[其它] Robust Adaptive Sampling For Monte-Carlo-Based Rendering

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发表于 2012-1-3 10:47:28 |只看该作者 |倒序浏览

1 Introduction

Computation times of Monte-Carlo-based rendering methods can

be lowered by correctly focusing processing power on the parts

of the image where convergence is harder to reach. This can be

done efficiently by estimating each pixel’s error, or better, using a

measure of quality of the pixel (such as entropy-based methods).

However, since these last methods do not uniformize the error over

the pixels, they are not very well suited for progressive or timeconstrained

computations.

As each pixel value is an estimation, a natural error measure is variance.

A problem is that tonemapping can make bright and dim

regions look similarly bright in the final image. Absolute variance

cannot be used directly and relative error measures should be used

instead. During Monte-Carlo rendering, error measures are often

computed from the previous xi radiance samples. Therefore, pixel

sampling depends on the error estimate, which itself depends on

pixel sampling, leading to a poor estimate of the actual error for

pixels whose initial error estimate is low.

In this work, we define a robust error estimator to obtain accurate

error estimates, additionally alternating between uniform and adaptive

sampling to consistently reduce the error estimate variance for

all pixels. We show in Figure 1 and its caption that our algorithm

is robust, and performs similarly to state-of-the-art entropy-based

methods [Xu et al. 2007], while leading naturally to uniform error

over the pixels.

2 Robust Error-Based Adaptive Sampling

Robust error estimate: A theoretical relative error measure of the

current estimate is er(Ip) = Var(Ip)=E[X]2 = Var(xi)=(Np

E[X]2), where Var(xi) is the experimental variance of the Np xi

radiance values and E[X] is the expected value of the radiance random

variable X. As the variance decreases linearly with the number

of samples, sampling according to er(Ip) tends to uniformize

the error over the pixels. E[X] being unknown, we need to estimate

it. Ip itself is the standard estimator for E[X], giving an error

fpajot,barthe,pauling@irit.fr

estimate ea(Ip) = Var(xi)=(Np I2

p ). However, when few samples

are much larger than the actual estimate – because of imperfect

importance sampling –, ea(Ip) largely under-estimates er(Ip).

Instead, we compute, for each pixel, an approximate medianMx of

the samples xi which are in a neighborhood of width h, and use it

to compute a robust error measure em(Ip) = Var(xi)=(NpM2

x).

h should be set to the width of the reconstruction filter, to naturally

handle visual edges, being caused by geometry, textures, shadows,

caustics, etc.. We compute Mx as the average of the medians computed

on small chunks of Nc elements of the sequence xi. We use

Nc = 10 to have a good estimate and a low memory overhead.

When Mx is 0, we resort to the standard ea(Ip) error measure.

Figure 1(a) shows that when increasing the number of outliers No

or their value during sampling, the ratio of errors ea(Ip)=er(Ip)

rapidly drops to zero, while our estimate remains a good approximation

of er(Ip) even for large outlier values.

Alternating between uniform and adaptive sampling: Instead of

using adaptive sampling and recomputing the probabilities every

Na samples, we alternate between adaptive sampling for Na samples,

and uniform pixel selection for Nu samples, with Nu larger

than the number of pixels in the image. The error estimates are then

updated once the Na + Nu samples have been computed. This ensures

that all error estimates receive samples, while still focusing

on pixels with larger errors. Note that alternation can be used with

any existing adaptive sampling algorithm to make it more robust.

Complete adaptive sampling algorithm: (1) a fixed number of

samples (for instance two) are shot per pixel. As we use a neighborhood

of pixels to evaluate the error at each pixel, only a few samples

are required to begin using adaptive sampling. (2) Compute error

estimates. Compute a maximum error such that 95% of the computed

errors are below. This avoids focusing processing power on

few pixels with very inaccurate and over-estimated errors. Set the

pixel probabilities accordingly to the clamped errors. (3) Compute

Na + Nu samples, using adaptive and uniform sampling. For each

sample, update the data required for the computation of the error

estimates. Loop back to step (2).

References

XU, Q., SBERT, M., XING, L., AND ZHANG, J. 2007. A novel

adaptive sampling by tsallis entropy. In CGIV ’07, 5–10.