11-23-08 - Hashing & Cache Line Size

A while ago I wrote about hash tables and reprobing and cache sizes. I mentioned Cliff Click's nonblocking hash table which is sort of interesting to me in theory but also scary and not something I really need at the moment. Anyhoo I was looking at : Some dude's C version of the lock free hash table and I noticed a little thing that's relevant to my old discussion.

He explicitly uses the cache line size and does +1 linear stepping along the cache line, then does a complex reprobe by rehashing, then does +1 again while in the cache line, then rehashes, etc. That seems like a pretty reasonable thing to do. I guess I considered that at the time, but didn't want to do it because it requires you to explicitly know the cache line size of the machine you're on. It literally requires a hard coded :

NumLinearSteps = CACHE_LINE_SIZE / sizeof(table_entry);

Some other : blogs about the lock free hash have extensive link lists.

11-22-08 - Rasterization

I just found this pretty sweet introduction article on barycentric rasterization from 2004. It's not super advanced, but it starts at the beginning and works through things and is very readable. There are some dumb things in the block checking, so if you care go to the last page and see the posts by "rarefluid".

BTW the "edge equations" are really 2d plane equations (edge cross products). Checking just the edge planes against 8x8 blocks is only a rough quick reject. You can have blocks that are right outside of one vertex at an acute corner, and those blocks are "inside" all the planes but outside of the triangle. The code they have posted also checks against the bounding box of the whole triangle which largely fixes this case. At most they will consider one extra 8x8 block which doesn't actually contain any pixels.

(it's also really not yet a full barycentric rasterizer, he's just doing the edge tests that way; from his other posts I figure he's doing interpolation using the normal homogenous way, but if you're doing the edge-tests like that then you should just go ahead and do your interpolation barycentric too).

This kind of block-based barycentric rasterizer is very similar to what hardware does. One of the nice things about it is the blocks can easily be dispatched to microthreads to rasterize in parallel, and the blocks are natural quanta to check against a coarse Z buffer.

The old Olano-Greer paper about homogenous coordinate rasterization is now online in HTML. Some other random junk I found that I think is just junk : Reducing divisions for polygon mappers & Triangle Setup .

This blog about Software occlusion culling is literally a blast from the past. As I've often suggested, if you care about that stuff, the SurRender Umbra technical manual is still the godly bible on all things occlusion. (or you can read my ancient article on it ). But I also still think that object-based occlusion like that is just a bad idea.

Coarse Z that's updated by the rasterizer is however a good thing. Doing your own on top of what the card does is pretty lame though. This is yet another awesome win from Larrabee. If we/they do a coarse Z buffer, it can get used by the CPUs to do whole-object rejections, or whole-triangle rejections, or macroblock rejections.

Apparently the guy who wrote that top article is Nicolas Capens ; he wrote "swShader" which was an open source DX9 software rasterizer, which got taken down and is now a commercial product (which was a silly thing to do, of course any customer would rather buy Pixomatic !!). I learned this from a random flame war he got in. Don't you love the internet?

11-21-08 - DXTC Part 4

So I finally implemented the end point lsqr fit from indeces thing that Simon does for myself. One thing immediately fell out - doing just 4 means and then end point fit pretty much equals the best mode of Squish. That's very cool because it's quite fast. Note that this is not doing all the searches of all possible clusterings - I just pick one clustering from 4 means and then optimize the end points for those indeces. (when I do 4 means I actually try a few different ways as I mentioned previously, I try all the permutations of putting the 4 means on the 4 palette entries, which is 4!/2 ways = 12 ways, but then I only optimize the best one of those, so it's still very fast).

One thing I noticed is that the lsqr fit really doesn't do much other than shift an end point by one. That is, the end points are in 565 already, you can do this nice optimization in floats, but when you quantize back to 565 you pretty much always hit the point you started with or at most a step of 1.

So the new "CB" modes are :

CB 1 = just 4 means then lsqr fit , faster than Squish, a bit slower than ryg. Quality is roughly competitive with Squish, but they have different strengths and weakness, so taking the best of the two might be reasonable. Squish never beats "CB 1" by a lot, but "CB 1" kills it on the weird "frymire" image.

CB 2 = CB 1 followed by simulated annealing.

CB 3 = Very slow heavy optimization. This is an attempt to see what "optimal" would get you. It tries "CB 2" and then it also tries using all 16 colors in the block as end points, so 16*16 trials, does the lsqr optimization on each trial, and then anneals the best of those. There are still various other things to try that might find a better result, but this is already pretty crazy slow. This is too slow to use in real production, the idea is simply to get an idea of how close "CB 2" is to optimal. Of course "CB 3" could still be far off optimal, I'm only conjecturing that it's close.

One of the interesting things to look at is the curve of diminishing returns from CB1 to 2 to 3. In most cases there's only a small improvement from 2 to 3, but there are exceptions, mainly in the weird degenerate images. kodim02 is one case (this is a photo but it's almost all red), and frymire of course. That meets expectations. On noisy natural images, the cluster of colors is pretty close to Gaussian noise, which works well with the PCA single line fit and the least-squares contiuous distribution approximation. On weird images with degenerate cases there can be stranger optimal solutions (for example : putting one of the color end points outside of the volume of original colors, so that one of the interpolated 1/3 colors can hit a certain value more precisely).

ADDENDUM : you might validly object and ask why the annealing is not getting closer to the global optimum. There are some approximations in the annealing that are hurting. For one thing I only try wiggling the ends by 1 step in 565. Then I don't really run it for very long, so it doesn't have a chance to make big walks and get to really different solutions. All it can really do is local optimizations with small steps to tunnel past local barriers to find better minima - it's not trying huge steps to other parts of the solution space. Theoretically if I ran a much longer annealing schedule with more time spent at higher temperatures it would do a better job of finding the global minimum. But I'm happy with this approach - the annealing is just an improved local optimization that steps bast small barriers, and to find drastically different global solutions I have to seed the trial differently.

The new numbers : (RMSE per pixel)

file	CB 1	CB 2	CB 3	Squish opt	Squish	ryg	D3DX8	FastDXT
kodim01.bmp	8.447	8.3145	8.2678	8.2829	8.3553	8.9185	9.8466	9.9565
kodim02.bmp	5.6492	5.4759	5.2864	6.1079	6.2876	6.8011	7.4308	8.456
kodim03.bmp	4.7533	4.6776	4.6591	4.7869	4.9181	5.398	6.094	6.4839
kodim04.bmp	5.5234	5.4286	5.3967	5.6978	5.8116	6.3424	7.1032	7.3189
kodim05.bmp	9.7619	9.6171	9.5654	9.6493	9.7223	10.2522	11.273	12.0156
kodim06.bmp	7.2524	7.1395	7.1086	7.15	7.2171	7.6423	8.5195	8.6202
kodim07.bmp	5.7557	5.6602	5.634	5.784	5.8834	6.3181	7.2182	7.372
kodim08.bmp	10.3879	10.2587	10.2056	10.2401	10.3212	10.8534	11.8703	12.2668
kodim09.bmp	5.3242	5.2477	5.2247	5.2935	5.3659	5.7315	6.5332	6.6716
kodim10.bmp	5.2564	5.1818	5.1657	5.2478	5.3366	5.7089	6.4601	6.4592
kodim11.bmp	6.7614	6.6503	6.6139	6.731	6.8206	7.3099	8.1056	8.2492
kodim12.bmp	4.8159	4.747	4.7308	4.7968	4.8718	5.342	6.005	6.0748
kodim13.bmp	11.0183	10.8489	10.7894	10.8684	10.9428	11.6049	12.7139	12.9978
kodim14.bmp	8.4325	8.3105	8.2723	8.3062	8.3883	8.8656	9.896	10.8481
kodim15.bmp	5.6871	5.5891	5.548	5.8304	5.9525	6.3297	7.3085	7.4932
kodim16.bmp	5.1351	5.0439	5.0274	5.065	5.1629	5.5526	6.3361	6.1592
kodim17.bmp	5.5999	5.5146	5.4976	5.509	5.6127	6.0357	6.7395	6.8989
kodim18.bmp	8.1345	8.0103	7.9791	7.9924	8.0897	8.6925	9.5357	9.7857
kodim19.bmp	6.6903	6.5979	6.5645	6.5762	6.652	7.2684	7.9229	8.0096
kodim20.bmp	5.4532	5.3825	5.3582	5.4568	5.5303	5.9087	6.4878	6.8629
kodim21.bmp	7.2207	7.1046	7.0718	7.1351	7.2045	7.6764	8.4703	8.6508
kodim22.bmp	6.52	6.4191	6.3933	6.4348	6.5127	7.0705	8.0046	7.9488
kodim23.bmp	4.9599	4.8899	4.8722	4.9063	5.0098	5.3789	6.3057	6.888
kodim24.bmp	8.5761	8.4633	8.4226	8.4299	8.5274	8.9206	9.9389	10.5156
clegg.bmp	14.8934	14.8017	14.6102	14.9736	15.2566	15.7163	21.471	32.7192
FRYMIRE.bmp	7.4898	7.3461	6.0851	10.7105	12.541	12.681	16.7308	28.9283
LENA.bmp	7.1286	7.0286	6.9928	7.1432	7.2346	7.6053	8.742	9.5143
MONARCH.bmp	6.6617	6.5616	6.5281	6.5567	6.6292	7.0313	8.1053	8.6993
PEPPERS.bmp	6.0418	5.9523	5.8026	6.4036	6.5208	6.9006	8.1855	8.8893
SAIL.bmp	8.4339	8.3077	8.2665	8.3254	8.3903	8.9823	9.7838	10.5673
SERRANO.bmp	6.7347	6.2445	5.9454	6.3524	6.757	7.0722	9.0549	18.3631
TULIPS.bmp	7.6491	7.5406	7.5065	7.5805	7.656	8.0101	9.3817	10.5873
lena512ggg.bmp	4.8961	4.8395	4.8241	4.8426	4.915	5.1986	6.0059	5.5247
lena512pink.bmp	4.6736	4.5922	4.5767	4.5878	4.6726	5.0987	5.8064	5.838
lena512pink0g.bmp	3.7992	3.7523	3.7455	3.7572	3.8058	4.2756	5.0732	4.8933
linear_ramp1.BMP	1.5607	1.348	1.3513	1.4035	2.1243	2.0939	2.6317	3.981
linear_ramp2.BMP	1.5097	1.2772	1.2772	1.3427	2.3049	1.9306	2.5396	4.0756
orange_purple.BMP	2.9842	2.9048	2.9074	2.9125	3.0685	3.2684	4.4123	7.937
pink_green.BMP	3.2837	3.2041	3.2031	3.2121	3.3679	3.7949	4.5127	7.3481
sum :	250.8565	246.2747	243.2776	252.3828	259.7416	275.5826	318.5562	370.8691

11-21-08 - More Texture Compression Nonsense

I guess the DX11 Technical Preview was just publicly released a few weeks ago. Unfortunately it's semi-crippled and still doesn't have information about BC7. From what I gather though BC7 does seem to be pretty high quality.

There're multiple different issues here. There's providing data to the card in a way that's good for texture cache usage (DXT1 is a big winner here, especially on the RSX apparently). Another is keeping data small in memory so you can hold more. Obviously that's a bigger issue on the consoles than the PC, but you always want things smaller in memory if you can. Another issue is paging off disk quickly. Smaller data loads faster, though that's not a huge issue off hard disk, and even off DVD if you're doing something like SVT you are so dominated by seek times that file sizes may not be that big. Another issue is the size of your content for transmission, either on DVD or over the net; it's nice to make your game downloadable and reasonably small.

I guess that the Xenon guys sort of encourage you to use PCT to pack your textures tiny on disk or for transmission. PCT seems to be a variant of HD-photo. MCT is I guess a DXT-recompressor, but I can't find details on it. I'm not spilling any beans that you can't find at MS Research or that end users aren't figuring out or PS3 developers are hinting at.

The "Id way" I guess is storing data on disk in JPEG, paging that, decompressing, then recompressing to DXT5-YCoCg. That has the advantage of being reasonably small on disk, which is important if you have a huge unique textured world so you have N gigs of textures. But I wonder what kind of quality they really get from that. They're using two different lossy schemes, and when you compress through two different lossy schemes the errors usually add up. I would guess that the total error from running through both compressors puts them in the ballpark of DXT1. They're using 8 bits per pixel in memory, and presumably something like 1-2 bits per pixel on disk.

Instead you could just use DXT1 , at 4 bits per pixel in memory, and do a DXT1-recompressor, which I would guess could get around 2 bits per pixel on disk. DXT1-recompressed is lower quality than JPEG, but I wonder how it compares to JPEG-then-DXT5 ?

If I ignore the specifics of the Id method or the XBox360 for the moment, the general options are :

1. Compress textures to the desired hardware-ready format (DXT1 or DXT5 or whatever) with a high quality offline compressor. Store them in memory in this format. Recompress with a shuffler/delta algorithm to make them smaller for transmisssion on disk, but don't take them out of hardware-ready format. One disadvantage of this is that if you have to support multiple hardware-ready texture formats on the PC you have to transmit them all or convert.

2. Compress texture on disk with a lossy scheme that's very tight and has good RMSE/size performance. Decompress on load and then recompress to hardware-ready format (or not, if you have enough video memory). One advantage is you can look at the user's machine and decide what hardware-ready format to use. A disadvantange is the realtime DXT compressors are much lower quality and even though they are very fast the decompress-recompress is still a pretty big CPU load for paging.

3. Hybrid of the two : use some very small format for internet transmision or DVD storage, but when you install to HD do the decompress and recompress to hardware-ready format then. Still has the problem that you're running two different lossy compressors which is evil, but reduces CPU load during game runs.

I don't think that having smaller data on disk is much of a win for paging performance. If you're paging any kind of reasonable fine-grained unit you're just so dominated by seek time, and hard disk throughput is really very fast (and it's all async anyway so the time doesn't hurt). For example a 256 x 256 texture at 4 bits per pixel is 32k bytes which loads in 0.32 millis at 100 MB/sec. (BTW that's a handy rule of thumb - 100k takes 1 milli). So the real interesting goals are : A) be small in memory so that you can have a lot of goodies on screen, and B) be small for transmission to reduce install time, download time, number of DVDs, etc.

One idea I've been tossing around is the idea of using lossy compressed textures in memory as a texture cache to avoid hitting disk. For example, a 256 X 256 texture at 1 bit per pixel is only 8k bytes. If you wanted to page textures in that unit size you would be ridiculously dominated by seek time. Instead you could just hold a bunch of them in memory and decompress to "page" them into usable formats. The disk throughput is comparable to the decompresser throughput, but not having seek times means you can decompress them on demand. I'm not convinced that this is actually a useful thing though.

BTW I also found this Old Epic page about the NVidia DXT1 problem which happily is fixed but I guess there are still loads of those old chips around; this might still make using DXT1 exclusively on the PC impossible. The page also has some good sample textures to kill your compressor with.

11-20-08 - DXTC Part 3

So we know DXT1 is bad compared to a variable bitrate coder, but it is a small block fixed rate coder, which is pretty hard to deal with.

Small block inherently gives up a lot of coding capability, because you aren't allowed to use any cross-block information. H264 and HDPhoto are both small block (4x4) but both make very heavy use of cross-block information for either context coding, DC prediction, or lapping. Even JPEG is not a true small block coder because it has side information in the Huffman table that captures whole-image statistics.

Fixed bitrate blocks inherently gives up even more. It kills your ability to do any rate-distortion type of optimization. You can't allocate bits where they're needed. You might have images with big flat sections where you are actually wasting bits (you don't need all 64 bits for a 4x4 block), and then you have other areas that desperately need a few more bits, but you can't gived them to them.

So, what if we keep ourselves constrained to the idea of a fixed size block and try to use a better coder? What is the limit on how well you can do with those constraints? I thought I'd see if I could answer that reasonably quickly.

What I made is an 8x8 pixel fixed rate coder. It has zero side information, eg. no per-image tables. (it does have about 16 constants that are used for all images). Each block is coded to a fixed bit rate. Here I'm coding to 4 bits per pixel (the same as DXT1) so that I can compare RMSE directly, which is a 32 byte block for 8x8 pixels. It also works pretty well at 24 byte blocks (which is 1 bit per byte), or 64 for high quality, etc.

This 8x8 coder does a lossless YCoCg transform and a lossy DCT. Unlike JPEG, there is no quantization, no subsampling of chroma, no huffman table, etc. Coding is via an embedded bitplane coder with zerotree-style context prediction. I haven't spent much time on this, so the coding schemes are very rough. CodeTree and CodeLinear are two different coding techniques, and neither one is ideal.

Obviously going to 8x8 instead of 4x4 is a big advantage, but it seems like a more reasonable size for future hardware. To really improve the coding significantly on 4x4 blocks you would have to start using something like VQ with a codebook which hardware people don't like.

In the table below you'll see that CodeTree and CodeLinear generally provide a nice improvement on the natural images, about 20%. In general they're pretty close to half way between DXTC and the full image coder "cbwave". They have a different kind of perceptual artifact when they have errors - unlike DXTC which just make things really blocky, these get the halo ringing artifacts like JPEG (it's inherent in truncating DCT's).

The new coders do really badly on the weird synthetic images from bragzone, like clegg, frymire and serrano. I'd have to fix that if I really cared about these things.

One thing that is encouraging is that this coder does *very* well on the simple synthetic images, like the "linear_ramp" and the "pink_green" and "orange_purple". I think these synthetic images are a lot like what game lightmaps are like, and the new schemes are near lossless on them.

BTW image compression for paging is sort of a whole other issue. For one thing, a per-image table is perfectly reasonable to have, and you could work on something like 32x32 blocks. But more important is that in the short term you still need to provide the image in DXT1 to the graphics hardware. So you either just have to page the data in DXT1 already, or you have to recompress it, and as we've seen here the "real-time" DXT1 recompressors are not high enough quality for ubiquitous use.

ADDENDUM I forgot but you may have noticed the "ryg" in this table is also not the same as previous "ryg" - I fixed a few of the little bugs and you can see the improvement here. It's still not competitive, I think there may be more errors in the best fit optimization portion of the code, but he's got that code so optimized and obfuscated I can't see what's going on.

Oh, BTW the "CB" in this table is different than the previous table; the one here uses 4-means instead of 2-means, seeded from the pca direction, and then I try using each of the 4 means as endpoints. It's still not quite as good as Squish, but it's closer. It does beat Squish on some of the more degenerate images at the bottom, such as "linear_ramp". It also beats Squish on artificial tests like images that contain only 2 colors. For example on linear_ramp2 without optimization, 4-means gets 1.5617 while Squish gets 2.3049 ; most of that difference goes away after annealing though.

RMSE per pixel :

file	Squish opt	Squish	CB opt	CB	ryg	D3DX8	FastDXT		cbwave KLTY		CodeTree		CodeLinear
kodim01.bmp	8.2808	8.3553	8.3035	8.516	8.9185	9.8466	9.9565		2.6068		5.757835		5.659023
kodim02.bmp	6.1086	6.2876	6.1159	6.25	6.8011	7.4308	8.456		1.6973		4.131007		4.144241
kodim03.bmp	4.7804	4.9181	4.7953	4.9309	5.398	6.094	6.4839		1.3405		3.369018		3.50115
kodim04.bmp	5.6913	5.8116	5.7201	5.8837	6.3424	7.1032	7.3189		1.8076		4.254454		4.174228
kodim05.bmp	9.6472	9.7223	9.6766	9.947	10.2522	11.273	12.0156		2.9739		6.556041		6.637885
kodim06.bmp	7.1472	7.2171	7.1596	7.3224	7.6423	8.5195	8.6202		2.0132		5.013081		4.858232
kodim07.bmp	5.7804	5.8834	5.7925	5.9546	6.3181	7.2182	7.372		1.4645		3.76087		3.79437
kodim08.bmp	10.2391	10.3212	10.2865	10.5499	10.8534	11.8703	12.2668		3.2936		6.861067		6.927792
kodim09.bmp	5.2871	5.3659	5.3026	5.4236	5.7315	6.5332	6.6716		1.6269		3.473094		3.479715
kodim10.bmp	5.2415	5.3366	5.2538	5.3737	5.7089	6.4601	6.4592		1.7459		3.545115		3.593297
kodim11.bmp	6.7261	6.8206	6.7409	6.9128	7.3099	8.1056	8.2492		1.8411		4.906141		4.744971
kodim12.bmp	4.7911	4.8718	4.799	4.9013	5.342	6.005	6.0748		1.5161		3.210518		3.231271
kodim13.bmp	10.8676	10.9428	10.9023	11.2169	11.6049	12.7139	12.9978		4.1355		9.044009		8.513297
kodim14.bmp	8.3034	8.3883	8.3199	8.5754	8.8656	9.896	10.8481		2.4191		6.212482		6.222196
kodim15.bmp	5.8233	5.9525	5.8432	6.0189	6.3297	7.3085	7.4932		1.6236		4.3074		4.441998
kodim16.bmp	5.0593	5.1629	5.0595	5.1637	5.5526	6.3361	6.1592		1.546		3.476671		3.333637
kodim17.bmp	5.5019	5.6127	5.51	5.6362	6.0357	6.7395	6.8989		1.7166		4.125859		4.007367
kodim18.bmp	7.9879	8.0897	8.0034	8.225	8.6925	9.5357	9.7857		2.9802		6.743892		6.376692
kodim19.bmp	6.5715	6.652	6.5961	6.7445	7.2684	7.9229	8.0096		2.0518		4.45822		4.353687
kodim20.bmp	5.4533	5.5303	5.47	5.5998	5.9087	6.4878	6.8629		1.5359		4.190565		4.154571
kodim21.bmp	7.1318	7.2045	7.1493	7.3203	7.6764	8.4703	8.6508		2.0659		5.269787		5.05321
kodim22.bmp	6.43	6.5127	6.4444	6.6185	7.0705	8.0046	7.9488		2.2574		5.217884		5.142252
kodim23.bmp	4.8995	5.0098	4.906	5.0156	5.3789	6.3057	6.888		1.3954		3.20464		3.378545
kodim24.bmp	8.4271	8.5274	8.442	8.7224	8.9206	9.9389	10.5156		2.4977		7.618436		7.389021
clegg.bmp	14.9733	15.2566	15.1516	16.0477	15.7163	21.471	32.7192		10.5426		21.797655		25.199576
FRYMIRE.bmp	10.7184	12.541	11.9631	12.9719	12.681	16.7308	28.9283		6.2394		21.543401		24.225852
LENA.bmp	7.138	7.2346	7.1691	7.3897	7.6053	8.742	9.5143		4.288		7.936599		8.465576
MONARCH.bmp	6.5526	6.6292	6.5809	6.7556	7.0313	8.1053	8.6993		1.6911		5.880189		5.915117
PEPPERS.bmp	6.3966	6.5208	6.436	6.6482	6.9006	8.1855	8.8893		2.3022		6.15367		6.228315
SAIL.bmp	8.3233	8.3903	8.3417	8.5561	8.9823	9.7838	10.5673		2.9003		6.642762		6.564393
SERRANO.bmp	6.3508	6.757	6.5572	6.991	7.0722	9.0549	18.3631		4.6489		13.516339		16.036401
TULIPS.bmp	7.5768	7.656	7.5959	7.8172	8.0101	9.3817	10.5873		2.2228		5.963537		6.384049
lena512ggg.bmp	4.8352	4.915	4.8261	4.877	5.1986	6.0059	5.5247				2.054319		2.276361
lena512pink.bmp	4.5786	4.6726	4.581	4.6863	5.0987	5.8064	5.838				3.653436		3.815336
lena512pink0g.bmp	3.7476	3.8058	3.7489	3.8034	4.2756	5.0732	4.8933				4.091045		5.587278
linear_ramp1.BMP	1.4045	2.1243	1.3741	1.6169	2.0939	2.6317	3.981				0.985808		0.984156
linear_ramp2.BMP	1.3377	2.3049	1.3021	1.5617	1.9306	2.5396	4.0756				0.628664		0.629358
orange_purple.BMP	2.9032	3.0685	2.9026	2.9653	3.2684	4.4123	7.937				1.471407		2.585087
pink_green.BMP	3.2058	3.3679	3.2	3.2569	3.7949	4.5127	7.3481				1.247967		1.726312

11-18-08 - DXTC Part 2

First of all, let's dispell the illusion that some people have that DXT1 is "pretty good". DXT1 is fucking awful. It compresses at 1.33333 bits per byte (4 bits per pixel). That's very large as far as image compressors are concerned. For typical images, around 4.0 bpb is true lossless, around 1.5 is perceptually lossless, and around 0.5 is "very good". In fact wavelet compressors can get as low as 0.1 bpb and acheive about the same quality as DXT1. Despite this I've heard smart people saying that "DXT1 is pretty good". Yes, it is a convenient fixed size block, and yes the decoder is extremely fast and simple, but as far as quality is concerned it is not even close. At 1.33333 it should be near lossless.

Here are some numbers on various DXT1 compressors. The numbers in the table are the RMSE (sqrt of the L2 error). The far right column is a wavelet compressor for comparison; it's not the best wavelet compressor in the world by a long shot, it's "cbwave" which is very old and which I designed for speed, not maximum quality. In any case it gives you an idea how far off DXT1 is. (BTW I always try to show results in RMSE because it is linear in pixel magnitude, unlike MSE or PSNR which is a very weird nonlinear scale). More discussion after the table...

(ADD: the "ryg" results shown here have a bug and should be ignored; see DXTC Post Summary )

file	Squish opt	Squish	CB opt	CB	ryg	D3DX8	FastDXT		cbwave
kodim01.bmp	8.2808	8.3553	8.352	8.6924	9.374	9.8466	9.9565		2.6068
kodim02.bmp	6.1086	6.2876	6.1287	6.3025	7.52	7.4308	8.456		1.6973
kodim03.bmp	4.7804	4.9181	4.8312	5.0225	5.855	6.094	6.4839		1.3405
kodim04.bmp	5.6913	5.8116	5.7285	5.9394	6.9408	7.1032	7.3189		1.8076
kodim05.bmp	9.6472	9.7223	9.707	10.112	10.8934	11.273	12.0156		2.9739
kodim06.bmp	7.1472	7.2171	7.1777	7.44	8.1005	8.5195	8.6202		2.0132
kodim07.bmp	5.7804	5.8834	5.8379	6.0583	6.8153	7.2182	7.372		1.4645
kodim08.bmp	10.2391	10.3212	10.346	10.747	11.3992	11.8703	12.2668		3.2936
kodim09.bmp	5.2871	5.3659	5.3306	5.5234	5.9884	6.5332	6.6716		1.6269
kodim10.bmp	5.2415	5.3366	5.2777	5.4633	5.9377	6.4601	6.4592		1.7459
kodim11.bmp	6.7261	6.8206	6.7643	7.0216	7.8221	8.1056	8.2492		1.8411
kodim12.bmp	4.7911	4.8718	4.8204	4.9863	5.6651	6.005	6.0748		1.5161
kodim13.bmp	10.8676	10.9428	10.925	11.4237	12.402	12.7139	12.9978		4.1355
kodim14.bmp	8.3034	8.3883	8.3398	8.6722	9.4258	9.896	10.8481		2.4191
kodim15.bmp	5.8233	5.9525	5.8568	6.0862	6.6749	7.3085	7.4932		1.6236
kodim16.bmp	5.0593	5.1629	5.0863	5.2851	5.8093	6.3361	6.1592		1.546
kodim17.bmp	5.5019	5.6127	5.5313	5.7358	6.4975	6.7395	6.8989		1.7166
kodim18.bmp	7.9879	8.0897	8.0192	8.3716	9.7744	9.5357	9.7857		2.9802
kodim19.bmp	6.5715	6.652	6.6692	6.91	8.0128	7.9229	8.0096		2.0518
kodim20.bmp	5.4533	5.5303	5.4895	5.6864	6.3457	6.4878	6.8629		1.5359
kodim21.bmp	7.1318	7.2045	7.1724	7.4582	8.1637	8.4703	8.6508		2.0659
kodim22.bmp	6.43	6.5127	6.4644	6.7137	7.8264	8.0046	7.9488		2.2574
kodim23.bmp	4.8995	5.0098	4.9244	5.0906	5.6989	6.3057	6.888		1.3954
kodim24.bmp	8.4271	8.5274	8.4699	8.8564	9.3906	9.9389	10.5156		2.4977
clegg.bmp	14.9733	15.2566	15.1755	15.7168	16.3563	21.471	32.7192		10.5426
FRYMIRE.bmp	10.7184	12.541	12.132	12.8278	12.989	16.7308	28.9283		6.2394
LENA.bmp	7.138	7.2346	7.1763	7.4264	8.1203	8.742	9.5143		4.288
MONARCH.bmp	6.5526	6.6292	6.5949	6.846	7.5162	8.1053	8.6993		1.6911
PEPPERS.bmp	6.3966	6.5208	6.4557	6.677	7.3618	8.1855	8.8893		2.3022
SAIL.bmp	8.3233	8.3903	8.3598	8.6627	9.8685	9.7838	10.5673		2.9003
SERRANO.bmp	6.3508	6.757	6.8385	7.9064	7.5303	9.0549	18.3631		4.6489
TULIPS.bmp	7.5768	7.656	7.6146	7.8786	8.4084	9.3817	10.5873		2.2228

Back to comparing the DXT1 encoders. BTW the test set here is the Kodak image set plus the Waterloo Bragzone image set. The Kodak set is all photographs that are pretty noisy, and there's not a huge difference in the coders. The Bragzone image set has some synthetic images with things like gradients which are harder to compress well, and there you can really dramatically see the bad encoders fall apart. In particular if you look at the results on "clegg" and "frymire" and "serrano" you can see how bad the "FastDXT" coder is.

The "Squish" in the table is the iterative cluster fit with uniform weighting. All coders work on linear RGB error; and the MSE is mean per pixel not per component.

The "CB" encoder is a simple 2-means fit. I seed the means by doing the PCA to find the best fit line. I put a plane perpendicular to that line through the average and take all the points on each half to be the two clusters, average the cluster to seed the means, and then iteratively refine the means by reclustering. Once I have the 2-means I do a simple search to find the best 565 DXT end points to find the two means. There are 3 cases to try :

1. put c0 and c1 = the two means

2. put c2 and c3 = the two means (so c0 and c1 are pushed out)

3. make (c0,c2) straddle mean 1 and (c3,c1) straddle mean 2 - this is best for gaussian clusters around the mean

A 2-means like this is slightly better than doing a pca "range fit" like Simon's fast method or the "ryg" method. If the data was all Gaussian noise, they would be equivalent, but of course it's not. You often get blocks that have a bunch of pixels at the low end which are all exactly the same color ( for example, all perfect black), and then a spread of a bunch of junk at the high end (like some orange, some yellow, etc.). You want to put one end point exactly on perfectly black and put the other endpoint at the center of the cluster on the high end.

"CB opt" and "Squish opt" take the results of the CB and Squish compressors and then improve them by iterative search. Simon Brown on his page mentions something about doing greedy endpoint refinement but claims it "doesn't converge because the indeces change". That's silly, of course it converges.

To do a greedy search : try wiggling one or both end points in 565 space. Find new best index fit for the new end points. Measure new error. If the error is lower, take the step.

Of course that works and it does converge and it's pretty simple and improves the encoding after Squish.

In fact you can do even better by doing simulated annealing instead of a plain greedy search. We should know by now that any time we have a greedy search like this where we can measure the error and it can get stuck in local minima, we can improve it with something like simulated annealing. I use 256 steps of annealing with a sinusoidal decreasing temperature pattern. I randomly pick a way to wiggle the two end points (you need to consider wiggling both, not just single wiggles). I try the wiggle and measure the error delta. Negative errors (improvements) are always taken, positive errors are taken probabilitistically based on the temperature. This is what was done in the "opt" coders above.

Most of the time the annealing search just makes a small improvement, but once in a while (such as on "frymire" and "serrano") it really finds something remarkable and makes a big difference.

Simon's cluster fit alg is very sweet. I didn't really understand it at first from the description of the code, so I'm going to redescribe it for myself here.

The basic idea is you find an assignment of indices, and from that solve a best fit to give you the end points for those indices. If you think about all the colors in the block living in 3x3 color space, assigning indices is like giving labels to each point. All the points with the same label are a "cluster". Then each cluster is fit with either an end point or one of the interpolated points.

Once you know the clusters, putting the best two end points is a simple linear optimization. You can solve the least squares analytically, it's not like a matrix iteration least squares problem or anything.

So the trick is how to decide the indices. If you tried them all, it would be 4^16 ways, which is way too many. So what Simon does is create a linear ordering of the points using the PCA axis, then try all clusterings that can be created by planes perpendicular to that linear axis. That's equivalent to just trying all groupings of the indeces sorted by their dot along the PCA axis. That is, the clusters are

[0,i) , [i,j) [j,k) [k,16)

for all i,j,k

which is a manageable amount to search, and gives you most of the interesting clusterings that you care about. Something that might improve Squish is tring a few different axes and picking the best.

BTW this end point optimization is very approximate. One issue is that it assumes the best index for each point doesn't change. It also of course just uses real number arithmetic to make the 1/3 points, not the actual integer rounding that's done. Those factors are actually pretty big.

11-18-08 - DXTC

(for links to all posts in this series see DXTC Post Summary )

I've been doing a little work on DXT1 compression. The main reason I'm looking at it is because I want to do something *better* than DXT1, and before I do that, I want to make sure I'm doing the DXTC right.

For now let's gather prior art :

The only real decision the coder gets to make is what the two 565 endpoints are. The other details you just have to get right - reconstruct the palette interpolants right, and assign the 16 indices to the closest palette entries right. For the moment I don't care about speed, I just want to make sure the indices are actually the best choices.

All the papers by Ignacio and J.M.P. van Waveren are modern classics. One thing of note : the "simplified" index finder that JMP has in his original paper is wrong. On page 12 he says "Evaluating the above expression reveals that the sub expression ( !b3 & b4 ) can be omitted because it does not significantly contribute to the final result" - apparently that is not true, because when I use the "rewritten" index finder it produces bad indexes. His original version of the index finder bit twiddle is fine.

One thing I've found with the index finding is that the integer and nonlinear effects are pretty strong and any ideas you have about planar geometry fail if you assume that your 4 points are colinear (they are not actually). For example, an obvious idea is to do a dot product test to pick between the 4 points. First dot product with the middle separating plane, then the two planes between the next pairs of points. This doesn't work. Part of the problem is because of the bad rounding in the generation of the 1/3 point. The JMP method is actually comparing vs the true distances, so that part is good.

In more detail, an obvious idea is to do something like this :

int dot = (color - c0) * (c1 - c0);

int bmid = (dot > mid01);
int b02 = (dot > mid02);
int b31 = (dot > mid31);

// using the DXT1 palette order label 0,2,3,1

int threebit = (bmid<<2) + (b02<<1) + b31;

int index = remap[threebit];

// remap is an 8-entry table that remaps from 3 bits to 2 bits to give an index

That "works" just as much as the "ryg" code works or the "Extreme DXT" method of finding indexes work - which is to say that it doesn't work.

FastDXT appears to be an implementation of the id "Real Time DXT" paper.

Ericsson Texture Compression (ETC2) is similar to DXTC but different; this is a pretty good paper, there are some interesting ideas here like the T and H blocks. It gets slightly better quality in some cases. It's obvious that you can beat DXTC by having a base color and a log-scaled delta color, rather than two end points. The two 565 end points is wasteful; you could for example do a 777 base color and a 444 log scaled delta.

Tight Frame Normal Map Compression is similar to DXT5 (really to 3Dc) with some small improvement on normal maps.

Extreme DXT Compression by Peter Uliciansky is indeed super fast. However, it has some errors. The division method that he uses to assign the indeces is not correct. You can improve it by offsetting the end points correctly to put the division quantization boundaries in the right place, but even then it's still not exactly correct unless the diagonal of the color bbox is along {1,1,1}. The error from this is pretty small and he's going for max speed, so it's semi reasonable what he does. Note that using the full range of the bbox for the division but insetting it to make the dxt1 colors (as he does) improves the indeces slightly, but it's not actually the correct range scale.

In more detail :

To find indices for the DXTC colors c0 and c1

You should use

base = c0 + (c0 - c1)/6

range = (c1 - c0) * 4/3

index = (color - base) * 4 / range;

or

index = (color - base) * 3 / (c1 - c0);

not :

index = (color - c0) * 4 / (c1 - c0);

But even if you do that it's still wrong because the planes through color space do not actually separate the colors from their closest palette entry.

MSDN compressed format documentation now has the right rounding for the 1/3 points, but the old DirectX8 docs have the wrong rounding. The old docs said

color_2 = (2*color_0 + 1*color_1 + 1)/3
color_3 = (1*color_0 + 2*color_1 + 1)/3

Which is actually better, but is not what the standard or the hardware does.

Simon Brown's Squish is very good. His Cluster Fit idea is elegant and clever and produces quite good quality.

There's a thread on MollyRocket about DXTC with some convenient code from "ryg". It's nice simple code, but it has some errors. The way he finds the indices is the dot product method which is wrong. Also the way he computes the 1/3 and 2/3 colors for the palette using LerpRGB is wrong, it rounds differently that the hardware really does.

One thing that some simple code gets wrong is the way to quantize from 24 bit to 565. You can't just take the top bits, because the 565 values are not reconstructed to the middle of their bins. You need to correct for the offsetting. You can either use the true range, which is something like -4 to 259, or you can skew the bits in the quantizer. The "ryg" code has a perfect quantizer that's very neat in the "As16Bit" function. An example is that the value 250 should map to 30 in 5 bits, not 31.

BTW the YCoCg-DXT5 method is pretty good on current hardware, but I'm not really looking at it. It's obviously inherently wasteful, it's just a way of making use of the current hardware, but it's not really what you'd want to be doing. It's also very large, at around 2.6 bits per byte (8 bits per pixel).

More advanced topics to come.

11-11-08 - REYES

Gritz' course on Renderman : To Infinity and Beyond is the best reference I've found on REYES. I'm still curious about the details of how to efficiently do the part from micropolygons to pixel colors.

The brute force way takes a micropolygon and puts a bound on it that cover its entire temporal sweep across the current frame. That is then binned into all the pixels it touches. In each pixel, 16 (or so) supersampling locations are checked with precomputed jittered sample locations and times. At each spot where a sample is found, the color & Z are added to a list for later blending. Note with fast moving objects and depth of field, it can get really inefficient, because micropolygons have to get tested in many pixels.

The Pixar Library is a nice collection of papers. There's a new paper I hadn't seen on point-based GI that's based on the Bunell disk method for realtime GI but enhanced for better quality. It seems extremely simple and the results look good. It's not super accurate in terms of solving the rendering equation with minimal error, but that's not really what we want anyway. Though actually now that I think about it it's just a variant on the ancient "render the world from each surfel's point of view" method for GI and doesn't really add much.

Anyway, back to REYES. Ignoring temporal sampling, the spatial sampling stuff can all be done with a normal rasterizer I'm pretty sure. The micropolygon subdivision and binning is just your fragment rasterizer (a rasterizer is just a device that creates micropolygons that are at most pixel size and makes a grid that's aligned with the screen grid). When you decide a fragment goes in a pel, you don't just stuff the color in the frame buffer, instead you stick it in the list to sample against the stochastic supersampled grid test just like REYES.

But it occurs to me that if you're building up a list of fragments in each pel like this, you may as well do analytic coverage than stochastic sampling. When you rasterize your triangles, compute the area of each pel that's covered. This can be done efficienctly incrementally just like a Wu antialiased line drawer for the case of only one triangle edge interesting a pel, if 2 or 3 edges intersect one pel it's more work.

Add the fragment to the pel list, consisting of {color, alpha, Z, area}. Once you have all these fragments sort them back-to-front and accumulate into the framebuffer. Fragments that have the same Z are first gathered together before being alpha-blended onto the current value in the frame buffer. That means that two triangles that share an edge act like a continuous surface and will correctly give you 100% area covers and will correctly not alpha blend with each other. This gives you exact anti-aliasing with alpha blending.

11-11-08 - Mapped Drives

Is there a way to make "Disconnected Network Drives" try to connect from DOS or from code? I'm talking about when you "net use" to make a network path to a drive letter, but you reboot or something so it's disconnected. It does a sort of lazy connect, where if you go click on it in Explorer it will hook right up, but if you go try to dir it from a CLI before you explore to it, you can't get to it.

I'd like to make my CLI file copy utils do something like "if target is a disconnect network drive, wake it up first".

---

The answer seems to rely somewhere in the functions RealDriveType, IsNetDrive, and WNetRestoreConnection.

These are functions that the DOJ forced MS to expose and the documentation is total balls. This page has better info than MSDN :

http://vbnet.mvps.org/index.html?code/network/isnetdrive.htm

I'm seeing some weird things with them though. For one thing, IsNetDrive is not a light query. It actually stalls and tries to wake up the drive right there, so most of the time all you have to do is call IsNetDrive and that actually will restore the connection. IsNetDrive can stall out for a very long time.

IsNetDrive also lies, as that page I linked details.

I haven't gotten WNetRestoreConnection to do anything useful. It always tells me "The local device name is already in use".

WNetRestoreConnection also pops up an interactive dialog box when it has errors which is fucking evil.

---

So ... in every case I've tried so far IsNetDrive actually refreshes the connection. I haven't yet found a need to actually call WNetRestoreConnection, which is good because it doesn't seem to work.

11-03-08 - Curves

I finally read Ignacio's talk on displaced subdivision with the NV geometry shader pipeline. You can find the whole talk with audio and such online but I like just the PDF . There's a lot of good stuff in there. A lot of the issues are general issues for low-high poly work, it's just that most people so far have been just ignoring the problems and creating meshes with bad discontinuities.

Ignacio's stuff is based on the Charles Loop paper "Approximating Catmull-Clark Subdivision Surfaces with Bicubic Patches". It's funny that it's really just a fancy version of the PN Triangles hack; the actual geometry surface made is not continuous, but there's a separate tangent field over it which makes the shading look smooth.

While there I noticed a bunch of Loop papers I hadn't read. The Loop-Blinn thing from 05 was particularly interesting because I've talked to a few people in the last N years about rendering fonts better. Thatcher worked on it a bit at Oddworld, and recently Sean's been doing it at RAD for our new GUI product. The Loop Blinn Paper described how to use a pixel shader to shade quadratic or cubic curves. The cubic stuff seems like rather a pain in the butt, but piecewise quadratic is certainly good enough for glyphs. Also they sort of gloss over the details of anti-aliasing, which is of course crucial; there's not much point in being this careful with curves unless you're going to get the antialiasing right.

The MDK Curvy Blues blog covers the same material and claims there are some mistakes in the Loop-Blinn paper. There's also a nice GLSL conics paper and demo app of the Loop-Blinn technique.

Zheng Qin, Michael McCool and Craig Kaplan seem to be doing a lot of work on this stuff at Waterloo. I sort of assume you're familiar with the simple Valve Alpha-test Distance Field thing ; the Qin work is the basis of the Valve technique and is more robust and general, but much more complicated. The Valve technique is probably okay in a game development scenarios where artists are monitoring how their work looks in game and are used to tweaking with weird algorithms, but it definitely has parameters that you have to tweak, and the use of single (or even two) distances can cause bad degeneracies if you're not careful, whenever you have details that are close to the size of a pixel in the distance map.