# Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

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1 Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each point in the pictue. The pupose of this lectue is to explain how to epesent colo and intensity numeically and then to develop illumination models that allow us to compute the colo and intensity at each point once we know the colo, location, and intensity of the light souces and the physical chaacteistics of the objects in the scene. 2. The RGB Colo Model Colo foms a 3-dimensional vecto space. The thee pimay colos -- ed, geen, and blue -- ae a basis fo this vecto space: evey colo can be epesented as a linea combination of ed, geen, and blue. The unit cube is often used to epesent colo space (see Figue 1). Black is located at one cone of the cube usually associated with the oigin of the coodinate system, and the thee pimay colos ae placed along the thee othogonal axes. Intensity vaies along the edges of the cube, with the full intensity of each pimay colo coesponding to a unit distance along the associated edge. Evey combination of colo and intensity is then epesented by some linea combination of ed, geen, and blue, whee the coefficients of each pimay lie between zeo and one. Thus fo each colo c thee is a unique set of coodinates (,g,b) inside the unit cube that epesents the colo c. The numeical values (,g,b) epesent intensities: the highe the value of o g o b, the geate the contibution of ed o geen o blue to the colo c. This model is known as the RGB colo model and is one of the most common colo models in Compute Gaphics. blue cyan geen magenta white yellow black ed Figue 1: The RGB colo cube. Evey colo and intensity is epesented by a linea combination of the ed, geen, and blue, whee the coefficients of each pimay colo lie between zeo and one.

2 Pais of pimay colos combine to fom colos complementay to the missing pimay. Thus ed+geen=yellow, which is the colo complementay to blue. Similaly, blue+geen=cyan, which is the colo complementay to ed, and blue+ed=magenta, which is the colo complementay to geen. Combining all thee pimaies at full intensity yields white=ed+geen+blue, which lies at the cone of the cube diagonally opposite to black. Shades of gay ae epesented along the diagonal of the cube joining black to white. In the emainde of this lectue we shall develop illumination models. These illumination models allow us to ende a scene by computing the (,g,b) colo intensities fo each suface point once we know the light souces and the physical chaacteistics of the objects in the scene. We shall conside thee illumination models: ambient light, diffuse eflections, and specula highlights. 3. Ambient Light Ambient light is light that is eflected into a scene off outside sufaces. Fo example, sunlight that entes a oom though a window by bouncing off neaby buildings is ambient light. Ambient light softens hash shadows geneated by point light souces. Thus ambient light helps to make scenes endeed by Compute Gaphics appea moe natual. Typically we assume that the intensity I a of the ambient light in a scene is a constant. What we need to compute is the intensity I of the ambient light eflected to a viewe fom each suface point in the scene. The fomula fo this intensity I is simply I = k a I a, (3.1) whee popety of the colo and mateial of the suface, which can be detemined expeimentally o simply set by the pogamme. Notice that the intensity I is independent of the position of the viewe. set 0 k a 1 is the ambient eflection coefficient. The ambient eflection coefficient k a is a Equation (3.1) is eally thee equations, one fo each pimay colo ed, geen, and blue. If we I a = (I a,ia g,ia b ) and k a = (k a,ka,ka ), whee (I a,ia,ia) ae the ambient intensities fo ed, geen and blue, and (k a,ka,ka ) ae the ambient eflection coefficients fo ed, geen, and blue, then Equation (3.1) becomes g g b b I = (,g,b) = (k aia,ka Ia,kaIa). (3.2) Notice that if the ambient light is white, then I a = (1,1,1), so the colo of the light eflected fom a suface is the colo of the suface. Howeve, if the colo of the ambient light is blue and the colo of the suface is ed, then I a = (0,0,1) and k a = (k a,0,0), so I = (,g,b) = (0,0,0) and the colo peceived by the viewe is black. 2

3 4. Diffuse Reflection Diffuse light is light eflected off dull sufaces like cloth. Light dispesed fom dull sufaces is eflected fom each point by the same amount in all diections. Thus the intensity of the eflected light is independent of the position of the viewe. Hee we shall compute the diffuse light eflected off a dull suface fom a point light souce. Let I p be the intensity of the point light souce. We need to compute the intensity I of the diffuse light eflected to a viewe fom each point on a dull suface. Let L be the unit vecto fom the point on the suface to the point light souce, and let N be the outwad pointing unit nomal at the point on the suface (see Figue 2). Then the fomula fo the diffuse intensity I is simply I = k d (L N)I p, (4.1) whee 0 k d 1 is the diffuse eflection coefficient. The diffuse eflection coefficient k d, like the ambient eflection coefficient k a, is a popety of the colo and mateial of the suface, which can be detemined expeimentally o simply set by the pogamme. L N Suface Figue 2: The unit nomal vecto N to a point on the suface, and a unit vecto L pointing in the diection of a point light souce. Equation (4.1) is a consequence of Lambet s Law. Conside a point light souce fa away fom a small suface facet (see Figue 3). Let I facet denote the intensity of light on the facet, and let I souce denote the intensity of the light souce. Then Light Beam Coss Section I facet = = Unit Aea Facet I Aea souce. But we can see fom Figue 3 that Beam Coss Section Facet Aea = cos(), whee is the angle between the nomal to the facet and the vecto to the light souce. Thus we aive at the following esult: 3

4 Lambet s Law: I facet = cos() I souce. (4.2) N Light Souce Beam Coss Section Figue 3: Lambet s Law: I facet = cos() I souce. In Equation (4.1), L and N ae unit vectos. Hence L N = cos(), whee is the angle between the nomal to the suface (N) and the diection to the light souce (L). Thus Equation (4.1) is simply Equation (4.2) attenuated by the facto k d. Equation (4.1) is eally thee equations, one fo each pimay colo ed, geen, and blue. If we set I p = (I p,i g p,i b p) and g k d = (k d,kd,kd b ), whee (I p,ip,ip) ae the intensities of the point light souce fo ed, geen and blue, and (k d,kd,kd ) ae the diffuse eflection coefficients fo ed, geen, and blue, then Equation (4.1) becomes I = (,g,b) = (k di g g b b p,kd Ip,kd Ip). (4.3) When thee ae many light souces, we simply add the contibutions fom each light souce. As with ambient light, the colo peceived by the viewe depends both on the colo of the light souce and the colo of the suface. If the light souce is white, then I p = (1,1,1), so the colo of the eflected light is the colo of the suface. Howeve, if the colo of the light souce is blue and the colo of the suface is ed, then I p = (0,0,1) and k d = (k d,0,0), so I = (,g,b) = (0,0,0) and once again the colo peceived by the viewe is black. 5. Specula Reflection Specula eflections ae highlights eflected off shiny sufaces. Unlike ambient and diffuse eflections, specula highlights ae sensitive to the position of the viewe. Hee we shall compute the specula highlights eflected off a shiny suface fom a point light souce. Let I p be the intensity of the point light souce. We need to compute the intensity I of the 4

5 specula highlight eflected to a viewe fom each point on a shiny suface. Let L be the unit vecto fom the point on the suface to the point light souce, and let N be the outwad pointing unit nomal at the point on the suface. Let R be the image of the vecto L as the light fom the point light souce in the diection L bounces off the suface -- that is, R is the vecto defined by setting the angle of incidence of L equal to the angle of eflection of R. Finally let V be a unit vecto fom the point on the suface to the viewe, and let α be the angle between V and R (see Figue 4). Then the fomula fo the intensity I of the specula highlight seen by a viewe in the diection V is given by I = k s cos n (α)i p = k s (R V ) n I p, (5.1) whee 0 k d 1 is the specula eflection coefficient and n 0 is the specula exponent, a constant that contols the concentation of the specula highlight. The specula eflection coefficient k s, like the ambient eflection coefficient k a and the diffuse eflection coefficient k d, is a popety of the colo and mateial of the suface, which can be detemined expeimentally o simply set by the pogamme. L N R V α Suface Figue 4: The unit nomal vecto N to a point on the suface, and a unit vecto L pointing in the diection of a point light souce. The unit vecto V points to the viewe, and the unit vecto R is the image of the light vecto L eflected off the suface. The angle is the angle between the vectos L and N, and the angle α is the angle between the vectos R and V. Notice that unlike ambient and diffuse eflections, specula highlights depend on the position of the viewe. The intensity of the specula highlight falls off apidly as the angle α between the eflected vecto R and the vecto to the viewe V inceases, since R V = cos(α) 0 as α 90 o. Moeove, inceasing the value of the specula exponent n futhe concentates the specula highlight aound the diection R, since high powes of cos(α) appoach zeo moe apidly than low powes of cos(α). As n, the suface appoaches a mio; only when α = 0 is any effect visible to the viewe. Convesely, when n 0, neithe the position of the viewe no the position of the light souce mattes, and the effect is much the same as fo ambient light. In Equation (5.1), the vectos L,V,N ae typically known, since the position of the light souce, the location of the viewe, and the shape of the suface ae contolled by the pogamme. The 5

6 paametes I p and k s -- the intensity of the light souce and the specula eflection coefficient -- ae also contolled by the pogamme. The eflection vecto R, howeve, must be computed. To find R, let L and L denote the components of L paallel and pependicula to N (see Figue 5). Then L = L + L and R = L L. Now ecall that L = (L N)N L = L L = L (L N)N. Theefoe R = 2(L N)N L. L N L L L R Suface Figue 5: Computation of the vecto R fom the vectos N and L: L = L + L and R = L L. In the calculation of specula eflection, we shall often assume that the light souces ae located at infinity. Notice that if the light souce is fa away and if the suface is plana, then L,N,R ae constants, and only R V needs to be ecalculated fo each point on the suface. Thus these assumptions help to speed up the calculation of specula eflections. Equation (5.1), like Equations (3.1) and (4.1), is eally thee equations, one fo each pimay colo ed, geen, and blue. If we set I p = (I p,i g p,i b p) and k s = (k s,ks,ks ), whee (I p,ip,ip) ae the intensities of the point light souce fo ed, geen and blue, and (k s,ks,ks ) ae the specula eflection coefficients fo ed, geen, and blue, then Equation (5.1) becomes I = (,g,b) = (k di g g b b p,kd Ip,kd Ip). (5.2) When thee ae many light souces, we simply add the contibutions fom each light souce. Specula highlights come in two types, depending on whethe the suface mateial is 6

7 homogeneous o inhomogeneous. When the suface mateial is homogeneous, like a pue metal, then the specula highlight takes on the colo of the suface much like diffuse eflections. Thus if the light souce is white, then I p = (1,1,1), so the colo of the specula highlight is the colo of the suface. Howeve, if the colo of the light souce is blue and the colo of the suface is ed, then I p = (0,0,1) and viewe. k d = (k d,0,0), so I = (,g,b) = (0,0,0) and no specula highlight is visible to the When the suface mateial is inhomogeneous, like a plastic, then unlike ambient light and diffuse eflections, the colo of the specula highlights peceived by the viewe depends only on the colo of the light souce and not on the colo of the suface. Fo example, if the colo of the light souce is blue and the colo of the suface is ed, then the colo of the specula highlight is blue not ed o black. 6. Total Intensity To find the total intensity at each point, we must add the ambient, diffuse, and specula components. Thus Total Intensity = Ambient Intensity + Diffuse Intensity + Specula Intensity. Fo each scene thee is only one ambient intensity, but fo the diffuse and specula intensities we need to add the contibutions fom each light souce. Theefoe Total Intensity = k a I a + p k d (L N)I p + pk s (R V ) n I p, (6.1) whee the sums ae taken ove all point light souces. As usual, Equation (6.1) is eally thee equations, one fo each pimay colo. Since in the RGB colo model evey colo and intensity is epesented by a point inside the unit cube, when thee ae many light souces it may be necessay to nomalize the total intensities to lie between zeo and one by dividing the intensity at each point by the maximum intensity in the scene. 7. Summay In the RGB colo model, each colo and intensity is epesented by thee coodinates, (,g,b), in the unit cube, each coodinate epesenting the contibution of the associated pimay colo ed, geen, o blue, to the given colo. Thee ae thee illumination models fo computing the colo and intensity at each suface point: ambient intensity, diffuse eflection, and specula highlights. The fomulas fo these illumination models ae summaized below: 7

8 Illumination Models Ambient Intensity = k a I a Diffuse Reflection = k d (L N)I p Specula Reflection = k s (R V) n I p Total Intensity = k a I a + p k d (L N)I p + k s (R V ) n p I p whee L is the unit vecto fom the point on the suface to the point light souce N is the outwad pointing unit nomal at the point on the suface R is the image of the vecto L as the light fom the point light souce in the diection L bounces off the suface (see Figue 5) V is a unit vecto fom the point on the suface to the viewe I a is the ambient intensity I p is the intensity of the light souce located at the point p k a is the ambient eflection coefficient k d is the diffuse eflection coefficient k s is the specula eflection coefficient n is the specula eflection exponent and the sums ae taken ove all the point light souces. The distinguishing popeties of these thee illumination models ae summaized in Table 1. Colo Viewe Location Ambient Intensity Suface Independent Diffuse Reflection Suface Independent Specula Highlights Suface Dependent (Homogeneous) Specula Highlights Light Souce Dependent (Inhomogeneous) Table 1: How the colo and intensity at a point depends on the position of the viewe as well as the colo of the suface and the colo of the light souce fo ambient, diffuse, and specula eflections. 8

9 Execises: 1. Suppose that the colo of the light souce is yellow and the colo of the suface is cyan. What is the colo peceived by the viewe fo a. diffuse eflection b. specula eflection (homogenous mateial) c. specula eflection (inhomogeneous mateial) 9

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