Kernel agnostic mcmc from deformable beta splatting

[RongLiu-Leo]

I’d like to begin by expressing our gratitude to the gsplat framework—our Deformable Beta Splatting (DBS) core is built on top of it. In developing DBS, we’ve taken care to introduce only minimal changes, so we hope our method can seamlessly contribute back to the gsplat repo.

I am initiating this PR to see the possibility of merging DBS into the main branch. This PR implements Section 3.4: Kernel-Agnostic Markov Chain Monte Carlo from the paper.
Theoretically, when primitive opacities are small, simply adjusting opacity guarantees distribution-preserved densification—regardless of the splatting kernel’s shape or the number of densification numbers.
Empirically, setting $\lambda_o = 0.01$ yields primitive opacities $o \sim \mathcal{N}(0.01, 0.03^2)$, which validates our small-opacity assumption.

This PR should reduce the training time and mem cost, and facilitate future arbitrary kernel explorations.

However, one unexpected observation is that removing the scale-adjustment step and eliminating binomial storage did not reduce training time or memory usage. I also experimented with a CUDA implementation that omits scale adjustment, but observed the same runtime and memory profile.

This needs further investigation.

Original Benchmark

Eval Stats (step 29999)

Scene	psnr	ssim	lpips	ellipse_time	num_GS
garden	27.30	0.855	0.104	0.03	1000000
bicycle	25.57	0.773	0.184	0.04	1000000
stump	26.99	0.789	0.162	0.05	1000000
bonsai	32.76	0.953	0.106	0.04	1000000
counter	29.43	0.924	0.131	0.05	1000000
kitchen	31.67	0.936	0.087	0.04	1000000
room	32.47	0.938	0.132	0.04	1000000
Mean	29.46	0.881	0.129	0.04	1000000

Train Stats (step 29999)

Scene	mem	ellipse_time	num_GS
garden	1.30	429.19	1000000
bicycle	1.32	416.98	1000000
stump	1.31	417.77	1000000
bonsai	1.38	571.39	1000000
counter	1.47	640.24	1000000
kitchen	1.39	578.49	1000000
room	1.42	574.03	1000000
Mean	1.37	518.30	1000000

This PR

Eval Stats (step 29999)

Scene	psnr	ssim	lpips	ellipse_time	num_GS
garden	27.30	0.854	0.105	0.04	1000000
bicycle	25.60	0.774	0.185	0.04	1000000
stump	27.01	0.790	0.164	0.05	1000000
bonsai	32.67	0.953	0.106	0.04	1000000
counter	29.51	0.924	0.131	0.04	1000000
kitchen	31.68	0.935	0.087	0.04	1000000
room	32.44	0.938	0.132	0.03	1000000
Mean	29.46	0.881	0.130	0.04	1000000

Train Stats (step 29999)

Scene	mem	ellipse_time	num_GS
garden	1.30	446.47	1000000
bicycle	1.32	433.73	1000000
stump	1.32	429.02	1000000
bonsai	1.39	573.84	1000000
counter	1.46	638.57	1000000
kitchen	1.39	586.67	1000000
room	1.44	649.40	1000000
Mean	1.38	536.82	1000000

[RongLiu-Leo]

Another practice is that only do the opacity regularization during densification to make distribution preserved. Otherwise the scene looks too transparent at the end. Cancelling this regularization won't hurt the metrics but produces more visual appealing results.

Change

if cfg.opacity_reg > 0.0:
        loss = (
            loss
            + cfg.opacity_reg
            * torch.abs(torch.sigmoid(self.splats["opacities"])).mean()
        )

to something like

if cfg.opacity_reg > 0.0 and iteration < cfg.strategy.refine_stop_iter:
        loss = (
            loss
            + cfg.opacity_reg
            * torch.abs(torch.sigmoid(self.splats["opacities"])).mean()
        )

[liruilong940607]

I like that it simplify the MCMC and still keep the same metrics!!!

But hurting runtime is less ideal. If we can get at least same runtime I would be happy to merge it.