The cone-tracing method of Amanatides (1984) was one of the first techniques for automatically estimating filter footprints for ray tracing. The beam-tracing algorithm of Heckbert and Hanrahan (1984) was another early extension of ray tracing to incorporate an area associated with each image sample rather than just an infinitesimal ray. The pencil-tracing method of Shinya, Takahashi, and Naito (1987) is another approach to this problem. Other related work on the topic of associating areas or footprints with rays includes Mitchell and Hanrahan’s paper (1992) on rendering caustics and Turkowski’s technical report (1993).

Collins (1994) estimated the ray footprint by keeping a tree of all rays traced from a given camera ray, examining corresponding rays at the same level and position. The ray differentials used in pbrt are based on Igehy’s (1999) formulation, which was extended by Suykens and Willems (2001) to handle glossy reflection in addition to perfect specular reflection. Worley’s chapter in Texturing and Modeling (Ebert et al. 2003) on computing differentials for filter regions presents an approach similar to ours. See Elek et al. (2014) for an extension of ray differentials to include wavelength, which can improve results with full-spectral rendering.

Two-dimensional texture mapping with images was first introduced to graphics by Blinn and Newell (1976). Ever since Crow (1977) identified aliasing as the source of many errors in images in graphics, quite a bit of work has been done to find efficient and effective ways of antialiasing image maps. Dungan, Stenger, and Sutty (1978) were the first to suggest creating a pyramid of prefiltered texture images; they used the nearest texture sample at the appropriate level when looking up texture values, using supersampling in screen space to antialias the result. Feibush, Levoy, and Cook (1980) investigated a spatially varying filter function, rather than a simple box filter. (Blinn and Newell were aware of Crow’s results and used a box filter for their textures.)

Williams (1983) used a MIP map image pyramid for texture filtering with trilinear interpolation. Shortly thereafter, Crow (1984) introduced summed area tables, which make it possible to efficiently filter over axis-aligned rectangular regions of texture space. Summed area tables handle anisotropy better than Williams’s method, although only for primarily axis-aligned filter regions. Heckbert (1986) wrote a good general survey of texture mapping algorithms through the mid-1980s.

Greene and Heckbert (1986) originally developed the elliptically weighted average technique, and Heckbert’s master’s thesis (1989) put the method on a solid theoretical footing. Fournier and Fiume (1988) developed an even higher quality texture filtering method that focuses on using a bounded amount of computation per lookup. Nonetheless, their method appears to be less efficient than EWA overall. Lansdale’s master’s thesis (1991) has an extensive description of EWA and Fournier and Fiume’s method, including implementation details.

More recently, a number of researchers have investigated generalizing Williams’s original method using a series of trilinear MIP map samples in an effort to increase quality without having to pay the price for the general EWA algorithm. By taking multiple samples from the MIP map, anisotropy is handled well while preserving the computational efficiency. Examples include Barkans’s (1997) description of texture filtering in the Talisman architecture, McCormack et al.’s (1999) Feline method, and Cant and Shrubsole’s (2000) technique. Manson and Schaefer (2013, 2014) have recently shown how to accurately approximate a variety of filter functions with a fixed small number of bilinearly interpolated sample values. This approach is particularly useful on GPUs, where hardware-accelerated bilinear interpolation is available.

Gamma correction has a long history in computer graphics. Poynton (2002a, 2002b) has written comprehensive FAQs on issues related to color representation and gamma correction. Most modern displays are based on the sRGB color space, which has a gamma of roughly 2.2 (International Electrotechnical Commission (IEC) 1999). See Gritz and d’Eon (2008) for a detailed discussion of the implications of gamma correction for rendering and how to correctly account for it in rendering systems.

Smith’s (2002) Web site and document on audio resampling gives a good overview of resampling signals in one dimension. Heckbert’s (1989) zoom source code is the canonical reference for image resampling. His implementation carefully avoids feedback without using auxiliary storage, unlike ours in this chapter, which allocates additional temporary buffer space to do so.

Three-dimensional solid texturing was originally developed by Gardner (1984, 1985), Perlin (1985a), and Peachey (1985). Norton, Rockwood, and Skolmoski (1982) developed the clamping method that is widely used for antialiasing textures based on solid texturing. The general idea of procedural texturing was introduced by Cook (1984), Perlin (1985a), and Peachey (1985).

Peachey’s chapter in Texturing and Modeling (Ebert et al. 2003) has a thorough summary of approaches to noise functions. After Perlin’s original noise function, both Lewis (1989) and van Wijk (1991) developed alternatives that made different time/quality trade-offs. Worley (1996) has developed a quite different noise function for procedural texturing that is well suited for cellular and organic patterns. Perlin (2002) revised his noise function to correct a number of subtle shortcomings.

Noise functions have received additional attention from the research community in recent years. (Lagae et al. (2010) have a good survey of work up to that year.) Building on Lewis’s observation that individual bands of Perlin’s noise function actually have frequency content over a fairly wide range (Lewis 1989), Cook and DeRose (2005) also identified the problem that 2D slices through 3D noise functions aren’t in general band limited, even if the original 3D noise function is. They proposed a new noise function that addresses both of these issues. Goldberg et al. (2008) developed a noise function that makes efficient anisotropic filtering possible, leading to higher quality results than just applying the clamping approach for antialiasing. Their method is also well suited to programmable graphics hardware. Kensler et al. (2008) suggested a number of improvements to Perlin’s revised noise function.

Lagae et al. (2009) have developed a noise function that has good frequency control and can be mapped well to surfaces even without a surface parameterization. Lagae and Drettakis (2011) showed how to compute high quality anisotropically filtered values of this noise function. More recently, Galerne et al. (2012) showed how to automatically determine parameters to this noise function so that the result matches example images. Further work on this topic was done by Du et al. (2013) and Gilet et al. (2014).

The first languages and systems that supported the idea of user-supplied procedural shaders were developed by Cook (1984) and Perlin (1985a). (The texture composition model in this chapter is similar to Cook’s shade trees.) The RenderMan shading language, described in a paper by Hanrahan and Lawson (1990), remains the classic shading language in graphics, though a more modern shading language is available in Open Shading Language (OSL) (Gritz et al. 2010), which is open source and increasingly used for production rendering. It follows the model of the shader returning a representation of the material rather than a final color value, like the approach introduced in Chapter 9. See also Karrenberg et al. (2010), who introduced the AnySL shading language, which was designed for both high performance as well as portability across multiple rendering systems (including pbrt).

See Ebert et al. (2003) and Apodaca and Gritz (2000) for techniques for writing procedural shaders; both of those have excellent discussions of issues related to antialiasing in procedural shaders.

The “Further Reading” section in Chapter 9 described approaches for anti-aliasing bump maps; a number of researchers have looked at the closely related issue of antialiasing surface reflection functions. Van Horn and Turk (2008) developed an approach to automatically generate MIP maps of reflection functions that represent the characteristics of shaders over finite areas in order to antialias them. Bruneton and Neyret (2012) surveyed the state of the art in this area, and Jarabo et al. (2014b) also considered perceptual issues related to filtering inputs to these functions. See also Heitz et al. (2014) for recent work on this topic.

Many creative methods for computing texture on surfaces have been developed. A sampling of our favorites includes reaction diffusion, which simulates growth processes based on a model of chemical interactions over surfaces and was simultaneously introduced by Turk (1991) and Witkin and Kass (1991); Sims’s (1991) genetic algorithm-based approach, which finds programs that generate interesting textures through random mutations from which users select their favorites; Fleischer et al.’s (1995) cellular texturing algorithms that generate geometrically accurate scales and spike features on surfaces; and Dorsey et al.’s (1996) flow simulations that model the effect of weathering on buildings and encode the results in image maps that stored the relative wetness, dirtiness, and so on, at points on the surfaces of structures. Porumbescu et al. (2005) developed shell maps, which make it possible to map geometric objects onto a surface in the manner of texture mapping.

A variety of texture synthesis algorithms have been developed in the past decade; these approaches take an example texture image and then synthesize larger texture maps that appear similar to the original texture while not being exactly the same. The survey article by Wei et al. (2009) describes work in this area through 2009 as well as the main approaches that have been developed so far. For more recent work in this area, see Kim et al. (2012), who developed an effective approach based on finding symmetries in textures, and Lefebvre et al. (2010), who attacked the specialized (but useful) problem of synthesizing textures for building facades.

References

Amanatides, J. 1984. Ray tracing with cones. Computer Graphics (SIGGRAPH ’84 Proceedings), 18, 129–35.
Apodaca, A. A., and L. Gritz. 2000. Advanced RenderMan: Creating CGI for Motion Pictures. San Francisco: Morgan Kaufmann.
Barkans, A. C. 1997. High-quality rendering using the Talisman architecture. In 1997 SIGGRAPH/Eurographics Workshop on Graphics Hardware, 79–88.
Blinn, J. F., and M. E. Newell. 1976. Texture and reflection in computer generated images. Communications of the ACM 19, 542–46.
Bruneton, E., and F. Neyret. A survey of nonlinear prefiltering methods for efficient and accurate surface shading. IEEE Transactions on Visualization and Computer Graphics 18 (2), 242–60.
Cant, R. J., and P. A. Shrubsole 2000. Texture potential MIP mapping, a new high-quality texture antialiasing algorithm. ACM Transactions on Graphics 19 (3), 164–84.
Collins, S. 1994. Adaptive splatting for specular to diffuse light transport. In Fifth Eurographics Workshop on Rendering, Darmstadt, Germany, 119–35.
Cook, R. L. 1984. Shade trees. Computer Graphics (SIGGRAPH ’84 Proceedings), 18, 223–31.
Cook, R., and T. DeRose. 2005. Wavelet noise. ACM Transactions on Graphics (Proceedings of SIGGRAPH 2005) 24 (3), 803–11.
Crow, F. C. 1977. The aliasing problem in computer-generated shaded images. Communications of the ACM 20 (11), 799–805.
Crow, F. C. 1984. Summed-area tables for texture mapping. Computer Graphics (Proceedings of SIGGRAPH ’84), 18, 207–12.
Dorsey, J., H. K. Pedersen, and P. M. Hanrahan. 1996. Flow and changes in appearance. In Proceedings of SIGGRAPH ’96, Computer Graphics Proceedings, Annual Conference Series, 411–20.
Du, S.-P., S.-M. Hu, and R. R. Martin. Semiregular solid texturing from 2D image exemplars. IEEE Transactions on Visualization and Computer Graphics 19 (3), 460–69.
Dungan, W. Jr., A. Stenger, and G. Sutty. 1978. Texture tile considerations for raster graphics. Computer Graphics (Proceedings of SIGGRAPH ’78), 12, 130–34.
Ebert, D., F. K. Musgrave, D. Peachey, K. Perlin, and S. Worley. 2003. Texturing and Modeling: A Procedural Approach. San Francisco: Morgan Kaufmann.
Elek, O., P. Bauszat, T. Ritschel, M. Magnor, and H.-P. Seidel. Spectral ray differentials. Computer Graphics Forum (Proceedings of the 2014 Eurographics Symposium on Rendering) 33 (4), 113–22.
Feibush, E. A., M. Levoy, and R. L. Cook. 1980. Synthetic texturing using digital filters. Computer Graphics (Proceedings of SIGGRAPH ’80), 14, 294–301.
Fleischer, K., D. Laidlaw, B. Currin, and A. H. Barr. 1995. Cellular texture generation. In Proceedings of SIGGRAPH ’95, Computer Graphics Proceedings, Annual Conference Series, 239–48.
Fournier, A., and E. Fiume. 1988. Constant-time filtering with space-variant kernels. Computer Graphics (SIGGRAPH ’88 Proceedings), 22, 229–38.
Galerne, B., A. Lagae, S. Lefebvre, and G. Drettakis. Gabor noise by example. ACM Transactions on Graphics (Processings of SIGGRAPH 2012) 31 (4), 73:1–73:9.
Gardner, G. Y. 1984. Simulation of natural scenes using textured quadric surfaces. Computer Graphics (SIGGRAPH ’84 Proceedings), 18, 11–20.
Gardner, G. Y. 1985. Visual simulation of clouds. Computer Graphics (Proceedings of SIGGRAPH ’85), 19, 297–303.
Gilet, G., B. Sauvage, K. Vanhoey, J.-M. Dischler, and D. Ghazanfarpour. Local random-phase noise for procedural texturing. ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia 2014) 33 (6), 195:1–195:11.
Goldberg, A., M. Zwicker, and F. Durand. 2008. Anisotropic noise. ACM Transactions on Graphics (Proceedings of SIGGRAPH 2008) 27 (3), 54:1–54:8.
Gritz, L., and E. d’Eon. 2008. The importance of being linear. In H. Nguyen (Ed.), GPU Gems 3. Boston, Massachusetts: Addison-Wesley.
Gritz, L., C. Stein, C. Kulla, and A. Conty. Open Shading Language. SIGGRAPH 2010 Talks.
Guthe, S., and P. Heckbert 2005. Non-power-of-two mipmapping. NVIDIA Technical Report, https://www.nvidia.com/object/np2_mipmapping.html.
Hakura, Z. S., and A. Gupta. 1997. The design and analysis of a cache architecture for texture mapping. In Proceedings of the 24th International Symposium on Computer Architecture, Denver, Colorado, 108–20.
Hanrahan, P., and J. Lawson. 1990. A language for shading and lighting calculations. Computer Graphics (SIGGRAPH ’90 Proceedings), 24, 289–98.
Heckbert, P. S. 1986. Survey of texture mapping. IEEE Computer Graphics and Applications 6 (11), 56–67.
Heckbert, P. S. 1989a. Image zooming source code. www-2.cs.cmu.edu/~ph/src/zoom/.
Heckbert, P. S. 1989b. Fundamentals of texture mapping and image warping. M.S. thesis, Department of Electrical Engineering and Computer Science, University of California, Berkeley.
Heckbert, P. S., and P. Hanrahan. 1984. Beam tracing polygonal objects. In Computer Graphics (Proceedings of SIGGRAPH ’84), 18, 119–27.
Heitz, E., D. Nowrouzezahrai, P. Poulin, and F. Neyret. Filtering non-linear transfer functions on surfaces. IEEE Transactions on Visualization and Computer Graphics 20 (7), 996–1008.
Igehy, H. 1999. Tracing ray differentials. In Proceedings of SIGGRAPH ’99, Computer Graphics Proceedings, Annual Conference Series, 179–86.
Igehy, H., M. Eldridge, and K. Proudfoot. 1998. Prefetching in a texture cache architecture. In 1998 SIGGRAPH/Eurographics Workshop on Graphics Hardware, 133–42.
Igehy, H., M. Eldridge, and P. Hanrahan. 1999. Parallel texture caching. In 1999 SIGGRAPH/Eurographics Workshop on Graphics Hardware, 95–106.
International Electrotechnical Commission (IEC). 1999. Multimedia systems and equipment—Colour measurement and management—Part 2-1: Colour management—Default RGB colour space—sRGB. IEC Standard 61966-2-1.
Jarabo, A., H. Wu, J. Dorsey, H. Rushmeier, and D. Gutierrez. Effects of approximate filtering on the appearance of bidirectional texture functions. IEEE Transactions on Visualization and Computer Graphics 20 (6), 880–92.
Karrenberg, R., D. Rubinstein, P. Slusallek, and S. Hack. AnySL: efficient and portable shading for ray tracing. In Proceedings of High Performance Graphics 2010, 97–105.
Kensler, A., A. Knoll, and P. Shirley. 2008. Better gradient noise. Technical Report UUSCI-2008-001, SCI Institute, University of Utah.
Kim, V. G., Y. Lipman, and T. Funkhouser. Symmetry-guided texture synthesis and manipulation. ACM Transactions on Graphics 31 (3), 22:1–22:14.
Lagae, A., and G. Drettakis. Filtering solid Gabor noise. ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH 2011) 30 (4), 51:1–51:6.
Lagae, A., S. Lefebvre, G. Drettakis, and P. Dutré. 2009. Procedural noise using sparse Gabor convolution. ACM Transactions on Graphics (Proceedings of SIGGRAPH 2009) 28 (3), 54:1–54:10.
Lagae, A., S. Lefebvre, R. Cook, T. DeRose, G. Drettakis, D. S. Ebert, J. P. Lewis, K. Perlin, and M. Zwicker. A survey of procedural noise functions. Computer Graphics Forum 29 (8), 2579–2600.
Lansdale, R. C. 1991. Texture mapping and resampling for computer graphics. M.S. thesis, Department of Electrical Engineering, University of Toronto.
Lefebvre, S., S. Hornus, and A. Lasram. By-example synthesis of architectural textures. ACM Transactions on Graphics (Proceedings of SIGGRAPH 2010) 29 (4), 84:1–84:8.
Lewis, J.-P. 1989. Algorithms for solid noise synthesis. In Computer Graphics (Proceedings of SIGGRAPH ’89), Volume 23, 263–70.
Manson, J., and S. Schaefer. Bilinear accelerated filter approximation. Computer Graphics Forum (Proceedings of the 2014 Eurographics Symposium on Rendering) 33 (4), 33–40.
Manson, J., and S. Schaefer. Cardinality-constrained texture filtering. ACM Transactions on Graphics (Proceedings of SIGGRAPH 2013) 32 (4), 140:1–140:8.
McCormack, J., R. Perry, K. I. Farkas, and N. P. Jouppi. 1999. Feline: fast elliptical lines for anisotropic texture mapping. In Proceedings of SIGGRAPH ’99, Computer Graphics Proceedings, Annual Conference Series, Los Angeles, 243–250.
Mitchell, D. P., and P. Hanrahan. 1992. Illumination from curved reflectors. In Computer Graphics (Proceedings of SIGGRAPH ’92), Volume 26, 283–91.
Norton, A., A. P. Rockwood, and P. T. Skolmoski. 1982. Clamping: a method of antialiasing textured surfaces by bandwidth limiting in object space. In Computer Graphics (Proceedings of SIGGRAPH ’82), Volume 16, 1–8.
Peachey, D. R. 1985. Solid texturing of complex surfaces. Computer Graphics (SIGGRAPH ’85 Proceedings), Volume 19, 279–86.
Peachey, D. R. 1990. Texture on demand. Pixar Technical Memo #217.
Perlin, K. 1985a. An image synthesizer. In Computer Graphics (SIGGRAPH ’85 Proceedings), Volume 19, 287–96.
Perlin, K. 2002. Improving noise. ACM Transactions on Graphics 21 (3), 681–82.
Porumbescu, S., B. Budge, L. Feng, and K. Joy. 2005. Shell maps. In ACM Transactions on Graphics (Proceedings of SIGGRAPH 2005) 24 (3), 626–33.
Poynton, C. 2002a. Frequently-asked questions about color. www.poynton.com/ColorFAQ.html.
Poynton, C. 2002b. Frequently-asked questions about gamma. www.poynton.com/GammaFAQ.html.
Shinya, M., T. Takahashi, and S. Naito. 1987. Principles and applications of pencil tracing. In Computer Graphics (Proceedings of SIGGRAPH ’87), Volume 21, 45–54.
Sims, K. 1991. Artificial evolution for computer graphics. In Computer Graphics (Proceedings of SIGGRAPH ’91), Volume 25, 319–28.
Smith, J. O. 2002. Digital audio resampling home page. http://ccrma.stanford.edu/~jos/resample/.
Suykens, F., and Y. Willems. 2001. Path differentials and applications. In Rendering Techniques 2001: 12th Eurographics Workshop on Rendering, 257–68.
Turk, G. 1991. Generating textures for arbitrary surfaces using reaction-diffusion. In Computer Graphics (Proceedings of SIGGRAPH ’91), Volume 25, 289–98.
Turkowski, K. 1993. The differential geometry of texture-mapping and shading. Technical Note, Advanced Technology Group, Apple Computer.
Van Horn, B., and G. Turk. 2008. Antialiasing procedural shaders with reduction maps. IEEE Transactions on Visualization and Computer Graphics 14 (3), 539–50.
van Wijk, J. J. 1991. Spot noise-texture synthesis for data visualization. In Computer Graphics (Proceedings of SIGGRAPH ’91), Volume 25, 309–18.
Wei, L.-Y., S. Lefebvre, V. Kwatra, and G. Turk. 2009. State of the art in example-based texture synthesis. In Eurographics 2009, State of the Art Report.
Williams, L. 1983. Pyramidal parametrics. In Computer Graphics (SIGGRAPH ’83 Proceedings), Volume 17, 1–11.
Witkin, A., and M. Kass. 1991. Reaction-diffusion textures. In Computer Graphics (Proceedings of SIGGRAPH ’91), Volume 25, 299–308.
Worley, S. P. 1996. A cellular texture basis function. In Proceedings of SIGGRAPH ’96, Computer Graphics Proceedings, Annual Conference Series, New Orleans, Louisiana, 291–94.