Rendering Equation

Synonyms

Digital image; Global illumination; Rendering

Definitions

Global Illumination:

Global interaction between the light sources and the surfaces of environment.

Pixel:

Abbreviation of the English term picture element.

Radiative Transport:

Area that studies the interaction of radiant energy with matter, on a macroscopic scale.

Transference Equation:

Integral-differential equation that describes the interaction of light with the participant medium and, given the appropriate boundary conditions, its interaction with arbitrary surfaces.

Transport Theory:

Field that embraces all macroscopic phenomena that result from the interaction of infinitesimal particles with the medium. Macroscopic behavior of photons, neutrons, and gas molecules are all within their scope.

Introduction

The basic and fundamental problem of Computer Graphics is to transform data into image. Digital image is the result of most of the processes involved in computer graphics. A mathematical model to describe an image...

Appendices

Appendix A

Mathematical Foundations

This appendix used as reference the work (Gomes and Velho 2008) and aims to highlight some important mathematical concepts for the study in question.

Spherical Coordinates

The unit sphere S ² ⊂ ℝ³, S ² = {(x, y, z) ∈ ℝ³; x ² + y ² + z ² = 1} has a natural parameterization, called spherical coordinates, which is based on the fact that each point of S ² is characterized by the angles of longitude θ and latitude ϕ, as illustrated in Fig. 13.

Fig. 13

Spherical coordinates

From figure:

$$ {\displaystyle \begin{array}{c}x=x\left(\theta, \phi \right)= cos\phi cos\theta; \\ {}y=y\left(\theta, \phi \right)= cos\phi sen\theta; \\ {}z=z\left(\theta, \phi \right)= sen\phi .\end{array}} $$

Solid Angle

The concept of solid angle is used in the definition of radiometric quantities by describing flows through a given region of a surface in a particular direction. To define a solid angle, one must define the concept of spherical projection.

Given a sphere S ⁿ⁻¹(r) of radius r in ℝⁿ with center at the origin, the spherical projection is the transformation p : ℝⁿ − {0} → S ⁿ⁻¹(r) defined by p(x) = rx/ ∣ x∣. Geometrically this transformation projects the point x on the surface of the sphere along the radius that leaves the origin 0 of ℝⁿ and passes through x, as illustrated in Fig. 14.

Fig. 14

Spherical projection

Consider a unit circle with center O. The concept of planar angle comes from the projection of an object in the circle. The measure of the arc of the circle containing the projection of the object corresponds to the planar angle.

The solid angle concept is an extension of the planar angle concept to measure the apparent area of objects in space. Consider a subset A of space, and a point O, which is the origin of observation. The measure of the solid angle ω determined by A is obtained as follows: we take a unit sphere S ² with center in O and the spherical projection p(A) of the set A on the sphere S ². The area of p(A) is the measure of ω. Figure 15a illustrates this concept.

Fig. 15

Notion of extended angle to measure the apparent area of objects

If it is necessary to take a sphere of radius r with center O, simply divide the projected area by the square of the radius. The measure of solid angle is given by:

$$ \omega =\frac{\mathrm{Area}\left(p(A)\right)}{r^2}. $$

(72)

The spherical projection determines a cone of vertex O and base A, as illustrated in Fig. 15b. This cone is the solid formed by the set of all semi-straight lines of the space originating from O and passing through points of the set A. The cone generalizes to three-dimensional space the geometric notion of the angular region of the plane geometry, and the Eq. 72 is a measure of this three-dimensional angular region.

According to Gomes and Velho (2008), both planar and solid angle are dimensionless quantities. In the first case, the arc length is divided by the radius of the circle, and in the second case an area is divided by the square of the radius length. However, the radian (rd) is used to indicate the measure of a planar angle and the stereo-radian (sr) as the unit of measure of a solid angle.

The light phenomena depend on the position of observer and therefore in general what matters is the projected area, which is the projection of the real area in the plane perpendicular to the radius of view. The projection of visualization is conical; however, if the area is very small in relation to the distance of the observer, it is possible to suppose that the projection is parallel (Fig. 16).

Fig. 16

Projected area

In this case, it is immediate to verify that the projected area is given by the product of real area A by cosine of the angle that the radius of vision does with the normal vector at the surface A:

$$ \mathrm{rea}\ \mathrm{projetada}= Acos\theta . $$

(73)

Solid Angle Element

Radiometric quantities are defined by the use of rates of change of radiant energy entering or leaving a surface. In this way, it is important to define the infinitesimal elements of area (dA) and of solid angle (dω). Consider an area element dA of a surface A of space, and let x ∈ A be a point of that area element at a distance r from the origin O, as illustrated in Fig. 17.

Fig. 17

Solid angle element

This element of area determines a solid angle element dω from the point O in direction u defined by the origin O and the point x. By the definition of solid angle, the measure of dω is given by:

$$ d\omega =\frac{\mathrm{Area}\left(p(dA)\right)}{r^2}. $$

(74)

Since these are infinitesimal magnitudes, it is possible to suppose that dA is a flat surface and to replace the spherical projection by a parallel projection along direction u. Using the Eq. 73 in the Eq. 74, we have that:

$$ d\omega =\frac{dAcos\theta}{r^2}. $$

(75)

The geometry of area dA can be arbitrary. In the case of a sphere, it is natural to represent the element of solid angle by a pyramid rather than a cone. Consider the sphere of radius r with spherical coordinates θ (azimuth) and ϕ (longitude) as illustrated in Fig. 18.

Fig. 18

Solid angle element in spherical coordinates

To determine the solid angle element at a point of spherical coordinates (ϕ, θ), one must define the area element dA of sphere. The planar angle element of a meridian passing through the point is dθ, so the corresponding arc element is rdθ.

The planar angle element of the equator is dϕ, projecting the point (ϕ, θ) in the plane of equator we have that the projected radius is rsenθ; therefore, the corresponding arc element is rsenθdϕ. The sphere area element dA is the product of these two arc elements:

$$ dA=\left( rd\theta \right)\left( rsen\theta d\phi \right)={r}^2 sen\theta d\theta d\phi . $$

(76)

Using Eq. 75, we obtain the measure of the infinitesimal element of the solid angle dω in spherical coordinates:

$$ d\omega =\frac{dA}{r^2}= sen\theta d\theta d\phi . $$

(77)

Integrating the volume element above into the whole sphere, one obtains the solid angle of the sphere:

$$ \varOmega ={\int}_0^{\pi }{\int}_0^{2\pi } sen\theta d\theta d\phi =4\pi . $$

(78)

Therefore, the solid angle of a hemisphere is 2π.

Cross-References

• Cascade Interactive Rendering for Water

• Comic Art in Games, Asset Production and Rendering

• Computer Representations of Trees for Realistic and Efficient Rendering

• Raycasting in Virtual Reality