Introduction

These Slides

They are available at ziotom78.github.io/raytracing_course/.

General Guidelines

• Monday theory lessons are recorded and uploaded to Ariel

• Attendance is required for laboratories (sign-in required)

• Questions outside of lessons:

  • If related to theory topics or of general interest, ask on the Ariel forum

  • If specific to your own code, contact the instructor (maurizio.tomasi@unimi.it)

Schedule

• Next week (March 3): no lessons as I will be at a conference
• Week of April 14: no lessons due to midterm exams
• Week after Easter: “special” lesson on April 23
• Week of June 2: no lessons
• Week of June 9: end of the course

Course Objectives

1. How to translate a physical model into numerical code?
2. How to write well-structured and documented code?
3. How to generate photorealistic images?

This is a multidisciplinary course: not reserved for cosmologists and astrophysicists!

Course Objectives

1. How to translate a physical model into numerical code?
2. How to write well-structured and documented code?
3. How to generate photorealistic images?

This is a multidisciplinary course: not reserved for cosmologists and astrophysicists!

Numerical Solutions

The Usefulness of Simulations

Planck: ESA space mission (2009–2013)

The Usefulness of Simulations

Assembly of the probe in ESA laboratories

The Usefulness of Simulations

The Usefulness of Simulations

Course Objectives

1. How to translate a physical model into numerical code?
2. How to write well-structured and documented code?
3. How to generate photorealistic images?

Public repositories

List of changes

Automatic tests

Bug tracking

Teamwork

Course Objectives

1. How to translate a physical model into numerical code?
2. How to write well-structured and documented code?
3. How to generate photorealistic images?

Image Generation

Vivian Maier (1926–2009), Self-portrait

Oceania (R. Clements, J. Musker, D. Hall, C. Williams, 2016)

«Cornell box»

«Cornell box»

Bibliography

• Physically Based Rendering: from Theory to Implementation (M. Pharr, W. Jakob, G. Humphreys, 4th ed.): quite complex but complete, it’s the gold standard for this topic. It’s available online.
• Advanced Global Illumination (P. Dutré, K. Bala, P. Bekaert, 2nd ed.): we’ll use it for the most “physical” parts.
• Realistic Ray Tracing (P. Shirley, R. K. Morley, 2nd ed.): old-fashioned, it’s useful just as an introductory text.

Radiometry

Light Propagation

• Quantum optics (not used in computer graphics)
• Wave model (diffraction, e.g., soap bubbles)
• Geometrical optics

(see Dutré, Bala, Bekaert)

Geometrical Optics

• Light propagates along straight lines (geodesics)
• The speed of light is assumed to be infinite
• The wavelength is assumed to tend to zero (frequency → ∞)
• Propagation is not affected by gravitational or magnetic effects

Why Do We Need Radiometry?

• In this course, we will deal with radiometry, the science that studies how radiation propagates through a medium.
• The important goal is to define quantities that characterize radiation as independently as possible from the instruments that measure it.

Radiometric Quantities

• Emitted energy (Joules)
• Radiant power, or flux (Energy passing through a surface per unit time)
• Irradiance, radiant emittance (flux normalized over a surface)
• Radiance (← the core of the course!)
• The definitions we will use are those used in computer graphics, but they may differ in other fields of physics! (For example, in astronomy, irradiance is called flux).

Flux

• Energy passing through a surface A per unit time: \Phi
• [\Phi] = \mathrm{W} (in physics, flux is instead measured as \mathrm{W}/\mathrm{m}^2!).
• Example: A is the surface of a detector, such as the human pupil or a camera lens.
• The larger the surface A, the greater the flux \Phi.

Irradiance/Emittance

• Flux \Phi normalized over the surface: I, E = \frac{\mathrm{d}\Phi}{\mathrm{d}A}, \qquad [I] = \mathrm{W}/\mathrm{m}^2.

• Irradiance I: what falls on \mathrm{d}A; emittance E: what leaves \mathrm{d}A.

Radiance

• This is what interests us!
• Flux \Phi normalized over the projected surface per unit solid angle: L = \frac{\mathrm{d}^2\Phi}{\mathrm{d}\Omega\,\mathrm{d}A^\perp} = \frac{\mathrm{d}^2\Phi}{\mathrm{d}\Omega\,\mathrm{d}A\,\cos\theta}, \qquad [L] = \mathrm{W}/\mathrm{m}^2/\mathrm{sr}.
• Like irradiance and emittance, radiance is a function of the point \mathbf{x} on the surface \mathrm{d}A: L = L(\mathbf{x}).

Solid Angles and Distance

• Since I \propto A^{-1} \propto d^{-2}, the irradiance on \mathrm{d}A at 3d is 1/9 of that at d.
• But \mathrm{d}\Omega = dA/d^2 \propto d^{-2}.
• Therefore, L \propto I/\mathrm{d}\Omega does not depend on d.

Radiance

• The ratio over the solid angle removes the dependence on the distance.
• The presence of \cos\theta removes the dependence on the orientation of \mathrm{d}A.

Notation for Radiance

• We will often use the notation L(\mathbf{x} \rightarrow \Theta) to indicate the radiance leaving a surface at point \mathbf{x} towards direction \Theta, which is associated with a solid angle \mathrm{d}\Omega.

• Similarly, L(\mathbf{x} \leftarrow \Theta) represents the radiance coming from direction \Theta and incident on \mathbf{x}.

Emission Spectra

• Each of the quantities discussed so far are actually wavelength-dependent.

• Different light sources have different spectra:

Spectral Radiance

• From radiance L(\mathbf{x} \leftrightarrow \Theta), we can define spectral radiance L_\lambda(\mathbf{x} \leftrightarrow \Theta), which refers to the wavelength interval [\lambda, \lambda + \mathrm{d}\lambda] and is denoted by the same letter L for convenience.

• It is defined by the equation

  L(\mathbf{x} \leftrightarrow \Theta) = \int_0^\infty L_\lambda(\mathbf{x} \leftrightarrow \Theta)\,\mathrm{d}\lambda, \ {} [L_\lambda(\mathbf{x} \leftrightarrow \Theta, \lambda)] = \mathrm{W}/\mathrm{m}^2/\mathrm{sr}/\mathrm{m}.

Properties of L

1. From L, we can derive \Phi, I, and E. For example:

   \Phi = \iint_{A, \Omega} L(\mathbf{x} \rightarrow \Theta)\, \cos\theta\,\mathrm{d}\Omega\,\mathrm{d}A_\mathbf{x},

2. In the absence of attenuation, it holds that L(\mathbf{x} \rightarrow \mathbf{y}) = L(\mathbf{x} \rightarrow \mathbf{z}), if \mathbf{x}, \mathbf{y}, \mathbf{z} lie on the same line; the same applies to L_\lambda.

3. The fact that L and L_\lambda do not depend on distance implies that the perceived color of an object at distance d does not change as d varies (if there is no attenuation).

Importance of L

L is what is measured by any light-sensitive sensor (camera, human eye).

Importance of L_\lambda

The behavior of L_\lambda as \lambda varies allows us to estimate color (hue):

(There is also a third parameter, saturation, which we will discuss shortly.)

Image Creation

Estimating L and L_\lambda together allows us to produce a color image:

Example

Consider a diffuse emitter, an object that emits light uniformly in all directions:

In this case, L(\mathbf{x} \rightarrow \Theta) = L_e\qquad\text{(constant)}.

Flux Calculation

\begin{aligned} \Phi &= \iint_{A, \Omega} L(\mathbf{x} \rightarrow \Theta)\,\cos\theta\,\mathrm{d}\Omega\,\mathrm{d}A =\\ &= \iint_{A, \Omega} L_e\,\cos\theta\,\mathrm{d}\Omega\,\mathrm{d}A =\\ &= L_e \int_A \mathrm{d}A \int_\Omega \cos\theta\,\mathrm{d}\Omega =\\ &= L_e \int_A \mathrm{d}A \int_0^{2\pi}\mathrm{d}\phi \int_0^{\pi/2}\mathrm{d}\theta \cos\theta\,\sin\theta =\\ &= \pi A L_e.\\ \end{aligned}

Light/Surface Interaction

“Cornell box”

The BRDF

The Bidirectional Reflectance Distribution Function (BRDF) is the ratio f_r(x, \Psi \rightarrow \Theta) between the radiance leaving a surface along \Theta and the irradiance (flux normalized over A, \mathrm{W}/\mathrm{m}^2) received from a direction \Psi:

\begin{aligned} f_r(x, \Psi \rightarrow \Theta) &= \frac{\mathrm{d}L (x \rightarrow \Theta)}{\mathrm{d}I(x \leftarrow \Psi)} = \\ &= \frac{\mathrm{d}L (x \rightarrow \Theta)}{ L(x \leftarrow \Psi) \cos(N_x, \Psi)\,\mathrm{d}\omega_\Psi }, \end{aligned} where \cos(N_x, \Psi) is the angle between the normal to \mathrm{d}A and the incident direction \Psi.

The BRDF

The BRDF is not defined as the ratio between two radiances, because f_r will be used to calculate the total outgoing radiance L_\text{tot} \sim \int_{2\pi} L\,f\,\mathrm{d}\omega. If f were a pure number or if A were “corrected” for the angle \theta, the integral would become more complicated.

Meaning of the BRDF

• Describes how a surface interacts with light;
• f_r \propto \cos^{-1}(N_x, \Psi): the inclination of the light source with respect to \mathrm{d}A is taken into account.
• f_r : \mathbb{R}^3 \times \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R} (x\times\Theta\times\Psi \rightarrow f_r), but in the most general case it also depends on \lambda and time t;
• It is a positive function: f_r \geq 0, and its unit of measure is 1/\mathrm{sr};
• The entire solid angle 4\pi is considered, because the BRDF is also used for (semi-)transparent surfaces.
• It assumes that light leaves the surface from the same point x where it encountered it (not true for subsurface scattering!).

Helmholtz Reciprocity

• The Helmholtz reciprocity holds for the BRDF: f_r(x, \Psi\rightarrow\Theta) = f_r(x, \Theta\rightarrow\Psi), that is, the BRDF does not change if the incoming direction is exchanged with the outgoing one.

• This property can be demonstrated using Maxwell’s equations, but the demonstration is long and not particularly interesting for our purposes.

Multiple Incidence Angles

Thanks to the superposition principle of e.m. fields, if there are multiple light sources i=1\ldots N that insist on a surface, it is sufficient to sum the components:

\begin{aligned} \mathrm{d}L_i(x \rightarrow \Theta) &= f_r(x, \Psi_i \rightarrow \Theta) \mathrm{d}E(x \leftarrow \Psi_i) =\\ &= f_r(x, \Psi_i \rightarrow \Theta) L(x \leftarrow \Psi_i)\,\cos(N_x, \Psi_i)\,\mathrm{d}\omega_{\Psi_i},\\ L_\text{tot}(x \rightarrow \Theta) &= \int_{\Omega_x} f_r(x, \Psi \rightarrow \Theta) L(x \leftarrow \Psi)\,\cos(N_x, \Psi)\,\mathrm{d}\omega_\Psi. \end{aligned}

Utility of the BRDF

Ideal Diffusive Surface

All incident radiation is distributed over the 2\pi hemisphere, so the BRDF is constant: f_r(x, \Psi \rightarrow \Theta) = \frac{\rho_d}\pi, where 0 \leq \rho_d \leq 1 is the fraction of incident energy that is reflected.

Other BRDFs

• A perfectly reflective surface is modeled by a Dirac Delta, that is, identically null except in the outgoing direction R given by the reflection law: R = 2(N \cdot \Psi) N - \Psi, where N is the normal (tangent) vector to the surface.

• Online libraries of BRDFs exist, usually obtained from laboratory measurements, almost all of which are paid.

The Rendering Equation

Formulation of the Equation

The equation we will study during the course is the following: \begin{aligned} L(x \rightarrow \Theta) = &L_e(x \rightarrow \Theta) +\\ &\int_{\Omega_x} f_r(x, \Psi \rightarrow \Theta)\,L(x \leftarrow \Psi)\,\cos(N_x, \Psi)\,\mathrm{d}\omega_\Psi, \end{aligned} where L_e is the radiance emitted by the surface at point x along the direction \Theta.