A Little about Radiosity Global Illumination

Radiosity: Hauntingly Beautiful

I saw “Radiosity Maps” option in Cinema 4D many years ago, but I didn’t understand what it was. Now, I understand, and I want you, faithful reader, to understand them as well. I’m basing this post on an old article I found on Libgen — an article from 1986 the trio of Donald P. Greenberg, Micheal F. Cohen, and Kenneth E. Torrance. I hope they don’t mind me basing my blog post on their article from 34 years ago!

We know that there’s three categories of light: ambient, diffuse, and specular. In Phong shading, which we talked about before, a simple formula is used to simulate lighting in the given scene. This, however, is not sufficient. It is enough for real-time rendering, where performance is the main concern, but when time is not of concern, we can use more complex formulae to calculate the lighting in our scene.

In 1979, Whitted published a paper in which he explained and talked about ray tracing, a method for producing images of excellent quality.

 

Raytracing, a noble way for noble men.

However, raytracing procedure is limited and can only model intra-environment reflectionss in specular direction. Additionally, shadows always exhibit sharp boundaries, and plus, since raytracing is view-dependant, for every view, there must be a new pass.

This, however, is not true of the radiosity method. This method renders Global Illumination independent of views.

In SIGGRAPH convention of 1984, Lady Cindy Goral exhibited a metohd that sparked a lot of interest and turned quite a few heads: Derived from thermal engineering, the field of Heat Transfer, they revealed a new Global Illumination method based on energy equilibrium, and called it the Radiosity Method.

 

The difference between Direct Illumination, i.e. Phong Lighting, and Radiosity is evident in the given picture. Bright lights, no more of the demented color bleeding which are caused by diffuse reflections, and softer shadows.

Two other papers were published in 1985 which introduced concepts such as hemi-cube which we’ll discuss later. For now, let’s talk about Radiosity, and what it exactly really is.

Radiosity Formulae

 The radiosity method explains energy equilibrium inside an enclosure. The light leaving the surface (its radiosity) consists of self-emitted light and reflected or transmitted incident light. The amount of light arriving at a surface requires complete specification of geometric relationships among all reflecting and transmitting surfaces, along with the light leaving every other surface. The formula of this relationship is:

Formula 1: Radiosity essence

B_i = E_i + \rho_i \sum_{j=1}^{N} B_j F_{ij} | i = 1 to N

Factors and coefficients are as follows:

Radiosity (B): The total rate of energy leaving a surface.

Emission (E): The rate of energy (light) emitted from a surface

Reflectivity (\rho): The fraction of incident light which is reflected back into the environment

Form-facor (F): The fraction of the energy leaving one surface which lands on another surface.

N = the number of discrete surfaces of patches.

What does this equation state, you may ask? Well, from this equation we understand that the amount of nergy leaving a particular surface is equal to the self-emitted light plus the reflected light. The reflected light is equal to the light leaving every other surface multiplied by both the fraction of that light which reaches the surface in question, and the reflectivity of the receiving surface.

We said that the form-factor is the fraction of the energy leaving one surface which lands on another surface. The formula for this is:

Formula 2: Form-factor essence

F_{ij} = \frac{1}{A_i} \int_{A_i} \int_{A_j} \frac{\cos\theta_i\cos\theta_j}{\pi r^2} HID_{ij}dA_jDA_i

Factors and coefficients can be seen in this figure:

 

Also, HID is a function which calculates the area of i and j.

The Hemi-Cube Method

We talked about the Hemi-Cube, but what is it? The Hemi-Cube Algorithm provides a numerical integration (you know, integrals) technique for evaluating Formula 2.  Instead of projecting into sphere, which is the normal procedure for lighting a scene, an imaginary cube is constructed around the center of the receiving patch. The environment is transformed to set the patch’s center at the origin with patch’s normal coinciding with the positive Z-axis (in other words, the tangent space!).

 

Then, the Hemi-Cube is divided into orhogonal mesh of pixels (the article puts “pixels” in quotes, as it was a novel word at the time!) at any desired resolution.

 

The Hemi-Cube divided into these so-called pixels… Witchcraft!

Ipso facto, the total value of the form-factor from the patch at the center of the hemi-cube to any patch j can be determined by the summation of these pixels.

Radiosity… GI?

As we’ve learned, Radiosity is a subset of GI, but it’s different from raytracing. Today, most 3D applications use Monte Carlo for their GI engine, which retains some aspects of Radiosity. Examples include Cinema 4D. It used to have Radiosity, but it changed it to GI, so people, mostly C4D users, think GI and Radiosity are different things.

Well, that is it! Dear reader, I appreciate your keen interest in my blog. Please leave a comment. If I’ve made an error, please alert me. I’m not perfect, I make mistakes.

Another thing. Please, if you’ve liked my blog, tweet it to your friends so they can enjoy it as well. Even if it marks you as the nerd of the group, please do it, chicks dig nerds these days. If you have a girlfriend, just take some MDMA and read my blog to her out loud, she’ll return the favor, I rather not say how. Besides tweeting you can reddit my blog. I post my blog in all the relevant subreddits, but I may miss one or two, or a million. So please, help me grow my blog’s readership. Thanks, Chubk.

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