The previous sections highlighted the physics of light transport; i.e. the way light interacts with the various types of materials. A Physically-Based system computes the light transport inside a scene by solving the primary light transport equation - the “rendering equation”.
The rendering equation (light transport equation) describes the energy equi- librium in a scene. Using an approach similar to the radiative heat transfer equations, James Kajiya derived the equation from the simple notion that the total intensity on a surface pointxis the summation of the total emitted intensity fromx and the incoming intensity atxfrom every direction over the hemisphere around x[Kajiya, 1986]. Thus, total outgoing Flux o(x) at pointx will be the sum of total emitted
flux e(x)and total incoming flux i(x)multiplied by the reflectance factor on point
x, C(x) (0C(x)1). The following section will describe the formulation of the Rendering Equation.
2.3.1 Formulation
According to the energy conservation law:
o(x) = e(x) +C(x) i(x)
Converting flux to irradiance by taking derivative w.r.t area A around pointx. d o(x)
Rewriting as an integral of all directions over hemisphere (⌦): Eo(x) = Z ⌦ Le(x!!)d!+ Z ⌦ C(x !)Li(x !)d! (2.13)
Figure 2.3: Transported light intensity from point A to B is the sum of all the incoming light at A and the emitted light from A towards B.
Equation 2.13 can be re-written as the equation 2.12 as an integral of radiance over all possible incoming angles over the hemisphere. L(x !) is the radiance incoming towards x in the direction!. Similarly,L(x!!) is the radiance emitting from x along the direction!. This introduces the concept of light flow along different directions. Accordingly, the reflectance factorC(x)from equation 2.12 has changed intoC(x !).
The final goal is to calculate outgoing light flow (outgoing radiance) from the pointxto a certain pointCin the direction . Figure 2.4 shows a visual schematic of the same. Taking derivative ofEo(x) with respect to outgoing angle calculates
the outgoing radiance towards pointC. dEo(x) d = Z ⌦ Le(x!!) d d!+ Z ⌦ C(x !) d Li(x !)d! (2.14) From equation 2.1, dE
d is the outgoing radiance Lo(x ! ). Moreover, intuitively,
R
⌦Le(x!!)d1 d! is the emitted radiance fromxtowards direction . Thus,
Lo(x! ) =Le(x! ) +
Z
⌦
C(x !)
d Li(x !)d! (2.15)
Recalling equation 2.2, the ratio of the outgoing and the incoming radiance is, by definition, called the BRDF fr(x, !). Compensating for the angle ✓ of the
surface normal NC of pointCand the direction ( !o) as shown in figure 2.4,
Lo(x! ) =Le(x! ) +
Z
⌦
fr(x, !)Li(x !)(NA· !o)d! (2.16)
Figure 2.4: The light ray in yellow is coming from the point X towards the point C (usually the camera position) with an angle with respect to the red Surface normal of the unit area A around point X. The blue arrows indicate incoming light rays from all around the environment towards point X.
Physically-Based rendering techniques because it satisfies the energy conservation laws and accurately approximates the light energy flow from one surface point to another. Next section will describe the various relevant Physically-Based rendering methods used to render photorealistic images.
2.3.2 Solving the Rendering Equation
The rendering equation is essentially an integral of the form (x, y) = (x, y1) +
Z
K(x, y2) (x, y2)dy2 (2.17) Here, (x, y)is an arbitrary function with two domain variablesx andy. K(x, y) is a kernel function also with domainsx and y. This type of equations have the same functions (x, y) on the left hand side and on the right hand side inside the integral. These integrals are called Fredholm integrals after Ivor Fredholm [Fredholm, 1903] and can not be solved analytically but withResolvent formalism techniques where a discretisable property of the function can be represented as an infinite series (for instance, Liouville-Neumannn series) in order to approximate the solution of the equation. The two most popular way of solving the rendering equation are Point sampling methods and Finite element methods.
2.3.2.1 Point Sampling Methods:
Point sampling methods sample the entire domain to pick a finite number of points at random and evaluate the Fredholm Integral equations based on these individual values on each of these points. The number of points used to evaluate the integral is inversely proportional to the error. Although the sampling is a random process, there are many different sampling techniques which increase the efficiency of point sampling approaches by carefully picking the probability density function inside the domain of the function.
Section 2.4 describes the Monte Carlo approach [Metropolis and Ulam, 1949] of solving such integrals and also describes the most relevant sampling strategies. Section 2.5 will in turn describe the “Ray tracing” [Takagi, 1997; Whitted, 2005] and “Path tracing” [Lafortune and Willems, 1993] which are the point sampling methods to solve the rendering equation.
2.3.2.2 Finite Element Methods:
Unlike the point sampling methods, finite element methods are not dependent on random sampling. Instead, these methods divide the problem domain into smaller sub-domains with simpler solutions. Following the evaluation of the elemental equa- tions of the sub-domains, the evaluated values are strategically recombined into a global system to put together the entire solution of the integral.
The next section will describe the Monte-Carlo method as a Point sampling technique to solve the rendering equation followed by the “Ray-tracing” and “Path- tracing” techniques. Section 2.5 will describe “Radiosity” [Cohen and Wallace, 2012] and “Instant Radiosity” as part of the finite element methods to solve the rendering equation.