Illumination Study of how different materials reflect light Definition of radiance, the fundamental unit of light transfer in computer graphics How the BRDF f r encapsulates the reflectance properties of a material Global illumination, describes how light gets from one local illumination context to another

BRDF-based Lighting

Physics of Light Color Radiometry BRDF Part I – Theory

How light interacts with matter? Complicated dynamics! This interaction depends on the physical characteristics of the light as well as the physical composition and characteristics of the matter.

Different Light Reflections Rough opaque surface (sandpaper)Smooth reflective surface (mirror)

Incoming Light Transmitted Light Reflected Light Scattering and Emission Internal Reflection Absorption Typical light-matter interaction scenario First, when light makes contact with a material, three types of interactions may occur: light reflection, absorption, and transmittance

Because light is a form of energy, conservation of energy tells us that

Opaque Materials The majority of incident light is transformed into reflected light and absorbed light. As a result, when an observer views an illuminated surface, what is seen is reflected light –i.e. the light that is reflected towards the observer from all visible surface regions.

Reflectance Distribution Model Most surfaces exhibit complex reflectances. –Vary with incident and reflected directions. –Model with combination: – + + = specular + glossy + diffuse = reflectance distribution

Radiometric Quantities Radiant Power or Flux (phi) Flux Density –Irradiance (E) –Radiant Exitance (M) or Radiosity (B) Radiance (L)

Irradiance The irradiance function is a two dimensional function describing the incoming light energy impinging on a given point.

What does Irradiance look like?

Radiance Radiance is a two dimensional function representing the light reflected from a surface.

Combining these Functions ! Bidirectional Reflectance Distribution Function (BRDF)

BRDF Approaches 1.Physically-based models 2.Measured BRDFs

What is a BRDF? A BRDF describes how much light is reflected when light makes contact with a certain material. BRDF : Bi-directional Reflectance Distribution Function

What is a BTDF? A BTDF describes how much light is transmitted when light makes contact with a certain material. BTDF : Bi-directional Transmission Distribution Function

The degree to which light is reflected (or transmitted) depends on the viewer and light position relative to the surface normal and tangent.

Key Point Light arriving from all possible incoming directions contributes to the quantity of light reflected towards an observer

What is a BRDF? Since a BRDF measures how light is reflected, it must capture this view- and light- dependent nature of reflected light. Consequently, a BRDF is a function of incoming (light) direction and outgoing (view) direction relative to a local orientation at the light interaction point

What is a BRDF? Additionally, when light interacts with a surface, different wavelengths (colors) of light may be absorbed, reflected, and transmitted to varying degrees depending upon the physical properties of the material itself. This means that a BRDF is also a function of wavelength

BRDF Suppose we are given an incoming light direction,  i, and an outgoing reflected direction,  o each defined relative to a small surface element. A BRDF is defined as the ratio of the quantity of reflected light in direction  o, to the amount of light that reaches the surface from direction  i ii oo didi dodo ii oo

BRDF

Different types of BRDFs

Modeling BRDF’s Mathematical derivation –Use laws of physics, geometry –Statistical model of idealized material Simulation –Model material directly –Render light reflected onto hemisphere Measurement –Reflect real light off of real material –Gonioreflectometer

Reflectance Data Painted Surfaces Human Skin Ceramics Blue Acrylic Sheet

Data Format X and Y components of the incident vector, X and Y components of the exitant (viewing) vector, computed irradiance, and computed exitant radiance.

Analytical Models and Acquired BRDF Data One question that often arises has to do with computing BRDFs for use in the general BRDF lighting equation. There are actually a couple of different ways to determine the value of a BRDF. –By evaluating mathematical functions derived from analytical models. –by resampling BRDF data acquired by empirical measurements of real-world surfaces.

The amount of light arriving from direction  i is proportional to the amount of light arriving at the differential solid angle. Suppose the light source in the figure has intensity L i. Since the differential solid angle is small, it is essentially a flat region on the hemisphere. As a result, the region is uniformly illuminated as the same quantity of light, L i, arrives for each position on the differential solid angle.

So the total amount of incoming light arriving through the region is The only problem is that this amount of light is with respect to the differential solid angle and not the actual surface element under consideration.

To determine the amount of light with respect to the surface element, the incoming light must be “spread out” or projected onto the surface element. This projection is similar to that which happens with diffuse Lambertian lighting and is accomplished by modulating that amount by This means that

ii oo didi dodo ii oo Value range?[0, >= 1]  division by cosine term Unit?inverse steradians (sr -1 )  division by solid angle

Classes of BRDF: Isotropic and Anisotropic Isotropic: BRDFs that represent reflectance properties that are invariant with respect to rotation of the surface around the surface normal vector. Anisotropic: BRDFs that describe reflectance properties that do exhibit change with respect to rotation of the surface around the surface normal vector.

Light interacts differently with different regions of a surface. This property, known as positional variance Most noticeably observed in materials such as wood that reflect light in a manner that produces surface detail. Both the ringing and striping patterns often found in wood are indications that the BRDF for wood varies with the surface spatial position.

Positional Variance Many materials exhibit this positional variance because they are not entirely composed of a single material Most real world materials are heterogeneous and have unique material composition properties which vary with the density and stochastic characteristics of the sub-materials from which they are comprised.

position-variant BRDFs depends on –incoming and outgoing directions, –wavelength of light, –positional variance A general BRDF in functional notation can be written as  the BRDF depends on the wavelength  i,  i,  the incoming light direction in spherical coordinates  o,  o  the outgoing reflected direction in spherical coordinates u and v  the surface position parameterized in texture space.

position-invariant (shift-invariant) BRDFs depends on –incoming and outgoing directions, –wavelength of light, A general BRDF in functional notation can be written as Valid for homogenous materials.

Keep in mind that the values produced by a BRDF do depend on the wavelength or color channel under consideration. In practice, the value of the BRDF function must be determined separately for each color channel (i.e. R, G, and B separately).

BRDF Attributes Helmholtz Reciprocity f r (  i,  r ) = f r (  r,  i ) –Materials are not a one-way street –Incoming to outgoing pathway same as outgoing to incoming pathway Conservation of Energy –When integrated, must add to less than one –Materials must not add energy (except for lights) –Materials must absorb some amount of energy

Properties of BRDF Reciprocity Conservation of Energy = The quantity of light reflected must be less than or equal to the quantity of incident light.

Conservation of Energy When considering the continuous hemisphere of all outgoing reflected directions, the sum becomes an integral over a hemisphere of all directions

The BRDF Lighting Equation Now let’s define a general lighting equation that expresses how to use BRDFs for computing the illumination produced at a surface point. Light arriving from all possible incoming directions contributes to the quantity of light reflected towards an observer

Illumination via the BRDF The Reflectance Equation The reflected radiance is –the sum of the incident radiance over the entire hemisphere –foreshortened –scaled by the BRDF rr Incident irradiance E i is the surface power density of light incoming from entire hemisphere 

the amount of light reflected in the outgoing direction is the integral of the amount of light reflected in the outgoing direction from each incoming direction. The continuous space of incoming directions amount of light reflected in direction  o from direction  i hemisphere of incoming light directions

The discrete space of incoming directions

For each incoming direction, the amount of reflected light in the outgoing direction is defined in terms of the BRDF. The intensity of the light reflected in the outgoing direction is defined by modulating the intensity of the light arriving at the surface point with the corresponding BRDF. ii oo amount of light arriving from direction  i

amount of light arriving from direction  i : the intensity of the light times the width of the cross sectional surface area on the unit sphere through which the light passes. The cross sectional area is the differential solid angle. indicates that all incoming directions are equally weighted

the contribution from direction  i to the intensity of the light reflected towards the observer is Consequently, the light reflected in the outgoing direction is

Key Point Light arriving from all possible incoming directions contributes to the quantity of light reflected towards an observer

Interactive Computer Graphics Tends not to consider the entire hemisphere of incoming directions when determining the illumination of a surface. The reason for this is that the math required to compute the lighting equation above is too expensive to compute for more than just a small number of directions!

Interactive Computer Graphics Use a small number of individual point light sources to compute the illumination of a surface. Rather than computing a sum of light contributions, individual point light sources define the set of incoming directions and light intensities to use in the evaluation of the lighting equation.

Phong X BRDF

incoming direction (  i ) outgoing direction (  o ) ii oo

Analytical Models Simple mathematical functions –INPUT: direction vectors (  i,  i) and (  o,  o) parameters that control the reflectance properties of the material) –OUTPUT: R, G, B and BRDF values. Examples for: –the Cook-Torrance model –the Modified Phong model –the Ward’s model

Parameterizations 6-D BRDF f r (  i,  r, x) –Incident direction L –Reflected direction V –Surface position x –Textured reflection (BTDF) 4-D BRDF f r (  i,  r ) –Homogeneous material –Anisotropic, depends on incoming azimuth –e.g. hair, brushed metal, ornaments 3-D BRDF f r (  i,  r,  i –  r ) –Isotropic, independent of incoming azimuth –e.g. Phong highlight 1-D BRDF f r (  i ) –Perfectly diffuse –e.g. Lambertian

Acquired BRDF Data Physical measurements made with BRDF measuring devices: –gonioreflectometer Many real-world BRDFs cannot be modeled easily using mathematical models.

Local Empirical Illumination (CPSC 453…)

Flat Shading One illumination calculation per polygon Assign all pixels inside each polygon the same color 1.Compute illumination I 2.ScanTriangle ( T, I )

Gouraud Shading Bilinearly interpolate colors at vertices down and across scan lines 1.Compute illuminations I1, I2, I3 2.ScanTriangle ( T, I1, I2, I3 )

Phong Shading Bilinearly interpolate surface normals at vertices down and across scan lines 1.Compute vertex normals N1, N2, N3 2.ScanTriangle ( T, N1, N2, N3, ) Compute illuminations at each interpolated point

BRDF(  i,  i,  o,  o)  G(  i,  i)·H(  o,  o).

BRDF terminology