Signal Processing - September 2016 - 73

absorbed for specific wavelength, and then exits the surface at
a range of angles with energy in wavelengths consistent with
what we perceive as the object's color. The process is complex
and difficult to model precisely. The diffuse and specular components of reflection are insufficient to model the rich visual
experience of real-world scenes including frosty snow, weathered copper, translucent marble, glossy paints, iridescent shells,
crushed velvet, woven burlap, aged asphalt, and metallic-flake
pigments. By measuring appearance directly, data-driven
reflectance models can be used to build representations that are
tuned to specific appearance classes.
The intensity of light reflected from a surface point when
illuminated at angle i i, z i and viewed from angle i v, z v is
described by the bidirectional reflectance distribution function
(BRDF) expressed as f (i i, z i, i v, z v) as illustrated in Figure 6.
The BRDF is defined as the ratio of the radiance exiting a
surface point to the irradiance incident on the surface point
[44]. The units of the BRDF are inverse steradians (sr-1), where
the steridian is the unit of solid angle. To parse these units,
consider that input light (irradiance) is the power per unit area
and has units of watts per meter squared (watts/m2). The total
output light intensity from a unit area of the surface has units
of watts/m2 but it radiates in a hemisphere of possible directions, so the output light in a particular direction has units of
watts/m2 per steridian. Each direction is represented by a solid
angle so that the integration over all directions represents the
entire 3-D space. Therefore the ratio of output light radiance
to input light irradiance as expressed by the BRDF has units
sr-1. To denote dependence on both viewing and illumination
angles, the BRDF is expressed as f (i i, z i, i v, z v) . Real-world
surfaces typically do not have a uniform BRDF due to both
surface markings and surface texture.
The bidirectional texture function (BTF) extends the BRDF
to characterize surface reflectance that varies spatially. The early
concept of BTF was introduced with the Columbia-Utrecht
Texture and Reflectance (CUReT) database [11], [12] and has
been used for numerous texture modeling and recognition studies. The BTF expressed as f (x, y, i i, z i, i v, z v) has dependence
on spatial parameters x, y and angular parameters. The BRDF
assumes a point-wise light transport relationship, incident light
at a point in a single direction results in exitant light at the same
point from multiple directions. However, because of subsurface
scattering, light incident on a point exits over a surface patch.
The BTF can model these effects by capturing patch-wise
reflection as shown in Figure 7. Light incident at a patch at a
particular direction results in reflected light from the patch over
a hemispherical range of directions. BTF modeling typically
assumes that incident light is uniform over the patch. In addition
to subsurface scattering, the BTF representation is useful for
capturing reflection from fine-scale geometry of textured surfaces such as bumps, wrinkles, and roughness. The fine-scale
shadowing, occlusions, shading, and foreshortening that affect
the pixel intensities of the recorded images become part of the
appearance model implicitly without knowledge of the surface
fine-scale geometric variation. The reflectance at each point
contains the nonlinearities of the shadowing and occlusions

of fine-scale geometry. For example, surface point at x, y may
be shadowed as the illumination direction changes from i i to
i i + d for some small angle d, causing an abrupt change in the
BTF to near zero reflectance. The BTF model can also be used to
texture-map a 3-D object represented by a polygonal mesh. The
3-D mesh is texture-mapped, not with a single image, but with
a BTF. Traditional texture mapping maps each 3-D vertex into a
two-dimensional (2-D) texture image parameterized by texture
coordinates u, v. The sampled BTF is a collection of images, so
that a 3-D object vertex is mapped to f (u, v, i i, z i, i v, z v) where
the illumination and viewing direction are defined with respect
to the mesh facet. BRDF/BTF measurements are spatially local
in their description, concentrating on describing the appearance
of a surface point or patch. Such a description is ideal for surfaces that exhibit spatial invariance where the appearance of the
patch is representative of the general appearance as in studies of
textured surfaces.
While BRDF is a pointwise reflectance measurement and
BTF is a patch-based reflectance measurement, the reflectance
of an entire scene can also be captured globally. Light fields and
reflectance fields describe the global reflectance of the entire
scene or entire object. Light fields are defined as radiance as
a function of position and direction [22], [32] and are fourdimensional (4-D) since they describe a spatial position with
two variables and ray orientation with two angles (polar angle
and azimuth angle). An eight-dimensional (8-D) reflectance
field [15] describes both the incident 4-D light field as well as
the 4-D exitant light field. Reflectance fields are analogous to
BRDF's since both representations are bidirectional, describing the direction of incident light and exitant light. However,
reflectance fields describe the input/output light over the global scene instead of a local point. Conceptually, light fields and
reflectance fields construct a closed surface such as a sphere
(or cube), surrounding the scene. The point on the closed surface can be parameterized by two variables that depict the
spatial position. With the assumption of a convex scene, a ray
emanating from each scene point can be constructed that intersects the closed surface. For each point on the closed surface,

BRDF

BTF

Figure 7. The BRDF describes light reflected from a surface point in a
hemisphere of possible directions, due to light incident to a surface
point at a particular angle. However, light incident at a point may be
interreflected and may be partially transmitted and scattered resulting in
light exiting the surface at multiple points. The BTF describes light exiting
a surface patch due to light incident on the patch at a specified angle accounting for interreflections and subsurface scattering.

IEEE SIgnal ProcESSIng MagazInE

September 2016

Table of Contents for the Digital Edition of Signal Processing - September 2016