Bidirectional Reflectance Function

The bidirectional reflectance distribution function (BRDF; $f_(\omega_,\, \omega_)$ ) is a function of four real variables that defines how light is reflected at an opaque surface. It is employed in the optics of real-world light, in computer graphics algorithms, and in computer vision algorithms. The function takes an incoming light direction, $\omega_$, and outgoing direction, $\omega_$ (taken in a coordinate system where the surface normal $\mathbf n$ lies along the ''z''-axis), and returns the ratio of reflected radiance exiting along $\omega_$ to the irradiance incident on the surface from direction $\omega_$. Each direction $\omega$ is itself parameterized by azimuth angle $\phi$ and zenith angle $\theta$, therefore the BRDF as a whole is a function of 4 variables. The BRDF has units sr⁻¹, with steradians (sr) being a unit of solid angle.

Definition

The BRDF was first defined by Fred Nicodemus around 1965.
The definition is:

$f_(\omega_,\, \omega_) \,=\, \frac \,=\, \frac\frac$

where $L$ is radiance, or power per unit solid-angle-in-the-direction-of-a-ray per unit projected-area-perpendicular-to-the-ray, $E$ is irradiance, or power per unit ''surface area'', and $\theta_$ is the angle between $\omega_$ and the surface normal, $\mathbf n$. The index $\text$ indicates incident light, whereas the index $\text$ indicates reflected light.

The reason the function is defined as a quotient of two differentials and not directly as a quotient between the undifferentiated quantities, is because other irradiating light than $\mathrm dE_(\omega_)$, which are of no interest for $f_(\omega_,\, \omega_)$, might illuminate the surface which would unintentionally affect $L_(\omega_)$, whereas $\mathrm dL_(\omega_)$ is only affected by $\mathrm dE_(\omega_)$.

Related functions

The Spatially Varying Bidirectional Reflectance Distribution Function (SVBRDF) is a 6-dimensional function, $f_(\omega_,\,\omega_,\,\mathbf)$, where $\mathbf$ describes a 2D location over an object's surface.

The Bidirectional Texture Function (BTF) is appropriate for modeling non-flat surfaces, and has the same parameterization as the SVBRDF; however in contrast, the BTF includes non-local scattering effects like shadowing, masking, interreflections or subsurface scattering. The functions defined by the BTF at each point on the surface are thus called Apparent BRDFs.

The Bidirectional Surface Scattering Reflectance Distribution Function (BSSRDF), is a further generalized 8-dimensional function $S(\mathbf_,\,\omega_,\,\mathbf_,\,\omega_)$ in which light entering the surface may scatter internally and exit at another location.

In all these cases, the dependence on the wavelength of light has been ignored. In reality, the BRDF is wavelength dependent, and to account for effects such as iridescence or luminescence the dependence on wavelength must be made explicit: $f_(\lambda_,\,\omega_,\,\lambda_,\,\omega_)$. Note that in the typical case where all optical elements are linear, the function will obey $f_(\lambda_,\,\omega_,\,\lambda_,\,\omega_) = 0$ except when $\lambda_ = \lambda_$: that is, it will only emit light at wavelength equal to the incoming light. In this case it can be parameterized as $f_(\lambda,\,\omega_,\,\omega_)$, with only one wavelength parameter.

Physically based BRDFs

Physically realistic BRDFs have additional properties,
including,
* positivity: $f_(\omega_,\, \omega_) \ge 0$
* obeying Helmholtz reciprocity: $f_(\omega_,\, \omega_) = f_(\omega_,\, \omega_)$
* conserving energy: $\forall \omega_,\, \int_\Omega f_(\omega_,\, \omega_)\,\cos d\omega_ \le 1$

Applications

The BRDF is a fundamental radiometric concept, and accordingly is used in computer graphics for photorealistic rendering of synthetic scenes (see the rendering equation), as well as in computer vision for many inverse problems such as object recognition. BRDF has also been used for modeling light trapping in solar cell

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