When generating photorealistic images in computer graphics, radiance calculations are often performed using integrals based on geometric optics. Now, let be the radiance of light incident from an angle (see Figure 1) at a point on a plane. It is known that the irradiance at that point is defined as:
Answer the following questions.
Figure 1
(1) Consider a situation where the distribution of radiance does not depend on . Show that in this case, the irradiance can be expressed as the following one-dimensional definite integral:
Note that this situation is considered in all subsequent questions.
(2) Consider values defined as . Figure 2 shows that the area of the rectangle defined in the interval is given by . Using this fact, write the formula for the approximate value of irradiance using the summation symbol . This method is generally called rectangular approximation.
Figure 2
(3) In general, given a random variable distributed according to a probability density function (where ) defined on , the expected value of a scalar function is defined as:
Now, assume that are random values generated according to (). At this time, using the fact that the approximation of the above expected value is given as
write the approximate value of irradiance using the summation symbol . This method is generally called Monte Carlo integration.
(4) Consider the error of the rectangular approximation and the expected value of the error of the Monte Carlo integration. Answer in what cases the error of each method becomes zero for the integrand (assuming ). Explain based on the solutions to question (2) and question (3). Note that for the rectangular approximation, trivial cases where becomes a step-like function as in Figure 2 are excluded.
(5) When the rectangular approximation or Monte Carlo integration was implemented using 32-bit floating-point numbers, the result started to drop towards zero at the point when exceeded a certain large number. Explain one possible cause for this phenomenon. Assume that is always counted correctly.
