International Journal
`of Computer Graphics
`
`Volume 2 Number 1
`January 1986
`
`’
`
`Springer
`Internation
`
`-_ ..
`.. _
`._L-*._.
`Page 1 of 9
`
`..
`
`--
`GIA EXHIBIT 1008
`
`Page 1 of 9
`
`GIA EXHIBIT 1008
`
`

`
`The
`
`isuaili
`Computer
`
`International Journal of Computer Graphics
`
`VoIume2 Numbert
`
`1986
`
`T.L. Kunii
`
`Editorial
`
`1
`
`S.W.Thomas
`
`Dlspersive refraction in raytracing
`
`3 4/"
`
`S.Kltaoka
`
`Experimental CSG environmentformodelling solid
`
`9 L/"'
`
`'
`
`T.L. Kunii, Y. Shirota,
`T. Noma
`
`A menu generatorfor audio visual networks
`
`15
`
`E Dehne, H. Noltemeier
`
`Clustering methods for geometric objects and applications to design
`L,
`problems
`31
`--
`
`F.Dehne
`
`D. Flappaport
`
`Opticalclustering 39 L/'
`
`Alinear algorithm for eliminating hidden-lines from a polygonal
`cylinder 44
`
`Cover: composition with windowpanes and crlstal ball by Donald Fl. Lelch. © Digital Effects Inc.
`
`Page 2 of 9
`
`

`
`International Journal of
`Computer Graphics
`Official Journal of the
`Computer Graphics Society, CGS
`
`The Visual Computer is dedicated
`to reporting on the state~oi-the-art
`technology in the fields of computer
`vision, graphics. and imaging with
`specific focus on applications.
`
`Specific areas of interest include:
`— Architecture and Design for
`Vision Systems
`Networks for Communicating
`Visual Data
`
`Thejourna|’s scope includes detec-
`tion and communication 01 visual
`data. intermediate data structures
`and processing techniques for
`visual data and computer graphics,
`and graphical representations of
`images.
`
`Contributions to the journal may be
`original papers. survey reports. or
`tutorial papers. A regular column
`carries product news from both
`academia and industry. A calendar
`of events provides timely information
`on upcoming meetings and sym-
`posia. Items for this column as well
`as for the product news section are
`welcome.
`
`Computer Vision and Pattern
`Recognition
`Graphics Workstations
`VLSI for Vision and Graphics
`Applications
`Languages for Vision and
`Graphics Applications
`Data Management and Struc-
`tures for Vision Information
`Geometric Modelling
`Robotics
`
`Image Generation Techniques
`Numerical Analysis for Computer
`Graphics
`Computer Animation
`CADICAM. CAE. CAT
`Graphics Applications for
`Business, Education. Medicine
`and the Sciences
`
`Ergonomics and Human Factors
`
`Springer International
`
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`1985
`
`Page 3 of 9
`
`

`
`ay tracing is a common method of accur-
`ately modelling the interaction of light
`with objects in a synthetic image (Whit-
`:
`ted 1980; Hall and Greenberg 1933;
`"
`‘
`Cook etal. 1984). It produces the most realistic
`looking images of any image synthesis technique
`to date. Most ray tracing implementations mode!
`reflection of light from shiny and matte surfaces,
`and refraction oflight through transparent objects.
`However,
`they make the simplifying assumption
`that the index of refraction ofan object is constant
`over the entire wavelength range of the visible spec-
`trum. Most refractive media do not satisfy this as-
`sumption, but
`instead refract different colors of
`light through different angles. We develop a simple
`method of modeling such dispersiire refraction, and
`outline an itnplernentation.
`
`Dispersive refraction
`
`Simple modeling of refraction and reflection, as
`has traditionally been done in ray tracing, is insuf-
`licient to model the “fire” of a diamond or the
`action of a prism. This comes from dr'sper.ri't2e re-
`fraction, that is, from the variation in the refractive
`index of the material with the wavelength of the
`light. In most dispersive materials, blue light rays
`are refracted more strongly than red rays. The dif-
`ference between the index of refraction ofa materi-
`al for a short, or blue, wavelength and that for
`a long, or red, wavelength is called the n'r'.rpersr'on
`of the material. Dispersive refraction ofa light ray
`by a prism is illustrated in Fig. 1. The variation
`of the index of refraction with wavelength for two
`types of crown glass is shown in Fig. 2, (data from
`the Harzdbook of Chenti'.s'try and Physirrs (Weast
`19791)). However, such detaiied information is sel-
`dom available. For example, tnost gemstone refer-
`ences cite only the index of refraction at the sodium
`D line, 589 nm, and indicate the dispersion as the
`difference of the indices of refraction at two other
`wavelengtlts, commonly the B (687 nm) and G
`(431 nm) lines. From this information, simple lin-
`ear interpolation can be used to find the approxi-
`mate index of refraction for any given wavelength.
`A phenotnenon familiar to most people is iota.’
`inter-nril reflection (Born 1975). This occurs when
`a light ray passing from a medium with a high
`index of refraction to one with a lower index of
`refraction would be refracted by more than 90 de-
`grees from the normal. In this case, none of the
`light is transmitted and the interface acts as a per-
`fect mirror.
`
`“Dispersive refraction
`in ray tracing *
`
`Spencer W. lffhomas
`
`Computer Science Department,
`University of Utah, Salt Lake City.
`UT 84112. USA
`
`[Dispersive refraction is the property that
`gives gemstones their fire, and that makes
`prisms produce a spectrum from white
`light. Modeling dispersion in a ray tracing
`environment
`requires solution of some
`new problems, but allows production of
`more exciting images. The mechanism of
`dispersive refraction is discussed, and its
`implementation is described. Pictures of a
`prism and of several diamonds are in-
`cluded.
`Images generated by this tech-
`nique are realistic, but are computation-
`ally expensive]
`
`Key words: Ray tracing — Refraction — Im-
`age synthesis
`
`* This work was supported in part by the National
`Science Foundation [DCR-8341'i'96 and MCS—
`8121750},
`the Defense Advanced Research Pro-
`jects Agency (DA/\Kll-84-K—0Ol7), and the Of-
`fice of Naval Research (N0t)t}14—82-K-0351). All
`opinions, findings, conclusions or reconnnendaw
`lions expressed in this document are those of the
`author and do not necessarily reflect the views of
`the sponsoring agencies
`
`'l'l1e Visual Computer (1986) 2: 3- 3
`'.<'-" Springer-Veriilg 1986
`
`3
`
`Page 4 of 9
`
`

`
`Fig. 1. Dispcrsix-'e rel‘rac1ion
`by a prism
`
`However, the transition is not an abrupt one from
`high transmission to no transmission; the amount
`oflight transmitted. and he11ce the amount oflight
`reflected, depends on the angle of incidence of the
`ray. This behavior is embodied by the Fi'e.wief equa-
`tsitm.-r, which are formulae for the reflection and
`transmission coefficients at
`the it1te1'face between
`
`two tnaterials. If both materials are totally trans-
`parent,
`the Fresnel equations {Born and Wolf‘
`19?5) can be written
`
`r _n2 cos ()1 —i.=, cos 02
`H
`H; cosfl. +u. cos 02
`
`.l._
`r _n, cos F}, —n2 cos {)2
`n, cos ()1 +n3 cos {)2
`
`R = in + i-ma
`T=1—R
`
`where H1 and R2 are the indices of refraction of
`the two materials, 0, and ()2 are the angle of inci-
`dence and refraction, respectively, r“ is the reflecti-
`vity for the component polarized parallel to the
`surface,
`ii‘,
`is
`the reflectivity for the component
`polarized perpendicular to the surface, R is
`the
`reflection coefficient, and 7' is the transmission co-
`efficient. The form of the Fresnel equation given
`above is particularly useful in computer graphics
`applications, as the angle eosines are easily calcu-
`lated. The equation for the reflection coefficient
`
`4
`
`in a different form,
`is also given,
`Torrance I982).
`
`in (Cook and
`
`i. High Dispersion Crown Glass
`.f.g?.in(‘. Croa-.'nG|at‘S»5 . ..
`.
`
`.
`
`.
`
`.
`
`.
`
`. .
`
`1.55
`
`IndexofFteiractson
`
`I
`I
`B 00
`5 U U
`Wavelength [urn]
`
`Fig. 2. Dispersion in glass
`
`Light and color
`
`Proper sitnulation of dispersion and color absorp-
`tion requires that colors be represented by the in-
`tensity of the wavelengths comprising the returned
`light ray. It is necessary to transform this .\';Jr*(‘trm’
`di‘.s‘rrfhiitt'rJi.r function to the red, green and blue in-
`tensities necessary for driving a color CRT display
`to make pictures. This can be accomplished by go-
`
`Page 5 of 9
`
`

`
`Msiaail —
`i .in'nptitcir
`
`Fig. 3. Ray clumping due to incorrect
`dispersion calculation
`
`ing through the CIE .'n‘{:‘r:eci- color space (Hall and
`Greenberg 1983). To transform a spectral distribu-
`tion into trilinear coordinates, X, Y, and Z, three
`
`integrals are calculated,
`
`x = j :12.) x(/".) .42.,
`Y = j :0.) yo.) (1).,
`z = j rot) go.) 42..
`The trilinear coordinates can then be transformed
`to RGB numbers by referring to the coordinates
`of each of the phosphors of the color CRT display.
`If the color lies outside the gamut of the display,
`one or more of the RGB coefficients will be out
`
`of range. In this case, it is necessary to desatur-ate
`the color to display it. The intersection of a line
`between the given color and a white of equal
`brightness with the boundary of the gamut of the
`display gives the appropriate desaturated color for
`display.
`
`Implementation
`
`Dispersion is modeled by breaking a ray, as
`needed, into several subrays, each covering only
`a portion of the spectrum, at refractive interfaces.
`As might be expected,
`this process can generate
`many rays, and it
`is not uncommon to fire over
`
`100 rays for a single pixel when modeling a highly
`dispersive medium. Each ray carries two -new
`pieces of information: the portion of the spectrum
`covered by the ray, and the angular spread of the
`ray. The spread is necessary when splitting a ray
`into subrays, and also enters into the decision
`of whether to split a ray. In addition to the an-
`gular spread, a spread vector, with magnitude
`tan {.spreadf2) defines the plane of divergence of
`the ray.
`
`Modeling dispersion
`
`Each object is given an index of refraction and
`a dispersion value. When a ray encounters an inter-
`face between two different indices of refraction,
`two new rays, one for each endpoint of the wave-
`length range covered by the incident ray, are re-
`fracted. The angular spread between the two re-
`fracted rays is computed and compared to the an-
`gular extent of the smallest light source. If it
`is
`larger than some fraction of the light source extent,
`the incident ray is subdivided into several rays so
`that the spread of each refracted ray will be small
`enough. Through experimentation,
`it was deter-
`mined that it
`is necessary for about six subrays
`to hit a light source. Iffewerrays are used, various
`
`5
`
`Page 6 of 9
`
`

`
`—' L,\'isual
`( antnptlter
`
`quantization effects appear in the resulting image.
`If both the low and high wavelength rays are to-
`tally reflected at the interface,
`the entire incident
`ray is reflected, and no other rays are generated.
`If the ray at one endpoint of the wavelength range
`is reflected and the other is refracted, the incident
`ray must first be split into two parts, one of which
`is totally reflected, and the other of which is both
`refracted and reflected. The wavelength that is the
`dividing point between total internal reflection and
`refraction is found by binary search, since the rela-
`tion between wavelength and refracted direction
`is nonlinear. After the incident ray has been so
`divided, it may be necessary to further subdivide
`the refracted part.
`If the direction of the incident ray is assumed con-
`stant over the entire range of wavelengths it repre-
`sents, anomalous results can occur. In particular,
`a clumping effect is visible in the refracted rays.
`Figure 3 demonstrates the problem. Furthermore,
`in this figure, the rays coming out of the diagonal
`of the prism should not cross. To correct for these
`effects, when an incident ray is subdivided the re-
`sulting rays must have their direction of incidence
`adjusted according to their position in the wave-
`length range. Figure 4 illustrates this process. In
`the left part of the figure, a ray incident on a sur-
`face is shown, with the angular spread of the ray
`shown by the lighter rays. If it is to be split into
`two subrays, each subray will be given a direction
`that is the average of the direction of the incident
`ray and one of the “spread” rays. These are shown
`by the dashed rays. To avoid recalculating the in-
`tersection of the ray and the surface, the subrays
`are assumed to be incident at the same point as
`the original. The effective result
`is illustrated in
`the right half of the figure. The new rays have
`been labeled so that their identity can be estab-
`lished between the two pictures.
`
`Fig. 4. Perlurbing the incidenlray
`
`Once the original ray has been subdivided, the re-
`fractive index for the central wavelength, /1, ofcach
`subray is calculated as
`
`where :19 is the refractive index at the sodium D
`wavelength, and 6 is the dispersion between the
`B and G wavelengths. Snell’s law of sincs
`
`rs, sin 0, =11; sin 02
`
`can then be applied to determine the direction of
`the refracted ray.
`If a ray does not intersect anything, its direction
`is then compared to the direction of each light
`source. If the ray,
`including its angular spread,
`overlaps the extent of a light source, the intensity
`of the light is prorated by the amount of overlap.
`A spectral distribution is constructed, assigning the
`calculated intensity to the range of the spectrum
`covered by the ray. This is returned as the color
`of the ray. To calculate the color of a ray that
`has been split, the spectral distributions of the sub-
`rays are added together. Note that when coml_3in-
`ing a reflected ray with a refracted ray, each must
`be multiplied by the appropriate reflection or
`transmission coefficient.
`
`Opaque, diffusely reflecting objects are treated in
`the same way as they are in standard ray tracing,
`except that their color must be modeled as a spec-
`tral distribution. The angular extents of the lights
`must be taken into account when computing the
`diffuse lighting component. The area covered
`by a
`li ht with an angular
`radius of
`5
`is
`2:rt(1—— 1—.r ). This is multiplied by the bright-
`ness of the light to get its contribution to the total
`illumination.
`
`To prevent a ray from bouncing around inside an
`object indefinitely, two pruning criteria are used.
`First, each ray has a significance factor, which is
`just the product ofall the reflection or transmission
`coefficients that will apply to the result returned
`by the ray. When this factor becomes very small,
`the ray is assumed to return black, because, in any
`case, its total contribution to the pixel color will
`be insignificant. This is similar to the “adaptive
`free-depth control" of (Hall and Greenberg 1983).
`As a back up, a maximum recursion depth can
`be set.
`
`The algorithms described above are given in a sim-
`plifled pseudo-code form below.
`
`Page 7 of 9
`
`

`
`ray_sotid intersects a ray with the solid and returns the ray color.
`:a5r_so11d(:ay, slgi
`it gig -< thrash then return black;
`ink.
`:- £.i.nd.__1.nt(:ay):
`II‘ null(LnI.-.1 then return ray_J.1ght(ny]
`else return fo:l:_rayst:ay. aiq.
`int}:
`
`ray_tr'ght cxanapures the contribution to the ray tram the right sources.
`3,-ay__11ght. tray]
`return for each light In light: sum
`o1rer1ap_amount'. tzly,
`light} ;
`for#_rays decides what happens at a particular ray-object
`rerurnsacolor.
`int.)
`£o:k.__raystray, sic.
`Get rays at limits at ray spread
`ray.lou].‘
`ray.loIr,
`lou-_ra5r
`:= perhurblrayi
`h!t.g'h___ray := pexrturbitlfu If-!y.h:‘|.gh.
`ra1r.hLgh1I
`Check for total internal rettectfon
`If
`:a£lactp(J.ov__:n1r,
`int)
`int] then
`and x-a£J.ec:t:.pth1gh_:ay,
`return t‘.ataJ.___:n£.'l.act:I'.:ay, aig,
`int}
`elaei! reflect-.p[1o\r_:ay,
`inc} then begin
`r_umrn := f!.nd_t'.ot'.al_ref1_wavat:ay..
`return fo1:k_ra§rs tperturbtray,
`:_u'ava..
`sig,
`int)
`+ tota1_:a£J.ect-.tpercurbtray,
`aig.
`int)
`
`intersection.
`
`int}:
`::ay.high',l,
`
`tay.1ou.. 1:_wave}.«
`
`Gfld
`int) then begin
`elsell re£1I.nat:p{h!.gh_:a3r,
`int]:
`r_\rava := find.__totaJ._:a£J._uava{ray,
`ratum £o:k_:|yu{pertutb(ra)r. zayulolr.
`:__wu-avu),
`eiq,
`int}
`+ totaJ.__:et:I.act{pa:I:u:b(ny,
`gig,
`int}
`
`r_ua\ra, 1:ay.hi.qh).
`
`and
`also begin
`Calculate number at new rays tram spread of dispersed rays
`ntaya := 1 + 6*[ray.apraad +
`d.i.spa:sa_spx:ead(1o\¢_:ay, hi,gh___:ay)]
`I|...’l.ght'._up:nad;
`d._uavu := (ray.h.igh—tay.laIr',|
`I nrayut
`return for J.
`:= 1 to :n:ay.1 aurn
`xnfizucttpcrturbtray, 1:ay.low+{.t.-1)"d._uavn,
`ra3,r.1cm+i.*d_uavuJ, Jig,
`int}
`
`and
`
`I
`
`retract generates the reflected and retracted rays and returns a color.
`reizect-. (ray, sig,
`int)
`Generate retracted ray using Snetls law
`:af:act_ra1r :- :o£:m:t{ra5r,
`int} :
`Calculate transmission coefficient
`int.) ;
`‘r
`:= ttesnel tray.
`rafnc-t__ny,
`* '1‘
`relum :ay_uo1idtre£ract_ra_v.r.
`'1“'a.i.g',|
`+ r.'a1r__so.'I.1.d.(ra£la<:t (ray,
`int.) .
`{Iv-1'} *a.i.g}
`- u—m::
`
`total_rett9clg9nerales a totally reflected ray and returns its color.
`totaJ._:a£1acI:(:.-air, Jig,
`int.)
`sxg} :
`int} ,
`a lg,
`return n:.r_.!o1!.dtx:e£1ect: truly.
`perturb generates a ray corresponding to the canton of the given ray between
`the low and high wavelengths.
`parturbiray,
`low, high}
`J.oII_di:
`:- [2* [low-:ay.1ou-I I [:|y.h$.gh-ray. low)
`I ray.apread_vec + 1.-ay.d1r;
`high_di.r.'
`:= l2* {high-ray. 1ou],H::ay.high-:ay.1oIr) — 1]
`* :ay..1praad_vec + :|':ay.d.i.:.'
`new_ray{dt‘r, tow, nigh, spread. spread_vec)
`return neu_ny tnozmlizat {J.ow_dir+h1gh_di.:} E2] .
`low, high,
`lenqtht th.i.gh_dl.:-ILoIr_d1:} £2] ,
`(h1gh_d:Lr—loIt_d.1.:) I2) :
`
`- 1}
`
`Results
`
`Two images are included to demonstrate the capa-
`bilities of the algorithm. The tirst is a simplc image
`of a prism with a piccc of paper behind it with
`
`isngul
`,t_b:1fiI.'~.';;itt:t‘_t..*-r"
`
`the word prism written on it. This is shown in
`Fig. 5. Note particularly the color fringing at the
`edges of the letters. The curved edges on the prism
`and sheet of paper are barrel distortion resulting
`from the close viewpoint and wide angle of view.
`
`it
`
`Modettng a diamond
`
`As a second demonstration, a diamond was mo-
`dcled. A polygonal model ofa diamond in the brit-
`ttrtttt cut (Bank 1973) was created. The brilliant
`cat has 5? facets, of which one is octagonal, 16
`are “diamond shapcd", and the remainder are tri-
`angular. Figurc 6 shows a line drawing of the mod-
`cl with the different facets labeled. The anglcs bc-
`tween the facets are critical
`to proper brilliancy
`of the cut gcm. They are measured from the plane
`of thc “girdlc", the broadest part of the diamond,
`and are also shown in the figure. The brilliant cut
`is designed so that almost all
`light entering the
`Front of the diamond (E.c., through the table, star,
`kite, or upper girdlc faccts) will be returned
`through those facets, and that no light will be lost
`through the back of the gem. It is the standard
`cut for diamonds and many other gems.
`'
`
`Fig. 5. A prism rcfracling a word
`
`The ray traccd image shown in Fig. 7 includes a
`“sctting" of seven diamonds, with six smallcr dia-
`monds arrangcd around a large central diamond.
`The group floats above a blue and white checkered
`cloth. The scene is lit by nine spot lights and onc
`hemispherical light simulating ambient 1'oom light.
`This scene demonstrates how much of our pe-rccp~
`tion of a diamond owes tobeing able to move
`
`7
`
`Page 8 of 9
`
`

`
`*"i",\'islt:il
`( AIIll[)lltCl°
`
`_ _ _ Upper Girdle Facet
`41.2‘
`Lower Girdle Facet
`46.3‘
`
`Pavilion Facet
`405'
`
`Fig. 6. A brilliant cut diamond
`
`Fig. 7. A group of diamonds
`
`it or ourselves while viewing it. In any fixed orien-
`tation, there are few “sparkles”, and most of the
`diamond reflects little light back to the observer.
`
`Conclusion
`
`A simple model of refractive dispersion leads to
`realistic looking pictures of gemstones and other
`dispersive media. A key feature of the algorithm
`is the the adaptive subdivision of the spectra! distri-
`bution of rays at dispersive interfaces. Like any
`ray tracing implementation, it is computationally
`expensive. It tends to be more expensive than stan-
`dard ray tracing because any ray-surface intersec-
`tion can generate many rays, instead of the two
`or three generated by a normal ray tracing algo-
`rithm.
`
`References
`
`Bank H (1973) From the World of Gemstones. Pinguin, Inns-
`bruck, Austria
`Born M, Wolf E (1975) Principles of Optics. Pergamon Press,
`Oxford
`Cook RL, Torrance KE (1982) A Rcflcctancc Model for Corn-
`puter Graphics. ACM Transactions on Graphics 1 :'!—24
`Cook RL, Porter, T, Carpenter L (1984) Distributed Ray Trac-
`ing. Comput Graphics (Proc. SIGGRAPH '34) 381137-145
`Hal] RA, Grcenberg DP (1983) A Testbed for Realistic Image
`Synthesis.
`IEEE Computer Graphics and Applications
`3: 10-20
`Weast RC {ed} (1971) Handbook of Chemistry and Physics.
`Chemical Rubber Company, Cleveland, Ohio
`Whilted T (1980) An Improved Illumination Mode] for Shaded
`Display. Comm ACM 23:343—349
`
`Page 9 of 9