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