Bidirectional reflectance distribution function

The bidirectional reflectance distribution function (BRDF; f r ( ω i , ω r ) {displaystyle f_{ ext{r}}(omega _{ ext{i}},,omega _{ ext{r}})}  ) 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, ω i {displaystyle omega _{ ext{i}}} , and outgoing direction, ω r {displaystyle omega _{ ext{r}}} (taken in a coordinate system where the surface normal n {displaystyle mathbf {n} } lies along the z-axis), and returns the ratio of reflected radiance exiting along ω r {displaystyle omega _{ ext{r}}} to the irradiance incident on the surface from direction ω i {displaystyle omega _{ ext{i}}} . Each direction ω {displaystyle omega } is itself parameterized by azimuth angle ϕ {displaystyle phi } and zenith angle θ {displaystyle heta } , 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. The bidirectional reflectance distribution function (BRDF; f r ( ω i , ω r ) {displaystyle f_{ ext{r}}(omega _{ ext{i}},,omega _{ ext{r}})}  ) 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, ω i {displaystyle omega _{ ext{i}}} , and outgoing direction, ω r {displaystyle omega _{ ext{r}}} (taken in a coordinate system where the surface normal n {displaystyle mathbf {n} } lies along the z-axis), and returns the ratio of reflected radiance exiting along ω r {displaystyle omega _{ ext{r}}} to the irradiance incident on the surface from direction ω i {displaystyle omega _{ ext{i}}} . Each direction ω {displaystyle omega } is itself parameterized by azimuth angle ϕ {displaystyle phi } and zenith angle θ {displaystyle heta } , 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. The BRDF was first defined by Fred Nicodemus around 1965. The definition is: f r ( ω i , ω r ) = d ⁡ L r ( ω r ) d ⁡ E i ( ω i ) = d ⁡ L r ( ω r ) L i ( ω i ) cos ⁡ θ i d ⁡ ω i {displaystyle f_{ ext{r}}(omega _{ ext{i}},,omega _{ ext{r}}),=,{frac {operatorname {d} L_{ ext{r}}(omega _{ ext{r}})}{operatorname {d} E_{ ext{i}}(omega _{ ext{i}})}},=,{frac {operatorname {d} L_{ ext{r}}(omega _{ ext{r}})}{L_{ ext{i}}(omega _{ ext{i}})cos heta _{ ext{i}},operatorname {d} omega _{ ext{i}}}}} where L {displaystyle L} is radiance, or power per unit solid-angle-in-the-direction-of-a-ray per unit projected-area-perpendicular-to-the-ray, E {displaystyle E} is irradiance, or power per unit surface area, and θ i {displaystyle heta _{ ext{i}}} is the angle between ω i {displaystyle omega _{ ext{i}}} and the surface normal, n {displaystyle mathbf {n} } . The index i {displaystyle { ext{i}}} indicates incident light, whereas the index r {displaystyle { ext{r}}} 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 d ⁡ E i ( ω i ) {displaystyle operatorname {d} E_{ ext{i}}(omega _{ ext{i}})} , which are of no interest for f r ( ω i , ω r ) {displaystyle f_{ ext{r}}(omega _{ ext{i}},,omega _{ ext{r}})} , might illuminate the surface which would unintentionally affect L r ( ω r ) {displaystyle L_{ ext{r}}(omega _{ ext{r}})} , whereas d ⁡ L r ( ω r ) {displaystyle operatorname {d} L_{ ext{r}}(omega _{ ext{r}})} is only affected by d ⁡ E i ( ω i ) {displaystyle operatorname {d} E_{ ext{i}}(omega _{ ext{i}})} . The Spatially Varying Bidirectional Reflectance Distribution Function (SVBRDF) is a 6-dimensional function, f r ( ω i , ω r , x ) {displaystyle f_{ ext{r}}(omega _{ ext{i}},,omega _{ ext{r}},,mathbf {x} )} , where x {displaystyle mathbf {x} } 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 ( x i , ω i , x r , ω r ) {displaystyle S(mathbf {x} _{ ext{i}},,omega _{ ext{i}},,mathbf {x} _{ ext{r}},,omega _{ ext{r}})} 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 and binned into RGB channels. 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 r ( λ i , ω i , λ r , ω r ) {displaystyle f_{ ext{r}}(lambda _{ ext{i}},,omega _{ ext{i}},,lambda _{ ext{r}},,omega _{ ext{r}})} . Note that in the typical case where all optical elements are linear, the function will obey f r ( λ i , ω i , λ r , ω r ) = 0 {displaystyle f_{ ext{r}}(lambda _{ ext{i}},,omega _{ ext{i}},,lambda _{ ext{r}},,omega _{ ext{r}})=0} except when λ i = λ r {displaystyle lambda _{ ext{i}}=lambda _{ ext{r}}} : that is, it will only emit light at wavelength equal to the incoming light. In this case it can be paramaterized as f r ( λ , ω i , ω r ) {displaystyle f_{ ext{r}}(lambda ,,omega _{ ext{i}},,omega _{ ext{r}})} , with only one wavelength parameter.