#### Sampling Theory and Aliasing

One of the best books on signal processing, sampling, reconstruction, and
the Fourier transform is Bracewell’s *The Fourier Transform and Its
Applications* (2000). Glassner’s *Principles of
Digital Image Synthesis* (1995) has a series of chapters
on the theory and application of uniform and nonuniform sampling and
reconstruction to computer graphics. For an extensive survey of the
history of and techniques for interpolation of sampled data, including the
sampling theorem, see Meijering (2002). Unser (2000) also surveyed recent
developments in sampling and reconstruction theory including the recent
move away from focusing purely on band-limited functions. For more recent
work in this area, see Eldar and Michaeli (2009).

Crow (1977) first identified aliasing as a major source of artifacts in computer-generated images. Using nonuniform sampling to turn aliasing into noise was introduced by Cook (1986) and Dippé and Wold (1985); their work was based on experiments by Yellot (1983), who investigated the distribution of photoreceptors in the eyes of monkeys. Dippé and Wold also first introduced the pixel filtering equation to graphics and developed a Poisson sample pattern with a minimum distance between samples. Lee, Redner, and Uselton (1985) developed a technique for adaptive sampling based on statistical tests that computed images to a given error tolerance. Mitchell investigated sampling patterns for ray tracing extensively. His 1987 and 1991 SIGGRAPH papers on this topic have many key insights.

Heckbert (1990a) wrote an article that explains possible pitfalls when using floating-point coordinates for pixels and develops the conventions used here.

Mitchell (1996b) investigated how much better stratified sampling patterns are than random patterns in practice. In general, the smoother the function being sampled, the more effective they are. For very quickly changing functions (e.g., pixel regions overlapped by complex geometry), sophisticated stratified patterns perform no better than unstratified random patterns. Therefore, for scenes with complex variation in the high-dimensional image function, the advantages of fancy sampling schemes compared to a simple stratified pattern are reduced.

Chiu, Shirley, and Wang (1994) suggested a *multijittered* 2D sampling
technique based on randomly shuffling the and coordinates of a
canonical jittered pattern that combines the properties of stratified and
Latin hypercube approaches. More recently, Kensler (2013) showed that
using the same permutation for both dimensions with their method gives much
better results than independent permutations; he showed that this approach
gives lower discrepancy than the Sobol pattern while
also maintaining the perceptual advantages of turning aliasing into noise
due to using jittered samples.

Lagae and Dutré (2008c) surveyed the state of the art in generating Poisson disk sample patterns and compared the quality of the point sets that various algorithms generated. Of recent work in this area, see in particular the papers by Jones (2005), Dunbar and Humphreys (2006), Wei (2008), Li et al. (2010), and Ebeida et al. (2011, 2012). We note, however, the importance of Mitchell’s (1991) observations that an -dimensional Poisson disk distribution is not the ideal one for general integration problems in graphics; while it’s useful for the projection of the first two dimensions on the image plane to have the Poisson-disk property, it’s important that the other dimensions be more widely distributed than the Poisson-disk quality alone guarantees. Recently, Reinert et al. (2015) proposed a construction for -dimensional Poisson disk samples that retain their characteristic sample separation under projection onto lower dimensional subsets.

`pbrt` doesn’t include samplers that perform adaptive sampling—taking
more samples in parts of the image with large variation. Though adaptive
sampling has been an active area of research, our experience with the
resulting algorithms has been that while most work well in some cases, few
are robust across a wide range of scenes. Since initial work in adaptive
sampling by Lee et al. (1985), Kajiya (1986), and Purgathofer (1987), a
number of sophisticated and effective adaptive sampling methods have been
developed in recent years. Notable work includes Hachisuka et al. (2008a),
who adaptively sampled in the 5D domain of image location, time, and lens
position, rather than just in image location, and introduced a novel
multidimensional filtering method; Shinya (1993) and Egan et al. (2009),
who developed adaptive sampling and reconstruction methods focused on
rendering motion blur; and Overbeck et al. (2009), who developed adaptive
sampling algorithms based on wavelets for image reconstruction. Recently,
Belcour et al. (2013) computed covariance of 5D imaging (image, time, and
lens defocus) and applied adaptive sampling and high-quality
reconstruction and Moon et al. (2014) have applied local regression
theory to this problem.

Kirk and Arvo (1991) identified a subtle problem with adaptive sampling
algorithms: in short, if a set of samples is both used to decide if more
samples should be taken and is also added to the image, the end result is
*biased* and doesn’t converge to the correct result in the limit.
Mitchell (1987) observed that standard image reconstruction techniques fail
in the presence of adaptive sampling: the contribution of a dense clump of
samples in part of the filter’s extent may incorrectly have a large effect
on the final value purely due to the number of samples taken in that
region. He described a multi-stage box filter that addresses this issue.

Compressed sensing is a recent approach to sampling where the required sampling rate depends on the sparsity of the signal, not its frequency content. Sen and Darabi (2011) applied compressed sensing to rendering, allowing them to generate high-quality images at very low sampling rates.

#### Low-Discrepancy Sampling

Shirley (1991) first introduced the use of discrepancy to evaluate the quality of sample patterns in computer graphics. This work was built upon by Mitchell (1992), Dobkin and Mitchell (1993), and Dobkin, Eppstein, and Mitchell (1996). One important observation in Dobkin et al.’s paper is that the box discrepancy measure used in this chapter and in other work that applies discrepancy to pixel sampling patterns isn’t particularly appropriate for measuring a sampling pattern’s accuracy at randomly oriented edges through a pixel and that a discrepancy measure based on random edges should be used instead. This observation explains why some theoretically good low-discrepancy patterns do not perform as well as expected when used for image sampling.

Mitchell’s first paper on discrepancy introduced the idea of using
deterministic low-discrepancy sequences for sampling, removing all
randomness in the interest of lower discrepancy (Mitchell 1992). Such
*quasi-random* sequences are the basis of quasi–Monte Carlo methods,
which will be described in Chapter 13. The seminal book on
quasi-random sampling and algorithms for generating low-discrepancy
patterns was written by Niederreiter (1992). For a more recent treatment,
see Dick and Pillichshammer’s excellent book (2010).

Faure (1992) described a deterministic approach for computing permutations
for scrambled radical inverses. The implementation
of the `ComputeRadicalInversePermutations()` function in this chapter
uses random permutations, which are simpler to implement and work nearly as
well in practice. The algorithms used for computing sample indices within
given pixels in Sections 7.4
and 7.7 were introduced by Grünschloß et al. (2012).

Keller and collaborators have investigated quasi-random sampling patterns for a
variety of applications in graphics (Keller 1996, Keller 1997, 2001). The
-sequence sampling techniques used in the
`ZeroTwoSequenceSampler` are based on a paper by Kollig and Keller (2002). -sequences are one instance of a general type of low-discrepancy
sequence known as -sequences and -nets. These are
discussed further by Niederreiter (1992) and Dick and Pillichshammer (2010). Some of Kollig and Keller’s techniques are based on algorithms
developed by Friedel and Keller (2000). Keller (2001, 2006) argued that
because low-discrepancy patterns tend to converge more quickly than others,
they are the most efficient sampling approach for generating high-quality
imagery.

The `MaxMinDistSampler` in Section 7.6 is
based on generator matrices found by Grünschloß and collaborators
(2008, 2009). Sobol (1967) introduced the family of generator matrices
used in Section 7.7; Wächter’s Ph.D. dissertation
discusses high-performance implementation of base-2 generator matrix
operations (Wächter 2008). The Sobol generator matrices our
implementation uses are improved versions derived by Joe and Kuo (2008).

#### Filtering and Reconstruction

Cook (1986) first introduced the Gaussian filter to graphics.
Mitchell and Netravali (1988) investigated a family of filters using
experiments with human observers to find the most effective ones; the
`MitchellFilter` in this chapter is the one they chose as the
best. Kajiya and Ullner (1981) investigated image
filtering methods that account for the effect of the reconstruction
characteristics of Gaussian falloff from pixels in CRTs,
and, more recently, Betrisey et al. (2000) described Microsoft’s ClearType
technology for display of text on LCDs. Alim (2013) has recently applied
reconstruction techniques that attempt to minimize the error between the
reconstructed image and the original continuous image, even in the presence
of discontinuities.

There has been quite a bit of research into reconstruction filters for image resampling applications. Although this application is not the same as reconstructing nonuniform samples for image synthesis, much of this experience is applicable. Turkowski (1990a) reported that the Lanczos windowed sinc filter gives the best results of a number of filters for image resampling. Meijering et al. (1999) tested a variety of filters for image resampling by applying a series of transformations to images such that if perfect resampling had been done the final image would be the same as the original. They also found that the Lanczos window performed well (as did a few others) and that truncating the sinc without a window gave some of the worst results. Other work in this area includes papers by Möller et al. (1997) and Machiraju and Yagel (1996).

Even with a fixed sampling rate, clever reconstruction algorithms can be useful to improve image quality. See, for example, Reshetov (2009), who used image gradients to find edges across multiple pixels to estimate pixel coverage for antialiasing and Guertin et al. (2014), who developed a filtering approach for motion blur.

Lee and Redner (1990) first suggested using a median filter, where the median of a set of samples is used to find each pixel’s value, as a noise reduction technique. More recently, Lehtinen et al. (2011, 2012), Kalantari and Sen (2013), Rousselle et al. (2012, 2013), Delbracio et al. (2014), Munkberg et al. (2014), and Bauszat et al. (2015) have developed filtering techniques to reduce noise in images rendered using Monte Carlo algorithms. Kalantari et al. (2015) applied machine learning to the problem of finding effective denoising filters and demonstrated impressive results.

Jensen and Christensen (1995) observed that it can be more effective to
separate out the contributions to pixel values based on the type of
illumination they represent; low-frequency indirect illumination can be
filtered differently from high-frequency direct illumination, thus reducing
noise in the final image. They developed an effective filtering technique
based on this observation. An improvement to this approach was developed by
Keller and collaborators with the *discontinuity buffer* (Keller 1998;
Wald et al. 2002). In addition to filtering slowly varying quantities
like indirect illumination separately from more quickly varying
quantities like surface reflectance, the discontinuity buffer uses
geometric quantities like the surface normal at nearby pixels to determine
whether their corresponding values can be reasonably included at the
current pixel. Kontkanen et al. (2004) built on these approaches to
build a filtering approach for indirect illumination when using the
irradiance caching algorithm.

Lessig et al. (2014) proposed a general framework for constructing quadrature rules
tailored to specific integration problems such as stochastic ray tracing,
spherical harmonics projection, and scattering by surfaces. When targeting
band-limited functions, their approach subsumes the frequency-space approach
presented in this chapter. An excellent tutorial about the underlying theory of
*reproducing kernel bases* is provided in the article’s supplemental
material.

#### Perceptual Issues

A number of different approaches have been developed for mapping out-of-gamut colors to the displayable range; see Rougeron and Péroche’s survey article for discussion of this issue and references to various approaches (Rougeron and Péroche 1998). This topic was also covered by Hall (1989).

Tone reproduction—algorithms for displaying high-dynamic-range images on low-dynamic-range display devices—became an active area of research starting with the work of Tumblin and Rushmeier (1993). The survey article of Devlin et al. (2002) summarizes most of the work in this area through 2002, giving pointers to the original papers. See Reinhard et al.’s book (2010) on high dynamic range imaging, which includes comprehensive coverage of this topic through 2010. More recently, Reinhard et al. (2012) have developed tone reproduction algorithms that consider both accurate brightness and color reproduction together, also accounting for the display and viewing environment.

The human visual system generally causes the brain to perceive that
surfaces have the color of the underlying surface, regardless of the
illumination spectrum—for example, white paper is perceived to be white,
even under the yellow-ish illumination of an incandescent lightbulb. A
number of methods have been developed to process photographs to perform
*white balancing* to eliminate the tinge of light source colors; see
Gijsenij et al. (2011) for a survey. White balancing is challenging, since
the only information available to white balancing algorithms is the final
pixel values. In a renderer, the problem is easier, as information is
available directly about the light sources and the surface reflection
properties; Wilkie and Weidlich (2009) developed an efficient method to
perform accurate white balancing in a renderer with limited computational
overhead.

For background information on properties of the human visual system, Wandell’s book on vision is an excellent starting point (Wandell 1995). Ferwerda (2001) presented an overview of the human visual system for applications in graphics, and Malacara (2002) gave a concise overview of color theory and basic properties of how the human visual system processes color.

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