(论文)[2020] Path Differential-Informed Stratified MCMC and Adaptive Forward Path Sampling

Posted on 2021-07-06 In CG.Paper Views:

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Tobias Zirr and Carsten Dachsbacher. 2020. Path Differential-Informed Stratified MCMC and Adaptive Forward Path Sampling. ACM Trans. Graph. 39,6, Article 246 (December 2020), 19 pages. https://doi.org/10.1145/3414685.3417856
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Path Differential-Informed Stratified MCMC and Adaptive Forward Path Sampling

we apply our theoretical framework to optimize an adaptive MCMC algorithm that only uses forward path construction.

MLT：Markov Chain Monte Carlo (MCMC) light transport
MCMC 在工业界用的比较少
- The quality of results is oftentimes hard to control and nonuniform.
对给出路径的方差估计
- We derive a novel analytic function that conservatively estimates variance caused by a given path sample.
As a proof of concept, we apply the gained insights to both stratification and optimal adaptive sampling in the context of MCMC:
- We stratify the target function of the Markov Chain (akin to Hoberock et al. [2010]) using our analytic variance bounds.
- We optimize the shape of proposal distributions in path space MCMC and by this push the capabilities of forward path tracing, to compete with other bi-directional methods.
- We complement the resulting small step based on forward path construction by embedding a full path tracer with minimal changes, importing established MC variance reduction and sample reuse, such as multiple importance sampling (MIS), next event estimation (NEE), and shared path prefixes.
Our MCMC renderer effectively becomes an adaptive path tracer, with the usual benefits of reduced algorithmic complexity, avoiding efforts to ensure reciprocity (e.g. for shading normals), and focussed sampling efforts due to tracing from the camera.

The goal of MC and MCMC rendering methods is to compute the light transported on all possible paths \(X=(\mathrm{x}_1,\cdots,\mathrm{x}_k\)) (for all vertex counts \(k\ge2\)) in a path space \(\Omega\) to the individual pixels of a sensor.
light trasport
For a given path \(X\) the measurement contribution \(f(X)\) [Veach 1998] (w.r.t. the area measure \(\mathrm{d}A\) on all scene surfaces M) is evaluated and the measurement for the j-th pixel with sensitivity \(h_j(X)\) is computed by a path integral in the rendering equation

\[ I_j=\int_{\Omega}h_j(X)f(X)\;\mathrm{dX},\;\; \mathrm{dX}=\prod_{i=1}^k\mathrm{dA_i} \]

随机抽样积分法
MC（蒙特卡洛积分）
- stochastic path sampling
- good importance sampling
  - in order to sample high-throughput paths frequently and to reduce variance of the result
- Multiple Importance Sampling (MIS) [Veach and Guibas 1995]
MCMC
- MCMC rendering algorithms, introduced as Metropolis Light Transport (MLT) [Veach and Guibas 1997], make use of the Metropolis-Hastings algorithm resp. the generalizing Reversible Jump (RJ) framework [Geyer 2003; Green 1995, 2003] to enable correlated path sampling while maintaining a controlled distribution that allows MC integration.
- MCMC + MIS（减小方差）
- Primary Sample Space MLT (PSSMLT) [Kelemen et al. 2002]

Stratification, the targeted distribution of samples across pixels and into regions (strata) with potential relevance to the image, is an important feature of MC estimators to accelerate the convergence of MC estimates.

Zsolnai and Szirmay-Kalos [2013] propose scene-wide automatic parameter tuning, which by its nature has limited local adaptivity.