Graphics Reference
In-Depth Information
plied. Figure 10.15 shows real-time rendering results using exit transfer matrices
containing subsurface scattering.
Sloan considered the 2003 paper, which dramatically improved the efficiency
and versatility of PRT rendering, as the tipping point of PRT research in terms of
its use in practical real-time rendering. Of course, as long as SH basis functions
are being used, high-frequency lighting and reflection remains a problem. Other
researchers were simultaneously developing methods to represent high-frequency
lighting where SH basis functions worked poorly.
10.2.4 Adaption to High-Frequency Lighting
As has been noted repeatedly, spherical harmonics have some serious drawbacks
when it comes to approximating real-world lighting. One problem arises at dis-
continuities, which occur at the edges of light sources and at shadow boundaries.
A discontinuity has an “infinite” frequency, in the sense that infinitely many SH
terms are needed to capture the discontinuity. A finite series approximating a
discontinuity usually exhibits “ringing” artifacts as described in Chapter 7. Ring-
ing is caused in part by the global nature of the SH basis functions: each basis
function covers the entire sphere, and therefore represents detail across the en-
tire sphere. In a convergent expansion, the SH coefficients approach zero as the
order increases; however, all the terms up to a certain degree are normally re-
quired to obtain a particular accuracy. Such problems with spherical harmonics
led researchers to investigate other basis functions for representing lighting and
reflection, particularly in high-frequency environments. A natural candidate were
wavelets
, which had been used for years in other areas of computer graphics.
Loosely speaking, wavelets are oscillatory functions of arbitrary frequency
that are nonzero only on a small part of the domain. The subset of the domain of a
function (including the boundary points) on which a function is nonzero is called
the
support
of the function. Wavelet functions are said to have
local
or
compact
support
—they are only nonzero on a closed bounded subset of the domain.
3
A
wavelet basis typically consists of a set of wavelet functions having shrinking
support: as the index increases, the region in which the basis function is nonzero
decreases. A smaller support means that the oscillation of the function occurs in
a smaller region, i.e., has high frequency. In this sense the decreasing support is
analogous to increasing frequency.
A wavelet basis is typically constructed from a “mother” wavelet function;
the remaining elements come from scaled and translated copies of this function.
3
The compact support property is not strictly necessary; it is just useful. Wavelets derived from
Gaussian functions are nonzero across the entire domain, but the values decay so quickly that they are
essentially zero far enough away from the peak.
