Efficient and accurate BRDF acquisition of real world materials is a
challenging research problem that requires sampling millions of incident light
and viewing directions. To accelerate the acquisition process, one needs to
find a minimal set of sampling directions such that the recovery of the full
BRDF is accurate and robust given such samples. In this paper, we formulate
BRDF acquisition as a compressed sensing problem, where the sensing operator is
one that performs sub-sampling of the BRDF signal according to a set of optimal
sample directions. To solve this problem, we propose the Fast and Robust
Optimal Sampling Technique (FROST) for designing a provably optimal
sub-sampling operator that places light-view samples such that the recovery
error is minimized. FROST casts the problem of designing an optimal
sub-sampling operator for compressed sensing into a sparse representation
formulation under the Multiple Measurement Vector (MMV) signal model. The
proposed reformulation is exact, i.e. without any approximations, hence it
converts an intractable combinatorial problem into one that can be solved with
standard optimization techniques. As a result, FROST is accompanied by strong
theoretical guarantees from the field of compressed sensing. We perform a
thorough analysis of FROST-BRDF using a 10-fold cross-validation with publicly
available BRDF datasets and show significant advantages compared to the
state-of-the-art with respect to reconstruction quality. Finally, FROST is
simple, both conceptually and in terms of implementation, it produces
consistent results at each run, and it is at least two orders of magnitude
faster than the prior art.

通过压缩感知技术中的稀疏表示方法，本论文提出了 Fast and Robust Optimal Sampling Technique (FROST) 用于设计一种能够以最小化恢复误差的方式放置采样方向的压缩采样算子，从而加速 BRDF 材料的获取，具有从压缩感知领域得到的强理论保证和与现有方法相比的显著优势。

FROST-BRDF: 快速且稳健的 BRDF 采样技术

FROST-BRDF: A Fast and Robust Optimal Sampling Technique for BRDF  Acquisition

We investigate the quantitative performance of affine-equivariant estimators
for robust mean estimation. As a natural stability requirement, the
construction of such affine-equivariant estimators has been extensively studied
in the statistics literature. We quantitatively evaluate these estimators under
two outlier models which have been the subject of much recent work: the
heavy-tailed and adversarial corruption settings. We establish lower bounds
which show that affine-equivariance induces a strict degradation in recovery
error with quantitative rates degrading by a factor of $\sqrt{d}$ in both
settings. We find that classical estimators such as the Tukey median (Tukey
'75) and Stahel-Donoho estimator (Stahel '81 and Donoho '82) are either
quantitatively sub-optimal even within the class of affine-equivariant
estimators or lack any quantitative guarantees. On the other hand, recent
estimators with strong quantitative guarantees are not affine-equivariant or
require additional distributional assumptions to achieve it. We remedy this by
constructing a new affine-equivariant estimator which nearly matches our lower
bound. Our estimator is based on a novel notion of a high-dimensional median
which may be of independent interest. Notably, our results are applicable more
broadly to any estimator whose performance is evaluated in the Mahalanobis norm
which, for affine-equivariant estimators, corresponds to an evaluation in
Euclidean norm on isotropic distributions.

我们研究了鲁棒均值估计的仿射等价估计器的定量性能，证明仿射等价性会导致恢复误差以一定的因子（sqrt (d)）定量恶化，并构建了一种新的近似最优的仿射等价估计器，该估计器基于一种高维中位数的新概念。