Page 222 - Contributed Paper Session (CPS) - Volume 4

CPS2201 Mikhail L.
ℱ is the set of all functions of 4 arguments on the 3-dimensional unit sphere
4
in ℝ .
4
3
In order to evaluate the influence of measurement errors and to be able to
measure the quality of fit of BRDF models, one needs a ”measure of distance”
between BRDFs. There are many choices of distances and quasi-distances
available: , 1 ≤ < +∞, , Sobolev distances, Kullback-Leibler
∞

information divergence Kullback and Leibler (1951), Mahalanobis (1936), chi-
squared distance used in correspondence analysis Langovaya et al. (2013). In
computer science literature on BRDFs, there are few papers that study the
quality of fit of BRDF models to real data. Most of these studies use the (most
standard) −norm. An alternative approach was taken in Langovoy et al.
2
(2014), where a perception-inspired quasi-metric for the space of BRDFs was
proposed.

4. Active manifold learning strategies
In BRDF sampling, the equispaced-angular grid pictured in Figure 1(a) is
the standard. However, as was shown in Langovoy et al. (2016), this choice of
measurement points leads to very inefficient sampling. Another strategy is in
using uniformly distributed points on a sphere, see Figure 1(c). Since it was
already understood in the community (see Höpe and Hauer (2010)) that the
standard grid is suboptimal, there were multiple heuristic attempts to propose
trickier grids that better reflect the typical structure of BRDF models. A good
example is shown in Figure 1(b). Ideally, the main goal of this research is to
find the best sampling strategy; this strategy has to retain its optimality at least
for a class of reasonable criteria, and for a sufficiently general classes of both
BRDFs as well as of estimating procedures.
On the other hand, any result showing that new strategy is better than the
default strategy, at least for one specific loss function, for one specific BRDF,
and one specific estimating procedure, is already instrumental in
understanding the general picture of learning BRDF manifolds from scarce
expensive data. This basic case is straightforwardly formulated in the language
of mathematical optimization, so we are able to obtain theoretical guarantees
on learning accuracy, at least for some special cases. Let us outline a possible
mathematical framework for BRDF sampling, in a basic case to begin with.
Consider BRDF ∈ ℱ . Suppose that is measured on the finite set
4
Ω ()

(4) Ω () = {( () , () , () , () ) |( () , () ) ∈ Ω , ( () , () ) ∈ Ω ()},

where

(5) = |Ω ()| = ∑|Ω ()|.
=1

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