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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 (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