are doing is what we call a DDA-Octree traversal algorithm. There is a paper by Kelvin Sung in the Eurographics Proceedings of 1991:

K. Sung, *A DDA Octree Traversal Algorithm for Ray Tracing*, Eurographics'91,
North Holland-Elsevier, ISBN 0444 89096 3, p. 73-85.

In this paper Kelvin compares the DDA-Octree some other traversal algorithms including the recursive bintre traversal, which he called ARVO. I started a discussion with him because his implementation was not optimal. A result of this was that he made a better ARVO implementation and published this in Graphics Gems. In the meantime I asked some students here to also make a comparison, which resulted in the figures that I mentioned in my previous message. Of course you can do some tuning and clever coding, and make one method look better than an other. (The other strategy is to implement other methods dummer than necessary, also an often applied strategy.)

(there's a followup to this article in the next RT News)

(there is also in-depth coverage of octree schemes in the next RT News)

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1) Form a grid structure, say 20x20x20, of pointers to linked lists. 2) Insert each object into the grid, e.g. find the bounding box for the object and for each grid cell overlapped by the box add the object. The object is added to the front of the linked list pointed to by the grid structure.

This grid structure is then accessed as is, walking through the linked list on each access. Linked list access is usually not as fast as pure indexing, i.e. it would be nice to just have a list of pointers at each grid cell to walk through, vs. grabbing "next" pointers. Even if of comparable speed, linked list nodes can be allocated in a fragmented fashion - some nodes are on this page of memory, some on that, etc. This leads to more paging than is necessary. Also, linked list nodes take up more memory than indexed lists. Finally, allocating a huge number of separate nodes is expensive both in time and in additional memory header costs. Some of these problems can be solved for linked lists by using a preallocated pool of memory which can then be used to quickly dole out link nodes. Nonetheless, linked lists are generally clunkier than need be, and are just an artifact of the way the grid structure is built. Also, indexed arrays allow such tricks as on-the-fly sorting: when an object in a cell is found to be intersected by a ray, you can swap that object to the top of the grid cell's indexed list, on the theory that it is likely to be the first thing hit next time the cell is accessed. This can buy you a few percent speed improvement (though I have occasionally seen performance get worse, strangely enough).

So, one method to get indexed lists at each cell of the grid structure is to post-process the linked lists, tossing out each linked list and replacing it with a list of pointers. That is, if a grid cell has a linked list consisting of 5 nodes, replace it with a single list of pointers to the 5 objects, plus a NULL pointer at the end of the list. Better yet, you could have kept a count of all the linked list nodes in use in the grid structure and allocate one hunk of memory of this length, plus additional space for the NULL end of list pointers. In this way a single piece of memory is allocated, minimizing both storage and page misses.

However, we can do even better, though at possibly a (generally small) additional cost. Allocating memory for linked list nodes can be costly, so this could be done away with by doing a two pass method. On the first pass, find out how many grid cells are overlapped by each object. Say you are doing the simple (but inefficient for ray tracing itself) method of seeing which grid cells are overlapped by the object's bounding box. It's trivial to compute how many grid cells are overlapped by each: sum up, and after you go through the objects once you know exactly how many list elements are needed in your array. Also allocate an array for the grid structure, to keep track of how many objects are in each grid cell. You then go through this array and can set where the beginning of each grid cell list begins in the large allocated list of elements. That is, for cell (0,0,0) in the grid you point at element 0. This cell has, say, 5 objects in it found in the first pass, so cell (0,0,1) will point at element 5+1 (+1 for the NULL end of list pointer). Add in this next cell's number of objects+1, this is where cell (0,0,2) is set to point, etc.

Once the grid structure's pointers are set up, go through the list of objects again, this time saving the pointer to the object in each overlapped grid cell's list. Since you know the exact number of list elements and how many are in each grid cell before the second pass, the object pointers will perfectly fit when stored. No additional memory (except for the piece which stored how many objects were in each grid cell; this piece could be reused as the array which holds the grid cell object lists) is allocated or freed.

Even if you do a more elaborate test than bounding box overlapping the grid, i.e. you precisely test whether the object itself overlaps each individual grid cell, two passes may still be worth it just for avoiding memory fragmentation or even to avoid running out of memory. You could also use a hybrid method, where the first pass just does box overlap testing to get an upper bound on the number of object pointers needed, then the second pass is more precise and so does not use all the memory allocated - it is still most likely a win over storing linked lists.

Anyway, perhaps this is more than you ever wanted to know, but it makes for a fairly efficient grid cell allocation scheme that avoids memory headaches and is simple to code, to boot. It's not for everyone, but is a way of thinking worth knowing about. There are a bunch of extra tricks to squeeze more memory out of the grid cell lists - using short or byte sized indices instead of pointers to the objects, using the sign bit or lowest bit of a word-aligned pointer to denote the end of an object list so you do not have to store NULL pointers, etc.

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Eric Haines / erich@acm.org