"Light Makes Right"
May 19, 2007
Volume 20, Number 1
Compiled by Eric Haines, Autodesk, 10 Brown Road, Ithaca, NY 14850 email@example.com. Opinions expressed are mine, not Autodesk's.
All contents are copyright (c) 2006,2007, all rights reserved by the individual authors
Archive location: http://www.acm.org/tog/resources/RTNews/html/
(text version at http://www.acm.org/tog/resources/RTNews/text/)
You may also want to check out the Ray Tracing News issue guide, the index to the Ray Tracing News, and the (ancient) ray tracing FAQ.
There was one ray-tracing related paper at I3D: "Interactive k-D Tree GPU Raytracing". A hybrid RTRT (real-time ray tracer), using the traditional GPU strengths for eye rays and for computing shading. The contributions are in the area of traversing a k-D tree with a GPU; GPUs don't do stacks well, so there are some clever techniques to limit the stack size and to recover from stack overflow.
Speaking of ray tracing, the second Symposium on Interactive Ray Tracing, aka RT07, will be September 10-12, 2007 at Ulm University, Germany. Papers due June 14, so get crackin': http://www.uni-ulm.de/rt07. Jacco Bikker will be the keynote speaker.
On to the articles. I'm feeling that this issue sometimes comes off as a bit curmudgeonly, with my rant about "free for non-commercial use" and whatnot. Well, just be happy I'm not rambling on about software patents, sales loads on mutual funds, or other typical mainstream sources of evil. Free-ish software is hardly evil, it's just unamiable. Next issue I promise I'll keep my tinfoil hat on so the satellite mind-control rays won't make me so peevish.
back to contents
A few of them also have PDFs for download or links for more information.
From Aaron Knoll:
Normally my papers are sort of vis-centric, but I thought you might like our recent tech report... it basically shows how to ray trace any arbitrary implicit pretty interactively using very simple 2x2 coherent SSE packets.
It's the top link there [well, now it's the second down. -EAH] -- there's also a video :)
Remember that first time you wrote a ray tracer (you haven't yet? Go do so!) and quickly added more code to get another cool image? Here's one person's account: http://www.superjer.com/pixelmachine/ - not a great read, but I appreciate the "one more bug to fix, one more feature to add" spirit.
So if you haven't written a ray tracer yet but would like to, Jacco Bikker (there's that name again) has a nice series of articles on the basics of coding up a ray tracer, see http://www.devmaster.net/articles/raytracing_series/part1.php
So what's the acceptance rate for various conferences? Now you know: http://vrlab.epfl.ch/~ulicny/statistics/
In RTNv18n1 Daniel Pohl's raytraced Quake 3 project got a quick mention here. Now he's doing Quake 4 ray-traced: http://www.theinquirer.net/default.aspx?article=36452
Back in RTNv18n1 I mentioned Farin and Hansford's new book, "Practical Linear Algebra". At the time their website wasn't quite working, now it is. Teaching materials (e.g. figures from their book) for this book are available at http://www.farinhansford.com/books/pla/pla-teaching.html. This site also contains additional chapters, such as a section on Principle Component Analysis. PCA is useful in graphics for finding tighter object-aligned (vs. axis aligned) bounding boxes.
An interesting something: http://www.redway3d.com/ - a commercial real-time ray tracer.
Last year the hot topic for RTRT was efficiency schemes for dealing with deformations and animation. Keenan Crane writes:
There are a couple more papers contributing to this "year of dynamic scenes"
- both of them work by updating bounding volume hierarchies with static
Unlike the BIH stuff, this approach only makes sense for small deformations.
Recommendation from Andrew Glassner:
Stock.xchng, http://www.sxc.hu/, has a huge collection of free stock photos. Their reuse rules are extremely liberal - basically, you can use the images in books, movies, etc., and about the only thing you can't do is sell or distribute the images themselves. See http://www.sxc.hu/forums/topic/12673 for a summary of the rules. Nice! If a quarter million free photos is not enough, here are more sites: http://www.nooti.com/design/free-stockphotos.php
Fractal art has been around forever, but it has improved considerably. Free programs like Apophysis, http://www.apophysis.org/, has done wonderful things in some users' hands, e.g. http://intergalacticart.blogspot.com/, and Ultra Fractal, http://www.ultrafractal.com/, a commercial program, also has its adherents: http://csdl2.computer.org/comp/mags/cg/2007/03/g3004.pdf
Who says wireframe is unrealistic rendering? http://news.windingroad.com/etc/wonder-woman-your-car-is-ready%E2%80%A6/
In the same spirit, a lovely NPR hotel room: http://www.luise-berlin.com/en/rooms/306.htm
Arvo's box/sphere intersection test has been improved: http://jgt.akpeters.com/papers/LarssonEtAl07/. Surprising, to me, as Arvo's test seemed pretty minimal. But the early out is to simply add the radius to the box's bounds and test if the center of the sphere is inside this extended box. This "extended box" is an approximation of the object formed by "adding" a sphere and box together, the Minkowski sum (e.g. http://www.cs.sunysb.edu/~algorith/files/minkowski-sum.shtml). Minkowski sums get used a lot for other object/object collision tests, e.g. frustum culling. I guess no one really noticed it could even be used to improve this seemingly "simple" test. Nice!
Lightsprint Vision: Realtime radiosity for your renderer, at http://lightsprint.com/. They have a downloadable demo, which looks nice enough. Anyone know who they are or what they do?
Perhaps you saw the mirror made of wood at SIGGRAPH some years back. More projects by the same artist: http://www.smoothware.com/danny/index.html
Very peculiar way to generate the inverse square root, quite amazing: http://www.lomont.org/Math/Papers/2003/InvSqrt.pdf
Finally, a 3D printer for candy: http://www.evilmadscientist.com/article.php/candyfab - POV-Ray is used for generating the 2D slices the 3D sugar printer needs for fabrication.
Fundamental Research Challenges in Computer Graphics:
- white papers from many luminaries in the field. They date from around 2003, but are still worth a look.
You know you've been in graphics too long if... http://www.ahajokes.com/com097.html
Make something cool for your kids, or friend's kids, or SO, or yourself: http://www.thevideosense.com/video/A-Must-See-Optical-Illusion/ - the http://www.grand-illusions.com/opticalillusions/dragon_illusion/ page has a link http://www.grand-illusions.com/images/articles/opticalillusions/dragon_illusion/dragon.pdf for making your own.
Here are a bunch of links:
http://www.sjbaker.org/teapot/index.html - history of the Newell teapot
http://www.bertsimons.nl/album/projects/clone/index.html - papercraft heads through computer graphics.
http://www.easyrgb.com/math.php - formulae for converting between many color spaces. There are other interesting bits on color here; explore!
http://www.ilm.com/theshow/ - fun little "Pirates of the Caribbean 2" special effects site.
http://www.spectrum.ieee.org/print/5049 - article by a member of the Microsoft text team, on their techniques for improving screen readability.
http://www.innovationlab.net/sw22811.asp - computer screen made from concrete
http://www.youtube.com/watch?v=Z2LUz2WVcek - display using falling water. After watching 20 seconds you're done, the rest is just repetitious.
http://www.sciencemag.org/cgi/content/short/313/5794/1729 - pretty. Larger image here: http://rsp.math.brandeis.edu/3D-XplorMath/Surface/gallery.html, from this site: http://rsp.math.brandeis.edu/3D-XplorMath/, which provides a free Mac program for exploring such surfaces.
http://newyorkdesign.hp.infoseek.co.jp/menu06.html - a perceptually tricky game - what's changing?
http://www.google.com/patents - Google's patent searcher
http://www.ks.uiuc.edu/Research/vmd/projects/ece498/ - interesting course, partly because it was co-taught by Dave Kirk, CTO of NVIDIA. Lectures downloadable from http://courses.ece.uiuc.edu/ece498/al/Syllabus.html. Lecture 18 is by John Stone, of Tachyon ray tracing fame, though it's not about ray tracing.
http://www.xrez.com/boston_giga.html - super-high res Boston photo.
http://www.docbert.org/ChicagoByNight - super-high res Chicago, with links to Sydney and Machu Picchu zoomables.
http://www.flickr.com/photos/21815268@N00/123158595/ - the real world really is complex and hard to model...
And sometimes the URL is just not what it seems: http://www.rendermagazine.com/
- from Neil Gatenby
back to contents
Problem: what is the relative probability of a random plane hitting a sphere of radius R vs. hitting a box with edge lengths of DxWxH? We know that the chance of a random ray hitting each is proportional to the surface area of each object (or any convex area, in fact). This is what gets called the Surface Area Heuristic, used for forming near-optimal kd-trees for ray tracing - see RTNv17n1. But what is the proportion for spheres vs. boxes when intersecting with a plane?
I'll even solve the easy part of the puzzle for you, the relative probability of any box compared to any other box. It turns out that if you sum the edge lengths of each box, this is the proportional chance that a random plane will hit it. So a 1x1x1 box has a measure of 1+1+1=3, a 1x2x3 box a sum of 1+2+3=6, so the second box is twice as likely to be hit by a random plane. This "sum of edges" measure is (misleadingly) called the "mean width" of the box. "Misleadingly" because it's not what I would think of as the mean width of anything. The "mean width" of a sphere is the sphere's diameter, but the "mean width" of a box is certainly not of the same scale. For example, a 2x2x2 box has a measure of 2+2+2=6; a sphere of diameter 2 just fits in this box and has a mean width of 2 (its diameter); a sphere of diameter 2*sqrt(3) circumscribes the 2x2x2 box and has a mean width of 2*sqrt(3)=3.46. So the 2x2x2 box must have a mean width between 2 and 3.46. So how does its measure of 6 relate to its true mean width, what's the scale factor? Prove it. Getting this scale factor then tells us how spheres' radii and boxes' edge sums relate.
Possible hint: the Discrete Differential Geometry course given at SIGGRAPH deals with this type of measurement early on, see the course notes at http://ddg.cs.columbia.edu/, second chapter (and make sure to get the 2006 version of the notes!). That said, I think there's a pretty easy way to solve this, but it does involve a double integral - I'm hoping one of you out there will have a nice intuitive answer & proof.
Bonus question: what applications are there for computing the relative probability of plane/box or plane/sphere intersection? I have one in mind (I'll mention it next issue), but would love to know of others.
back to contents
> Techniques such as Phong highlighting, environment mapping, and Gouraud interpolation could all have been rejected because they were just "tricks that look good" and not science.
If you check a bibliography, you'll notice there is a twelve years hiatus between the publication of the Bresenham's algorithm for lines (published 1965, submitted 1963, first presentation 1962), and the publication of Bresenham's algorithm for circles (CACM, 1977). Now, if you talk to Jack, he'll explain that it's not that it took him twelve years (or even fifteen) to get the idea for circles. After all, once you have the first idea, the second is an obvious extension [though the circle-drawing algorithm itself is not obvious. - EAH].
No, the problem is that it took him twelve years to get the second paper accepted for publication. It kept being rejected as "there's no science in that", or "it's useless"... So it seems problems in getting new ideas accepted are not something new in Computer Science.
I should credit those who told this story to me, Jean-Jacques Bourdin and Vincent Boyer. They are two guys (from the University of Paris) who worked on line drawing algorithms for quite some time. I remember at EG 99, in Milano, JJ Bourdin giving a talk that began with, "Today, I'm going to show you an algorithm twenty times faster than Bresenham's algorithm". He got the audience's full attention. :-) [Paper is at http://www.ai.univ-paris8.fr/~jj/Rech/SI/eg99.pdf - EAH]
[I contacted Jean-Jacques and Vincent about whether Nicolas remembered the details correctly. Vincent said "Jack confirmed this story after my PhD. thesis presentation", and provided a minor correction to the text above. I also attempted to contact Jack Bresenham, and, happily, he replied. The explanation is perhaps a bit long, but I found it worth including in full: the simple storyline is concise and dramatic, but reality is more nuanced. People get busy with other things, interest in a particular type of research wanes and waxes, and so on. Jack's tale also puts things in some sort of perspective: a two-year publication turnaround seems like nothing compared to 14 years delay overall. - EAH]
Yes, the story is essentially correct but wanders a slight bit from fully being factual. One outright rejection and almost a second rejection is correct; most of the long delay in successfully submitting the circle paper for publication was primarily my responsibility. ACM contributed only a couple of years delay once I finished the paper to their satisfaction.
I did develop the circle algorithm in 1962 or 1963 just after the line algorithm. Both were done as an employee at IBM's San Jose development laboratory working in Dr. E. E. Lindstrom's computation laboratory. The IBM technical report TR_02.286 I have is dated 01-27-64 and was done in what then was IBM's General Products Division Development Laboratory, San Jose, California.
My recollection is that the circle work was essentially complete, but not formally documented, and IBM 1401 code to drive the Calcomp plotter working in the latter part of 1962 around the time I returned to Stanford or early 1963 shortly after I was again taking classes full time. Owing to personal matters, publication of a formal TR was delayed.
The circle algorithm's publication history stretches over a long time. I proposed to submit it to the IBM Systems Journal in the early 1960s. I can not recall a specific date; it would certainly have been after the August 1963 ACM national conference in Denver, which is where an IBM Systems Journal editor approached me to ask to publish the line drawing algorithm I presented at the conference. The circle paper was reviewed within IBM and one reviewer pointed out an error in my claim to minimize the difference between the true radius and the radius to each point of the digitized incremental circle when, in fact, I demonstrated that the algorithm minimized the difference between the square of the true radius and the square of the radius to each point of the digitized incremental circle. That is, I claimed an error metric of: min: |R-r| when what I actually proved was min: |R^2-r^2|.
The period of time 1963-64 is a bit fuzzy for me and I am not sure whether the review was an internal one in the San Jose lab before it was submitted to the IBM Systems Journal or if it was an actual IBM Systems Journal review; my recollection is that it was submitted and the reviewers were those from the Systems Journal. Personal matters made it infeasible for me to spend time correcting the error metric so publication effectively terminated owing to my non-response. My guess is that original submission of the circle paper to the Systems Journal likely was in 1964 as I do recall when we spoke in Denver in August 1963, the editor told me the journal would as well be interested in the circle algorithm when written.
Some years later, either late 1973 or early 1974, I chanced to read Mike Pitteway's BCS Computer Journal November 1967 article: "Algorithm for Drawing Ellipses or hyperbolae with a digital plotter". I'd not followed much of computer graphics for quite a while so was pleasantly surprised to read a reference to my line drawing article from 1965 and decided it might still be of interest to publish the circle algorithm as by then I'd worked out, as an avocational interest, what became the appendix of my February 1977 incremental circle article in Communications of the ACM. I put together a new version of the paper and submitted it to the IBM Systems Journal. The IBM SJ editor responded that the journal no longer was interested in publishing such material. The SJ rejection is what my casual conversation remarks referred to in France at the University of Paris-8 dissertation defense.
I submitted the circle paper to Communications of the ACM in June 1974. W. M. Newman had it reviewed, accepted it for publication subject to my changing a few things, then subsequently revised the acceptance saying he'd become aware of other circle algorithms and was sending it out for additional review. The others were not incremental digital circles and, as one would expect, the second set of reviewers pointed out the new context of digital circles for plotters & CRT raster displays and the paper was again accepted for publication.
ACM was significantly far, far behind in publishing papers so the circle paper had at its conclusion when finally published February 1977 the publisher's note: "Received June 1974; revised September 1975" as was standard ACM practice. A final twist was that ACM editorship changed before publication when Jim Foley took over from William Newman. I probably mentioned in Paris, as the context was light and relaxed, that ACM also nearly rejected the paper and took the unusual step of having a full second panel of reviewers.
In my keynote talk at the "11th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision 2003", I made more formal remarks with the intent of encouraging young computer scientists to never be discouraged by rejection or delay if they believe they have a worthwhile idea. A pdf copy of my remarks can be seen at: http://wscg.zcu.cz/wscg2003/Papers_2003/Bresenham.pdf
back to contents
Many books for courses are just that, texts for the classroom. They cover each topic just enough to get the point across to most students, and assume a teacher is around to fill in the gaps and help along any other students who didn't quite understand the book. A reasonable assumption, certainly, but of necessity it means the text will skim over areas in order to cover every major topic in a field and so appear "complete".
The book "Ray Tracing from the Ground Up", by Kevin Suffern, published by A.K. Peters, will be available at SIGGRAPH 2007. I've skimmed through most of the chapters (not as a paid editor, but rather just to comment), so can offer up an initial impression. This book has a perfect title for it. Theory and code snippets are blended to show how to make a classical or stochastic ray tracer from scratch. It assumes the reader has just about no knowledge of graphics and at most some understanding of calculus. Each chapter tackles some topic: perspective, reflection, intersection, etc. The text has many excellent figures and illustrative renderings, along with C++ code snippets that are explained when presented (vs. the often slap-dash nature of many code-laden books that present long, weakly commented listings that fill half the book). The book will come with a CD that includes a basic ray tracer, as well as code for generating various scenes (there's no scripting language front-end for the ray tracer).
Overall, the book is somewhat "old school". With the exception of a few newer topics, e.g. ambient occlusion, most of the material presented dates back to the 80's and early 90's. But this is as it should be for a text of this sort: fundamentals are established and built upon, with the author doing his best to make sure the reader truly understands what is going on each step of the way, (hopefully) without the need of a teacher to fill in the gaps. Through examples and extensive illustrations the author attempts to build not only a basic understanding but present mental models and give some intuition as to what various equations and algorithms represent. For example, I've never seen a clearer explanation of the ray/box (slabs) intersection method - it's done as it deserves to be done, walking through the various types of hits and misses and showing (through excellent colored figures and ray-traced test images) how the algorithm actually works. This is not a text for researchers or advanced students, but truly for the novice, the hobbyist, the enthusiastic amateur.
The style is informal and approachable, with the author normally speaking in the first person singular or plural, e.g. "I'll use the same representation for the BRDFs", "We need an expression for the primary rays". He assumes you're going to make a ray tracer, and he leads you through what you need to know and gets you coding it up. He points out variations and elaborations along the way. This approach is perfectly in the spirit of writing your own ray tracer, in which you normally have the drive of adding "just one more feature" that keeps you up until 5 AM. He even points out common pitfalls and ways to debug various features.
The book is not without its limitations. The coverage of some topics sometimes ends a little too quickly for my tastes. For example, the basics of efficiency grid creation and traversal are presented, but simple efficiency improvements such as mailboxing are not mentioned. Admittedly, mailboxing is not useful for multiprocessor systems, but I think it's worth mentioning as a handy idea in general.
As a test, I chose two terms, "radiance" and "Fresnel", and searched through the book to see how these are treated. The book does well with radiance, as it does a reasonable job defining the various types of radiometric units and draws the important connection between radiance and a sample ray. For Fresnel it mostly focuses on the Fresnel equation's effect on reflectance vs. transmittance as a function of angle. This is fairly important for a ray tracer, though the text rightly points out that it's often less noticeable than you might think. Where I find it important is for things like the surface of a pool or pond, where the effect of reflectance is low looking directly into the bottom while it increases as you look more towards the horizon.
The book presents the technique of making the specular color match the diffuse color to give a metallic look, vs. using a white specular color for plastics. It would have been nice to note that the Fresnel equation also is important in how metal and plastic differ in appearance. The Fresnel reflection effect, where at grazing angles all surfaces approach becoming perfectly reflective, is briefly mentioned indirectly when shading models are discussed. The book is interesting in that it does a thorough job of reconciling the Phong shading model, which is not energy-conserving, by reformulating it properly as a BRDF. However, Phong is as complex a shading model as is presented in the book. And this makes perfect sense within the context of what the author is trying to do: the "80% of the way, good enough for a start" Phong shading model is presented and put into the proper theoretical context. The author gives brief explanations and a number of references to more elaborate shading models. The focus of the book is to get the reader to the 80% level in a wide range of areas, with pointers where to go for more information if an area is of particular interest.
I could easily imagine this textbook being the basis for a basic or mid-level undergraduate computer graphics course. Such a course would necessarily ignore GPUs entirely, but the advantage would be in teaching first principles (light transport, BRDFs, sampling theory) and focusing on the scientific and mathematical concepts used in rendering as a whole. There are any number of areas that are not addressed by the book, such as tone mapping or BSP tree formation or procedural bump mapping. However, the basics are all there, and each teacher can elaborate on their own areas of interest. Those basics are carefully covered, with the proper theory and equations being presented without any dumbing down of the material. Questions and exercises are provided at the end of every chapter. The informative illustrations alone make the book worth purchasing by anyone planning on teaching or understanding more about the essentials of ray tracing.
(A little) more information about the book can be found at http://www.akpeters.com/product.asp?ProdCode=2728.
back to contents
I'd like to pass on my perspective when I see this sort of "non-commercial use" notice. As a programmer for 30-odd years and a commercial developer for 24 of those, I've never pursued getting a license for any of these smaller bits of software - instead I ignore them. I don't have a budget on-hand and it's not worth my time to negotiate a price for a small contribution; just getting reimbursed for buying a shareware utility can be a chore.
I can understand how some content such as free music could easily be repackaged and sold as-is, and so a Creative Commons license (http://creativecommons.org/licenses/) that said "non-commercial only" has a logic to it for those items. But usually an algorithm is such a tiny part of a commercial application that there's little point in protecting such relatively small bits of code. As a commercial developer I can assure you I've never thought, "yes, I finally have a spline/line intersection algorithm in hand, at last I can accumulate a vast personal fortune!"
So, please, unless your code size is large enough and comprehensive enough and robust enough that it makes sense as a separate standalone DLL (a thing which I have licensed), don't bother saying "not for commercial use etc." on it. It's more annoying than anything, and usually there's a similar piece of software (with more liberal licensing) somewhere else on the web. I've always honored licenses that say, "please give us credit," if that's your concern. If you've truly contributed something new and want recognition, instead consider submitting an article (and code, if you wish) to the "journal of graphics tools"; as a bonus, you become a known expert on the topic.
If your goal is simply to hasten the collapse of the capitalist system by limiting reuse to non-commercial developers, good luck with that, let me know how it works out. Personally, I'd rather have you protest for more vacation days - hmmm, that could be an interesting licensing term: "by using this software, you have saved 3 days of development time, so must be given an additional vacation of this length by your company as payment". Or just go with the LGPL and call it a day.
I should note that all the code snippets in the Ray Tracing News are free only for commercial use; hobbyists and academics are not permitted to use them. So, how did that make you feel? Just kidding, of course - any code bits here are copyright by their original authors, but as far as I'm concerned are free for reuse, with bugs being Your Problem (though please let us know), and attribution to the original author being A Nice Thing. Oddly, I probably won't send lawyers after you if you don't do so. Which of course is the case with 99% of "free for non-commercial reuse", since any dishonest commercial developer could simply use the code anyway and no one would be the wiser. "When algorithms are outlawed, only outlaws will use algorithms".
followup here, search on "rant".
back to contents
First, POV-Ray is officially the first ray tracer in orbit: http://www.oyonale.com/iss/english/makingof.htm
POV-Ray 3.7 beta, http://www.povray.org/beta/, has two significant features: SMP (symmetric multiprocessing) support, which means it will use all the processors on a multicore machine, and BSP tree support, so that you can experiment with this scheme vs. bounding volume support.
I got their code and experimented with tuning the BSP tree generator settings a fair bit, trying to find some optimal numbers for a range of scenes. What was interesting was how little the knobs I twiddled seems to affect overall performance. At the extremes I'd see a fall-off in speed, but for the most part the BSP trees formed all had about the same efficiency. That said, I did this tweaking last year, and there have been a number of improvements since then to their code, so perhaps things have changed. My take-away was that doing some sort of cost function was a win, but after this is in place the various weighting factors for the cost function were not all that critical. Which is a good thing, in my opinion, as it means the BSP tree is pretty robust and forgiving.
There was also a scene dependence: sometimes the original BVH code would be faster, sometimes the BSP tree would. Unlike some of the highly optimized RTRT systems out there that render only triangles, POV-Ray is very much an object-oriented system. I think this flexibility gives it a different feel for efficiency tuning: sort of an all-terrain vehicle vs. a Formula One car.
[Chris kept sending me interesting tidbits on POV-Ray development, so I've glued together some of his correspondence. - EAH]
POV-Ray is now an official SPEC CPU2006 benchmark, see http://www.spec.org/cpu2006/Docs/453.povray.html. SPEC is a non-profit organization that creates standardized sets of benchmarks for testing computer systems.
It appears POV is being used a lot for benchmarking the new multi-core CPU's, since as it turns out our SMP scaling is quite good. A few links in case you're interested:
I threw a little real-time-raytracing demo together using the beta for use at IDF, it turned out rather fun really. I modified our internal image class (which we use for, amongst other things, holding files that have been read from disk for use as image maps) to accept a live stream from a webcam. I image-mapped that onto the sky sphere and then set up an internal continuous render loop with a fixed scene (BSP gave us about 40% on that particular scene) and a moving camera that simulated an infinite zoom over a blobby mirrored landscape (think of zooming into a Menger sponge - it looks infinite but it's only about 20 frames repeated).
The net result was fairly hypnotic, dunno why, but you could see a live view of yourself reflected in a strange landscape moving underneath, hard to describe (I had output turned off for speed). But it was interesting enough I will probably add the feature back in properly for folks to play with now that multi-core systems are getting affordable and are fast enough to experiment with RTR at viewable resolutions. FWIW I was getting about 7 FPS at 320x240 no AA on that scene using a four cores. (I also did a zoom into a mirrored menger, which was even more hypnotic, but a little too slow for the demo (4 FPS max).)
If you're interested in our RTR demo, here it is:
For best effect, run it on a machine that has a webcam (the image is mapped into the scene). Otherwise be sure to turn off 'use_vidcap' in the scene files.
With four cores I get about 10-12 FPS with rtr-menger.pov at 320x240.
Two POV-Ray RTR related benchmarking articles:
back to contents
Imagine a scene with three objects: say a room with two chairs in it, in opposite corners of the room. You want to make a bounding volume hierarchy for it in some form, with each object included only once inside the tree. In searching for a splitting plane, you find one that divides the two chairs, one from another. However, if you use the traditional method of putting objects in either one child or the other (and so not letting them attach separately to the parent node), the room itself must be included with one child or the other. So one child bounding box is as large as the parent bounding box! What good will such a child bounding box do? Intersecting it gives no additional new information, since you already know at that point that the parent bounding box is intersected.
Imagine another scene, where you have a room containing 1023 dice. At each level of the BVH, the set of objects is split into two groups. For example, from the root parent node there are two children, one containing 512 dice, the other containing 511 dice and the room object itself. This second child has a bounding box the same size as the parent's. Now split that box into two children. Again, one child will have 255 dice and the room, so the same size bounding box again. This goes on all the way down to the leaf nodes, at which point the room finally is separate from all the dice; all the room's ancestors have the same bounding box, so are a waste to intersect. For our 1023 dice room, there are then 9 of these child boxes, each a parent of the next, that must always be tested for intersection whenever a ray hits the scene's root bounding box, and never provide new information when intersected. A total waste.
The problem is that the room itself must always get inserted in one child bounding volume or another. So in this case it seems entirely justified to have the room itself (or its bounding box) be one of the two children of the root node. Say we find a better split for a parent bounding box's objects by using surface area heuristics (SAH), like Goldsmith & Salmon use, but doing top-down processing. One idea to catch such big objects in the parent is to check the relative size of any objects "split" by (i.e. overlapping) this splitting plane. An object hit by this splitting plane could be either large or small. In our room with dice example, the room object is always hit by the splitting plane, and so is large. A die hit by the splitting plane is small. If you allow child bounding boxes to overlap a bit, any dice hit by the split plane could be instead placed inside one child box or the other without seriously increasing the surface area of the child box (i.e., without increasing the chance the child box is hit).
For example, you might try a rule of checking if a "split" object would extend either new child bounding box to be, say, 10% larger in surface area than it would be without the "split" object. Any objects that could not fit in a slightly-expanded child bounding box could then be put in their own separate bounding box. All of the other objects (unsplit, or split but small) are then put in a separate bounding box, which is then divided normally, using the computed split plane. This preserves the binary tree structure of the hierarchy while also leaving larger objects in the levels of the tree closer to the root. Doing so allows the smaller objects to be efficiently bound by further bounding box subdivisions.
Root +--------------------------+ Large, Split Objects Unsplit and Smaller Split Objects +--------------------------------------+ Objects Below Split Plane(*) Objects Above Split Plane(*)
(*) plus small split objects that can fit inside the child without increasing the bounding box too much.
Havran et al. point out that Guenther and Gaede talk about the problem of different sized objects and that they propose putting such objects in the node itself as separate children, i.e. children of an internal node. This is qualitatively the same as (though probably faster than) using an unbalanced binary tree, really, as such a binary tree can "simulate" having more than two children for a parent by having an "extra" bounding box between, i.e.
A +--+--+ B C D
can be simulated by:
A +-----+ B E +---+ C D
This is essentially what we're doing here, putting large objects in B and creating node E to perform the actual split of the smaller objects.
This algorithm of checking if an object is too large and should be separated out early on works for the room of dice case: the room is noted as being quite large and is immediately made a child of the root node. It is always tested whenever the root node is intersected, which avoids wastefully intersecting 9 bounding boxes each ray.
Another attribute that does not get used by most "split in the middle" strategies is the effect of the split on the resulting child bounding boxes. For example, imagine this is the data set, each "*" being an object:
+-------------------------------+ |***************A***B | | | | | | ************| +-------------------------------+
Evaluating this set of objects along the horizontal axis by a simple split cost function would cause a split to occur around point A. What we really would prefer is a split at point B, as the two children boxes will then tightly bound the two separate lines of objects. Adding code to track the total size of the bounding volumes formed by each split plane is less additional work than it might appear. You simply need a separate pass over the data, going from right to left and adding each object in turn, storing the right-side bounding box for each potential split plane. Then when you sweep from left to right you build up the left-side bounding box incrementally. At each split plane location then tested you will have the exact left-side bounding box of the objects left of the split, and can retrieve the exact right-side bounding box, so giving a much better estimate of how good a splitting plane would perform.
My point is simply this: I believe the rules we have for forming bounding volume hierarchies are still somewhat weak and do not take full advantage of the fact that bounding volumes can overlap and can trim space in all three dimensions when splitting a set of objects in two. Splitting kd-trees optimally is well researched, with the surface area heuristic working quite well. (Though even then, theoretical and practical results can differ: Havran's thesis points out that in theory we should look at the log of the number of objects on each side of the splitting plane, since we're going to further subdivide this set of objects and so access it in a log(N) fashion. As Havran discovered, splitting works better in practice when weighted by the number of objects, not its log.) Splitting BVHs optimally, and deciding whether to allow inclusion of children at internal nodes (vs. leaf node boxes only), doesn't feel fully solved to me, or even solved just for today's current CPU architectures. The additional effect of the split along one axis and the fact that it often decreases the size of one or both child bounding boxes along the other two axes is also not accounted for in a simple split evaluation system.
Here's an example of a scene, call it the pick-up sticks scene, in which splitting planes (or, for that matter, any other scheme I know) for forming BVH's will get the wrong answer:
| | | | | | | | | | | | | | ------+---+---+---+------- | | | | -----+---+---+---+------ | | | | ------+---+---+---+------- | | | | -----+---+---+---+------ | | | | | | | | | | | | | | PWith each horizontal and vertical line representing a separate object, a "sort the centroids" splitting plane strategy might make a vertical split at location P, producing two child boxes with just about the same dimensions as the original parent box, because the horizontal objects' centroids force these objects into both boxes, forcing them to be larger. Surface areas can't really get used properly by sweeping a plane along either axis. The right answer is to group all the horizontal lines into one child box and all the verticals into the other box: both boxes overlap considerably, but both are smaller and lead to better subdivision further down the tree. How to figure this out, short of brute force exploration of all groups (for 8 objects a mere 2**7-1, 127 pairs of sets, in this case), is not clear to me.
Last issue I pointed out a bunch of work by different research groups, all on the idea of BIH; see RTNv19n1. This research is an interesting hybrid of bounding volumes and kd-trees. Havran et al. call the structure spatial kd-trees (SKD-trees), first developed by Ooi et al. in 1987. Woop calls them bounded kd-trees (B-KD trees). Wald et al. call them AH-trees. I'm choosing BIH only because it's quick to type and looks quite different.
Think about what a pair of child bounding volumes does on a single axis. A parent bounding box goes from point A to B on an axis:
A pair of children in can cause up to four more planes to become relevant for testing rays. That is, each child has two planes for its own new bounding box, so four planes are added between A and B, inclusive:
A--------------------------B | left | | right |
The left and right boxes could also overlap, of course:
A--------------------------B | left | | right |
In actuality, at the first division the two children can add only two planes total, if the parent box tightly bounds its contents and if all objects are put into one child or the other. The left box must have its left plane be at "A", the right must have its right plane at "B" (again, they could overlap):
A--------------------------B | left | | right |
If this were not the case, then the original parent bounding box could have been smaller to begin with: why would it encompass space to the left of the left box, for example, if nothing was in that space? However, the four plane system does make more sense after a few levels of BIH, as nodes can be split on other axes such that the outer two planes have an effect.
BIH and related work favor using this type of "two interior plane" approach for subdividing a scene. It's a fascinating hybrid approach, drawing on the strengths of kd-trees and bounding volumes. The split plane does not create duplicate pointers to the same (split) object, as it does with the kd-tree: split objects get put in the left or right child. What's clever about BIH is that it gets the benefit of bounding volumes by avoiding object duplication at the split and by tightly bounding each child if there are no objects that get split, ignoring the effect of how the bounding volumes shrink along the other two axes. One nice feature is that the amount of memory needed is known in advance, since no object is listed in more than one cell. This lack of duplication also means that BIH can work well for dynamic data in animated scenes, an important topic in real-time ray tracing. The traversal gets much of the benefit of kd-trees, with closest hits causing termination at some point, as there are just a few extra test cases involved. The children essentially inherit the left and right planes of the parent, so these do not need to be intersected again when each child is visited.
However, after just one BIH division, it's quite possible to have empty space between parent and child. For example, say you have this scene:
+-------------+ |PPPP OOOOO | |PPPP OOOOO | | | | | | QQQQQ RRR| | QQQQQ RRR| +-------------+
with four objects O, P, Q, R. You split along the vertical axis, forming two horizontal planes bounding the two groups:
+-------------+ |PPPP OOOOO | |PPPP OOOOO | ^-------------------^ | | | | V-------------------V | QQQQQ RRR| | QQQQQ RRR| +-------------+
Now each child will benefit from a vertical split, P separated from O and Q from R. But there is blank space that is to the right of the O object, and to the left of the Q object, that is not trimmed away by the two-plane BIH approach:
< > +----|-|--------+ |PPPP| |OOOOO | |PPPP| |OOOOO | ^-------------------^ | | | | V-------------------V | QQQQQ| |RRR| | QQQQQ| |RRR| +--------|--|---+ < >
Using four separate planes would help here, trimming away the empty space. Some research has been done with these, but overall the extra complication during testing seems to outweigh the benefits. Really, in the case above, only 3 planes for each child (not 4) give some benefit, so there seems to be a sort of "case by case basis" as to what is best.
Just to give some counterpoint to the interest in this new type of algorithm, one critique I've heard of the BIH family of techniques is that it performs well on data where the triangles are about the same size, e.g. laser-scan data, but not very well when the triangle sizes differ. This in fact ties in with the pick-up sticks and the room of dice examples: the room itself is large compared to the dice. By allowing only leaf node bounding boxes (or BIH structures) to hold objects, the room essentially makes either the left or right segment in any split to be as large as the parent. I can imagine under these circumstances that larger objects cause the overall efficiency of the hierarchy to be weakened by the "room of dice" effect.
It's obvious for any given ray tracer and scene there's an optimal BVH for it, one that runs the fastest. Finding this optimal BVH via brute force quickly becomes undoable for more than a few objects. The question I still have after all these years is, "What is the difference between this ideal optimal BVH and ones formed by the various algorithms used today?" Basically, are we close to optimal today with some particular formation and traversal algorithms? If not, how much more could be gained? How scene and data dependent are the current schemes? We've certainly learnt that grid efficiency schemes are good when the distribution of objects themselves is fairly isotropic (no clustering in a single cell, e.g. teapot in a stadium or even just high density of similar sized objects in one location). How does the distribution of object sizes affect bounding volume hierarchies and their efficiency? I suspect when objects are about the same size and orientation we do quite well achieving near-optimal BVHs. It's the pick-up sticks and room of dice scenes that can cause us problems.
Data structures like skd-trees were originally developed for sets of points, not objects with volumes. When the objects are point-like (i.e. relatively small compared to the current bounding box being subdivided), skd-trees work quite well. Where the problem gets interesting is when objects do not follow this assumption. The pick-up sticks and room of dice cases could be shrugged off as pathological, but I have to disagree. Synthetic scenes will often have simple ground planes or other large pieces of geometry around the fringe. Objects like pipes are indeed long objects that extend a considerable distance in a single direction.
Also, remember that this sort of problem can happen at any box in the hierarchy. 5 levels down you might run into miniature pick-up sticks or room with dice situations (or a hundred others; I've presented only two) that break the "point-like" assumption for your current bounding box. I've offered a possible direction for research to improve efficiency for the room of dice situation, but do not see any easy way to cope with the pick-up sticks scene efficiently at this point.
(This article continues in RTNv21n1.)
back to contents
In their excellent paper "A Cross-Platform Framework for Interactive Ray Tracing" (http://www.uni-koblenz.de/~cg/publikationen/cp_raytrace.pdf), Markus Geimer and Stefan Müller propose a branchless SIMD friendly variation of the slab test. It's extremely fast but there's a catch, under some conditions (say (box_min-pos) == 0 while inv_dir = inf) a NaN is produced and due to the way SSE min/max works, it ends up with a bogus result. Here's an attempt to fix that corner case while keeping things tight. It also works with flat voxels and take around 33 cycles on a good day (compared to the 17 cycles of the original version), if you have a decent compiler (wink wink, nudge nudge).
The code on my page is a bit ugly. The basic idea is to filter NaNs in a typical slab test made branchless with mere min[ss|ps]/max[ss|ps] due to the exotic Intel specific feature that says that for those instructions: "If a value in the second operand is an SNaN, that SNaN is returned unchanged to the destination (that is, a QNaN version of the SNaN is not returned). If only one value is a NaN (SNaN or QNaN) for this instruction, the second operand (source operand), either a NaN or a valid floating-point value, is written to the result." That's much faster than any of those ugly SSE conditional move sequences.
The scalar version is sub-optimal, I haven't put much effort into it and given the nature of SSE it's a lost cause. But it works really nicely for packets (i.e. 4 rays vs 1 box, or 2 vs 1 with doubles).
No, I don't have recent timing to back up my claims.
I contacted Geimer & Müller about this fix, they took notice and produced an Altivec version but I've never seen it. On the other hand Syoyo Fujita has published such version http://lucille.atso-net.jp/blog/?p=14 which was further refined as: http://lucille.atso-net.jp/svn/angelina/simd/raybox_intersection/
back to contents
For some time, I've been using a very fast (but statistically unsound) random number generator which I made thread-safe, and portable. It worked very well in most cases, but I started seeing various sampling artifacts when using large numbers of antialiasing samples, and particularly when also performing ambient occlusion lighting with a goodly number of samples. I suspected my RNG or at least my use of it, and indeed with some testing and using separate number streams for AO lighting, depth of field, and antialiasing samples I got better results.
After these initial experiments, I began looking into recent papers on RNGs which evaluated them based on both performance and their statistical properties. I've been fascinated both by what I learned about the statistical performance of various RNGs, and also the speed of various implementations people have come up with.
This is one of my favorite references thus far (see the paper PDF at the bottom of the page):
I've also been harvesting various usenet snippets posted by George Marsaglia, who is the author of the "Diehard" RNG test suite:
As one might guess, implementing and using the various published RNGs in practice puts a different light on them. One of the RNGs (Mersenne Twister) that is very popular for its combined statistical and computational performance characteristics when tested standalone ended up having a noticeably detrimental effect on performance when I tested it in actual use in Tachyon. This problem is presumably due to extra cache misses caused by references to an internal 624 element state vector used in the process of generating random numbers. For small test scenes, the MT RNG was faster than the others. When rendering large scenes where cache utilization really counts, MT lost as much performance relative to others as it had gained in the case of the small test scenes.
Some of the other RNGs which are very fast in many evaluations I've seen lose much of their advantage by the time you make them thread safe. Some of the most efficient formulations I've seen are implemented as preprocessor macros which access global or static data, and so they don't translate well into a multithreaded world.
After testing various RNGs for a couple of weeks now, I now see that there are even more avenues I haven't explored. There are RNGs specifically designed to generate floating point numbers, avoiding much of the usual integer arithmetic and taking pains to avoid integer to floating point conversions which are very slow on some platforms (e.g. pre-SSE x86).
Here's a nice paper about RNG methods that directly generate floating point random number values, avoiding the speed hit:
One of the other complications I've run into during testing is that when one runs a ray tracer or other code in parallel, and you want the different CPUs working with independent random number streams that aren't correlated with each other, you have to come up with a scheme to reliably seed the RNGs such that each CPU/thread/etc gets its own random number stream. In playing with the faster/simpler RNGs I know about, I discovered that some of them had big weaknesses in terms of the ease of seeding them arbitrarily. Some of them produce very bad streams with certain seeds, which showed up while testing my code on a machine with a large CPU count requiring a lot of independent streams. There are serious random number generation libraries out there that do all of this for you, but I'd previously hoped it wasn't so tricky to get it right. I'm now starting to think that it's not a coincidence that there are serious libraries for this purpose... :-) One of the libraries that some people using monte carlo methods seem to like is SPRNG:
I'm currently holding out for a simpler RNG that I can use without linking in significant extra library like SPRNG, but we'll see how it pans out.
One "cheap sleazy hack" I've done was to use different RNGs for different parts of rendering code based on the quality and relative lack of correlation of the streams needed for different algorithms.
All this has led me to wonder what RNGs other people in the rendering field have used, like, dislike, etc...
(Thread continues in RTNv21n1.)
[I asked John whether float to long conversion was still a slow operation on CPUs, and how SSE gets around this problem. - EAH]
Yes, apparently it remains problematic for the main x86 FPU due to the need to change the FPU control word, which causes a long pipeline stall. This page has a description of the problem, the new C99 'lrint' and 'lrintf' functions which are intended to address the problem, and benchmark results using gcc:
Intel added SSE instructions for integer truncation and other operations to improve performance. Agner Fog's web site has several useful performance optimization documents that contain detailed information on this. See page 117 of the "Optimizing software in C++" PDF, the "Conversions between floating point numbers and integers" section.
Also, see the "Optimizing subroutines in assembly language" PDF, section 17.5 "Converting from floating point to integer (all processors)":
While we're on these interesting subjects, here's another good read. This is an interesting page with fun low-level floating point tricks that Rick LaMont (RenderDotC) pointed out to me a while back:
back to contents
First, the final algorithm given by Pete appears as the answer to question 6.08 in the comp.graphics.algorithm FAQ and seems to be based on the newsgroup post wmEgVDG00VUtMEhUwg@andrew.cmu.edu (reproduced at http://www.math.niu.edu/~rusin/known-math/96/sph.rand)
Second, the algorithm can be easily modified to produce unit vectors uniform over the sphere and constrained to be within a cone near a given unit vector. If h is the cosine of the half-angle of the cone, then changing the cosT line to:
cosT = h + (1-h)*r
will produce unit vectors constrained to be as close to the +Z axis as desired, depending on h. Then a simple change of basis will re-orient them as needed. The nice thing about this approach is that the cost is constant regardless of how tight the cone is. Setting h=-1 samples all directions uniformly over the sphere; h=0 samples over the hemisphere; h just below 1 gives a very small, tight cone. I've used this to quickly implement simple glossy reflections with controllable fuzziness.
back to contents
I tested this book in a number of ways, since I don't have the time or energy to read it all. First I looked up spatial kd-trees (the new hot topic in ray tracing, see "BIH" elsewhere in this issue), and there were a few references scattered about, though with only a brief mention or explanation at each. With an average of two references per page, it's hard to do much more than a brief mention of most articles. The information presented was correct, but there was fairly little information on why spatial kd-trees might offer some benefit.
My favorite odd data structure, the loose octree, was not in the index. However, Thatcher Ulrich's paper about it was in the references. About half of the references point to the sections in which the reference was mentioned, a feature I like and that should become standard in all books with references. Looking at these, I found Ulrich's structure was noted as a loose quadtree instead. The useful properties of the loose octree ("instant" insertion and easy deletion) were alluded to in an exercise for the section but otherwise not mentioned. This data structure was contrasted with the MX-CIF quadtree, but to do so properly you had to flip between two sections more than 200 pages apart. This happened elsewhere in the book, sections will often reference other closely related sections that are far away. The approach of the book is to categorize all data structures into those dealing with point data, object-based and image-based image representations, intervals and small rectangles, and high-dimensional data. In practice there is a fair bit of overlap in these areas, hence the large jumps.
My next test was to look up ray tracing in the index - very little there, two brief sentences in the book itself. There is also a reference keyword index in the book, with many articles related to ray tracing listed there. This index could be useful for literature searches on a particular topic (and I did find loose octree here, once I noticed this other index existed). That said, some of the references that I consider key to the field of ray tracing efficiency structures, e.g. Goldsmith & Salmon, Kay & Kajiya, or Arvo's summary chapter in "An Introduction to Ray Tracing", are not in the set of references. Other major topics also tend to be mentioned in passing, such as GIS. The focus of the book is the data structures themselves, the fields they are used in are often ancillary. This leads to a sometimes scattered mixing of papers using similar schemes in quite different areas.
I flipped through the book reading snippets here and there. Numerous key data structures are described in detail, with variants covered. For example, classical kd-trees (used in 2D for point location) get over 40 pages of coverage, with many variants covered in other sections. I found the text to be concise, but sometimes too terse if you did not already understand the idea. For example, from page 2, a "trie" is:
a branching structure in which the value of each data item or key is treated as a sequence of j characters, where each character has M possible values. A node at depth k-1 (k>=1) in the trie represents an M- way branch depending on the value of the kth character (i.e., it has M possible values). Some (and at times all) of the data is stored in the leaf nodes, and the shape of the trie is independent of the order in which the data is processed.
I'm good with the first sentence, pretty sure about the second, but then, I admit it, this explanation mostly lost me at the leaf nodes, and there was no figure or example in the book to illustrate the concept. Which may be something of an exception, since there is on average about one figure per page in this book. Reading about tries on Wikipedia (http://en.wikipedia.org/wiki/Trie) and looking at the example there cleared up what a "trie" is (though I dislike the Wikipedia's example that makes a trie look somewhat like a binary tree), as well as explaining where the name came from.
This is perhaps an extreme example of how the book left me behind, many other parts of the book are not this opaque, but I found it disconcerting to hit such a snag on just the second page. I personally tend to best learn from a concrete example presented along with the abstract definition, so whenever these were lacking I felt in over my head.
There are many exercises in the book, though it seems a bit of a stretch to use this book as a primary textbook as it's a bit too comprehensive; the exercises could be useful for providing ideas to instructors, however. As mentioned before, there is a fair bit of cross-referencing, which is good. Also worth mentioning is that there is also an author and credit index.
This is in many ways an incredible work, it's astounding that one person could write it. It is a fine reference book for serious researchers of spatial data structures: if you want to find most or all of the work related to a particular 2D or 3D data structure, this book is the place to start for gathering references to read. If you are already well-versed in a particular data structure's uses and benefits, you may find the book's presentation useful for pointing out nuances or connections with other fields. Fields like computer graphics often gain from cross-pollination of ideas from other areas, so I can imagine a diligent reader gaining new insights into existing problems.
The breadth of the book is amazing, skipping from polygon simplification to nearest neighbor to boundary representation to discrete Fourier transforms to Voronoi diagrams, to name but a few. However, it is this breadth that ultimately weakens this book for me as a source for casual learning. There is so much to cover that much of it is explained in a lean and clipped fashion. If you already understand the data structure and area being discussed, you'll be fine, but otherwise you might slog through and learn much of the "how" but miss out on the "why". The problem is that the book wants to be comprehensive, pulling in articles from everywhere, over all time periods (computer architectures and bottlenecks change, so some techniques become dated because of this evolution). These articles often come from disparate fields with different concerns, and there is little analysis of the strengths and weaknesses of all the various alternate schemes. Nor can there be, as this would make for a book at least twice the length or more. So, there is a tension in the book of wanting to teach about a data structure, while also wishing to be comprehensive across all related fields, while also knowing there's a page limit.
It's a difficult book to just dip into: many sections build on previous sections, on down to sub-sections, e.g. Section 18.104.22.168.2. Without already knowing what was in the previous sections, it is usually difficult to skip around and sample much of the book; you have to start at major sections to know what is going on. Which is often true to some extent in many books, but this interconnectedness makes it a bit difficult to learn just the basics of a particular data structure. Most spatial data structures are separable entities, they can be explained without needing extensive background in other structures. This book is less of a reference than as an attempt to give a unified view of all these related data structures.
My overall reaction is that if I want to learn about a particular area and spatial data structure, I would instead first seek out a book that is specifically about that field and so explains the data structure's use in that context. As it is, this book is a bit like 5 different books from 5 different fields spliced together into one volume, with connections made between various sections along the way. You never quite know what the next section will bring.
These are merely my own initial impressions, based on spending just a few hours with the book. In the final analysis, I recommend you check it out yourself and form your own opinion. There is more information about the book at http://www.cs.umd.edu/~hjs/quadtree.
(Another review of the same book is in RTNv21n1.)
back to contents