4.2 Working with Radiometric Integrals

A frequent task in rendering is the evaluation of integrals of radiometric quantities. In this section, we will present some tricks that can make it easier to do this. To illustrate the use of these techniques, we will take the computation of irradiance at a point as an example. Irradiance at a point normal p Subscript with surface normal bold n Subscript due to radiance over a set of directions normal upper Omega  is

upper E Subscript Superscript Baseline left-parenthesis normal p Subscript Baseline comma bold n Subscript Baseline right-parenthesis equals integral Underscript normal upper Omega Endscripts upper L Subscript normal i Baseline left-parenthesis normal p Subscript Baseline comma omega right-parenthesis StartAbsoluteValue cosine theta EndAbsoluteValue normal d omega Subscript Baseline comma
(4.7)

where upper L Subscript normal i Baseline left-parenthesis normal p Subscript Baseline comma omega right-parenthesis is the incident radiance function (Figure 4.5) and the cosine theta factor in the integrand is due to the normal d upper A Subscript Superscript up-tack factor in the definition of radiance. theta is measured as the angle between omega Subscript and the surface normal bold n Subscript . Irradiance is usually computed over the hemisphere script upper H squared left-parenthesis bold n Subscript Baseline right-parenthesis of directions about a given surface normal  bold n Subscript .

Figure 4.5: Irradiance at a point normal p Subscript is given by the integral of radiance times the cosine of the incident direction over the entire upper hemisphere above the point.

The integral in Equation (4.7) is with respect to solid angle on the hemisphere and the measure normal d omega Subscript corresponds to surface area on the unit hemisphere. (Recall the definition of solid angle in Section 3.8.1.)

4.2.1 Integrals over Projected Solid Angle

The various cosine factors in the integrals for radiometric quantities can often distract from what is being expressed in the integral. This problem can be avoided using projected solid angle rather than solid angle to measure areas subtended by objects being integrated over. The projected solid angle subtended by an object is determined by projecting the object onto the unit sphere, as was done for the solid angle, but then projecting the resulting shape down onto the unit disk that is perpendicular to the surface normal (Figure 4.6). Integrals over hemispheres of directions with respect to cosine-weighted solid angle can be rewritten as integrals over projected solid angle.

Figure 4.6: The projected solid angle subtended by an object is the cosine-weighted solid angle that it subtends. It can be computed by finding the object’s solid angle, projecting it down to the plane perpendicular to the surface normal, and measuring its area there. Thus, the projected solid angle depends on the surface normal where it is being measured, since the normal orients the plane of projection.

The projected solid angle measure is related to the solid angle measure by

normal d omega Subscript Baseline Superscript up-tack Baseline equals StartAbsoluteValue cosine theta EndAbsoluteValue normal d omega Subscript Baseline comma

so the irradiance-from-radiance integral over the hemisphere can be written more simply as

upper E Subscript Superscript Baseline left-parenthesis normal p Subscript Baseline comma bold n Subscript Baseline right-parenthesis equals integral Underscript script upper H squared left-parenthesis bold n Subscript Baseline right-parenthesis Endscripts upper L Subscript normal i Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis normal d omega Subscript Baseline Superscript up-tack Baseline period

For the rest of this book, we will write integrals over directions in terms of solid angle, rather than projected solid angle. In other sources, however, projected solid angle may be used, so it is always important to be aware of the integrand’s actual measure.

4.2.2 Integrals over Spherical Coordinates

It is often convenient to transform integrals over solid angle into integrals over spherical coordinates left-parenthesis theta comma phi right-parenthesis using Equation (3.7). In order to convert an integral over a solid angle to an integral over left-parenthesis theta comma phi right-parenthesis , we need to be able to express the relationship between the differential area of a set of directions normal d omega Subscript and the differential area of a left-parenthesis theta comma phi right-parenthesis pair (Figure 4.7). The differential area on the unit sphere normal d omega Subscript is the product of the differential lengths of its sides, sine theta normal d phi Subscript and normal d theta Subscript . Therefore,

normal d omega Subscript Baseline equals sine theta normal d theta Subscript Baseline normal d phi Subscript Baseline period
(4.8)

(This result can also be derived using the multidimensional transformation approach from Section 2.4.1.)

Figure 4.7: The differential area normal d omega Subscript subtended by a differential solid angle is the product of the differential lengths of the two edges sine theta normal d phi Subscript and normal d theta Subscript . The resulting relationship, normal d omega Subscript Baseline equals sine theta normal d theta Subscript Baseline normal d phi Subscript , is the key to converting between integrals over solid angles and integrals over spherical angles.

We can thus see that the irradiance integral over the hemisphere, Equation (4.7) with normal upper Omega equals script upper H squared left-parenthesis bold n Subscript Baseline right-parenthesis , can equivalently be written as

upper E Subscript Superscript Baseline left-parenthesis normal p Subscript Baseline comma bold n Subscript Baseline right-parenthesis equals integral Subscript 0 Superscript 2 pi Baseline integral Subscript 0 Superscript pi slash 2 Baseline upper L Subscript normal i Baseline left-parenthesis normal p Subscript Baseline comma theta comma phi right-parenthesis cosine theta sine theta normal d theta Subscript Baseline normal d phi Subscript Baseline period

If the radiance is the same from all directions, the equation simplifies to upper E Subscript Superscript Baseline equals pi upper L Subscript normal i .

4.2.3 Integrals over Area

One last useful transformation is to turn integrals over directions into integrals over area. Consider the irradiance integral in Equation (4.7) again, and imagine there is a quadrilateral with constant outgoing radiance and that we could like to compute the resulting irradiance at a point normal p Subscript . Computing this value as an integral over directions omega Subscript or spherical coordinates left-parenthesis theta comma phi right-parenthesis is in general not straightforward, since given a particular direction it is nontrivial to determine if the quadrilateral is visible in that direction or left-parenthesis theta comma phi right-parenthesis . It is much easier to compute the irradiance as an integral over the area of the quadrilateral.

Differential area normal d upper A Subscript on a surface is related to differential solid angle as viewed from a point normal p Subscript  by

normal d omega Subscript Baseline equals StartFraction normal d upper A Subscript Baseline cosine theta Over r squared EndFraction comma
(4.9)

where theta is the angle between the surface normal of normal d upper A Subscript and the vector to normal p Subscript , and r is the distance from normal p Subscript to normal d upper A Subscript (Figure 4.8). We will not derive this result here, but it can be understood intuitively: if normal d upper A Subscript is at distance 1 from  normal p Subscript and is aligned exactly so that it is perpendicular to normal d omega Subscript , then normal d omega Subscript Baseline equals normal d upper A Subscript , theta equals 0 , and Equation (4.9) holds. As normal d upper A Subscript moves farther away from normal p Subscript , or as it rotates so that it is not aligned with the direction of normal d omega Subscript , the r squared and cosine theta factors compensate accordingly to reduce  normal d omega Subscript .

Figure 4.8: The differential solid angle normal d omega Subscript subtended by a differential area normal d upper A Subscript is equal to normal d upper A Subscript Baseline cosine theta slash r squared , where theta is the angle between normal d upper A Subscript ’s surface normal and the vector to the point normal p Subscript and r is the distance from normal p Subscript to normal d upper A Subscript .

Therefore, we can write the irradiance integral for the quadrilateral source as

upper E Subscript Superscript Baseline left-parenthesis normal p Subscript Baseline comma bold n Subscript Baseline right-parenthesis equals integral Underscript upper A Endscripts upper L Subscript Superscript Baseline cosine theta Subscript normal i Baseline StartFraction cosine theta Subscript normal o Baseline normal d upper A Subscript Baseline Over r squared EndFraction comma

where upper L Subscript Superscript is the emitted radiance from the surface of the quadrilateral, theta Subscript normal i is the angle between the surface normal at normal p Subscript and the direction from normal p Subscript to the point normal p prime on the light, and theta Subscript normal o is the angle between the surface normal at normal p prime on the light and the direction from normal p prime to normal p Subscript (Figure 4.9).

Figure 4.9: To compute irradiance at a point normal p Subscript from a quadrilateral source, it is easier to integrate over the surface area of the source than to integrate over the irregular set of directions that it subtends. The relationship between solid angles and areas given by Equation (4.9) lets us go back and forth between the two approaches.