Appearance Models

To further study the appearance of objects, we will introduce two fundamental radiometric quantities. The radiance is the amount of light that passes through a particular area and falls within a given solid angle in a specified direction. In

100 5. PHOTO-REALISTICTEXTURERECONSTRUCTION

(a) Rendered geometry of an indoor scene.

(b) Texture mapped onto the geome-try.

Figure 5.1: Texture mapping is one of the most successful techniques to create high quality and visually realistic 3D models. Although the geometric model (a) gives us a good idea about the scene, we can get a much more realistic picture by reconstructing and mapping a texture onto the mesh surface (b).

contrast, the irradiance represents the amount of light falling onto a surface. For example, if an ideal point light source radiates uniformly in all directions and there is no absorption, then the irradiance is reduced in proportion to the inverse square of the distance to the source.

The surface appearance can be thought of as a combination of the both concepts.

Whenever a light sources illuminates a surface we can measure the irradiance at a particular point. Based on its material, the surface emits all or only a portion of the irradiance, and we can measure the emitted light at a view point. The emitted light can vary with direction (directional distribution), and we are interested in the amount of light emitted per unit surface area. Hence, we arrive at the definition of radiance: namely, the power emitted per unit area (perpendicular to the viewing direction) per unit solid angle. This is perhaps the most fundamental unit in com-puter vision and graphics. It is easy to show that the irradiance on the observer’s retina or a camera sensor is proportional to the radiance of the observed surfaces.

One way to model the surface appearance is by describing reflectance properties.

Unfortunately, reflectance itself is a complex phenomenon. In general, a surface may reflect a different amount of light at each position, and for each possible direction of incident and reflected light (see Figure 5.2(a)). A common way to model the reflection is the Bidirectional Reflectance Distribution Function (BRDF)

5.1. INTRODUCTION 101 [Nicodemus, 1965], which defines the reflection at an opaque surface. This 4-dimensional function

fr(ωi, ωo) = Lo(ωo)

Li(ωi) (5.1)

returns the ratio of reflected radiance L_oexiting along the outgoing direction ω_o, to the irradiance incident L_i on the surface from the incoming light direction ω_i. Each direction ω is itself parameterized by the two polar angles φ and θ with respect to the surface normal n. Following this definition, the reflected radiance at a surface point p is calculated with:

Lo(ωo) = Z

Ω_p

Li(ωi) fr(ωi, ωo) cos θidωi, (5.2)

where the integral is defined over the 3D upper hemisphere Ωp at the point p.

Here θi denotes the angle made between ωi and the surface normal n. There exists a large variety of lower dimensional approximations for the BRDF exploit-ing common material properties such as homogeneity and isotropy (radiance is unchanged if the incoming and outgoing vectors are rotated by the same amount around the surface normal). Two common examples are the Lambertian BRDF [Wolff et al., 1992] which assumes fr := const. resulting in a matte or diffuse appearance, and the Blinn-Phong BRDF [Blinn, 1977], which models the reflec-tions as a lobe centered around the direction of an ideal mirror reflection for each incident angle that contains significantly more energy than the rest. This model is designed to represent glossy materials. As we can see, the reflection functions embody a significant amount of information. They can tell us whether a surface is shiny or matte, metallic or dielectric, smooth or rough. Knowing the reflectance function of a surface allows us to make complete predictions of how that surface appears under any possible lighting. For real surfaces the BRDF is spatially vary-ing since the surface material itself is varyvary-ing. This spatial variation adds two more dimensions to the BRDF and is then called the Spatial Varying Bidirectional Reflectance Distribution Function (SVBRDF).

Even the SVBRDF is not enough to characterize all materials. Many surfaces exhibit translucency: a phenomenon in which light enters the object, is reflected inside the material, and eventually re-emerges from a different point on the sur-face. Such sub-surface scattering can have a dramatic effect on appearance.

The Bidirectional Scattering-Surface Reflection Distribution Function (BSSRDF) models the phenomenon of light leaving the surface at a different point than the one at which it entered. This is done by adding two more dimensions to the SVBRDF:

BSSRDF (pi, ωi, po, ωo) (5.3)

102 5. PHOTO-REALISTICTEXTURERECONSTRUCTION

(a) Reflection. (b) Sub-surface scattering.

Figure 5.2: The reflection at an opaque surface can be thought of as a function of the incoming light direction ωiand the outgoing ωolight direction (left). Sub-surface scattering is a phenomenon in which light enters the object, is reflected inside the material, and eventually re-emerges from a different point on the surface (right).

Unfortunately, even the BSSRDF is not enough. Moreover, some surfaces are fluorescent: they emit light at different wavelengths than those present in the in-cident light. Some other surfaces may have appearance that changes over time because of chemical changes or physical processes such as drying or weathering.

Other surfaces might capture light and re-emit it later leading to phosphorescence and other such phenomena. Thus, a complete description of light scattering at a surface needs to add at least two wavelength (λi, λ_o) and two time dimensions (t_i, t_o) to the BSSRDF. Thus, the full description of the material appearance is the 12-dimensional scattering function. A taxonomy of different reflection and appearance models is presented in Figure 5.3.