Representations of Visual AppearanceRepresentations of Visual Appearance

COMS 6160 [Fall 2006], Lecture 2

Ravi Ramamoorthi

http://www.cs.columbia.edu/~ravir/6160

OutlineOutline

Basic preliminaries: Light Field, Radiance, Irradiance

Plenoptic Function and BRDF

Reflection Equation

Radiance Radiance

Power per unit projected area perpendicular to the ray per unit solid angle in the direction of the ray

Symbol: L(x,ω) (W/m2 sr)

Flux given by dΦ = L(x,ω) cos θ dω dA

Radiance propertiesRadiance properties

Radiance is constant as it propagates along ray Derived from conservation of flux Fundamental in Light Transport.

1 21 1 1 2 2 2d L d dA L d dA d

2 21 2 2 1d dA r d dA r

1 21 1 2 22

dA dAd dA d dA

r

1 2L L

Radiance propertiesRadiance properties

Sensor response proportional to surface radiance (constant of proportionality is throughput) Far away surface: See more, but subtends smaller angle Wall is equally bright across range of viewing distances

Consequences Radiance associated with rays in a ray tracer All other radiometric quantities derived from radiance Acquire functions of incoming and outgoing radiance

Irradiance, RadiosityIrradiance, Radiosity

Irradiance E is the radiant power per unit area

Integrate incoming radiance over hemisphere Projected solid angle (cos θ dω) Uniform illumination:

Irradiance = π [CW 24,25] Units: W/m2

Radiosity Power per unit area leaving

surface (like irradiance)

OutlineOutline

Basic preliminaries: Light Field, Radiance, Irradiance

Plenoptic Function and BRDF

Reflection Equation

Plenoptic FunctionPlenoptic Function

Radiance at each wavelength in every direction for every spatial location at every time instance

7D function (x,y,z,θ,φ,λ,t)

Measured appearance is ratio of outgoing radiance to incoming (ir)radiance. 14D function in general

This course is about subsets of this 14D function Acquisition of appropriate slices (BRDFs one example) Efficient Representation for rendering, editing, storage

BRDFBRDF

Reflected Radiance proportional to Irradiance

Constant proportionality: BRDF [CW pp 28,29] Bidirectional Reflection Distribution Function (4 Vars)

Reflectance Equation [CW pp 30]

( )( , )

( ) cos

r ri r

i i i i

Lf

L d

( ) ( , ) ( ) cosi

r r i r i i i iL f L d

Specular Term (Phong)Specular Term (Phong)

Specular Term (Blinn-Phong)Specular Term (Blinn-Phong)

( , ) ( , ) ( )sr r e r i hL x L x L n

Reflected Light(Output Image)

Emission Incident Light (fromlight source)

Sum over all light sources

Blinn-Phong model(using half-angle)(s is shininess)

| |i o

hi o

i r

x

nh

OutlineOutline

Basic preliminaries: Light Field, Radiance, Irradiance

Plenoptic Function and BRDF

Reflection Equation

Reflection Equation

ir

x

( , ) ( , ) ( , ) ( , , )( )r r e r i i i r iL x L x L x f x n Reflected Light(Output Image)

Emission Incident Light (fromlight source)

BRDF Cosine of Incident angle

Reflection Equation

ir

x

( , ) ( , ) ( , ) ( , , )( )r r e r i i i r iL x L x L x f x n Reflected Light(Output Image)

Emission Incident Light (fromlight source)

BRDF Cosine of Incident angle

Sum over all light sources

Reflection Equation

ir

x

( , ) ( , ) ( , ) ( , , ) cosr r e r i i i r iiL x L x L x df x

Reflected Light(Output Image)

Emission Incident Light (fromlight source)

BRDF Cosine of Incident angle

Replace sum with integral

id