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Local Illumination. CS 319 Advanced Topics in Computer Graphics John C. Hart. Local Illumination. Study of how different materials reflect light Optics Geometric (yes) Wave (not in this class) Definition of radiance , the fundamental unit of light transfer in computer graphics

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local illumination

Local Illumination

CS 319

Advanced Topics in Computer Graphics

John C. Hart

local illumination2
Local Illumination
  • Study of how different materials reflect light
  • Optics
    • Geometric (yes)
    • Wave (not in this class)
  • Definition of radiance, the fundamental unit of light transfer in computer graphics
  • How the BRDF fr encapsulates the reflectance properties of a material
  • Global illumination, handled later, describes how light gets from one local illumination context to another
radiometric terms
Radiometric Terms
  • Photon (g)
    • Quantum of EM radiation
    • Color given by frequency (n) [1/s]
  • Radiant Energy (Q)
    • Measured in Joules (J)
    • Energy carried by a photon

Q = hn [J]

  • Flux (F)
    • Light energy passing through a region per unit time
    • Measured in Watts (Joules/sec)

F = dQ/ds [W=J/s]

Planck’s constanth = 6.6262  10-34J s

60W light bulb meansthat 60J of energy passing through the glass each second.

flux density
Flux Density
  • Measures distribution of flux (power) across surface area
  • Irradiance (E) – Incoming flux density

E = dF/dA [W/m2]

    • How many photons reach a given surface area in a given amount of time

E = dQ/dAds [W/m2]

  • Radiosity (B) – Outgoing flux density

B = dF/dA [W/m2]

    • a.k.a. exitance
hemispherical projection
Hemispherical Projection

dw

r

  • Use a hemisphereW over surface to measure incoming/outgoing flux
  • Replace incoming light with its image on hemisphere
  • Projected area forms a solid angle dw
  • Solid angle measured by steradian (sr) which corresponds to an area of r2 on sphere of radius r
  • Area of sphere = 4pr2
    • 4p steradians in a sphere
    • 2p in a hemisphere
  • Hemispherical projection of source a distance d away is proportional to 1/d2

A

d

~A/d2

A

A

1

intensity
Intensity
  • Intensity (I) – Power (flux) per unit solid angle (W/sr)

I = dF/dw

  • Flux is the intensity distributed over a solid angle

dF = Idw

  • The power of a point light source is its intensity integrated across the sphere
  • For an isotropic point light source

I = F/4p

Can increase the intensity of a lightbulb by covering some of it with a mirrored paint, forcing more photons (some of them reflected) through a smaller solid angle.

4p sr

FW

foreshortening

dq1=dq2 but dA1dA2

Foreshortening

Equal Intensities

dq1

  • Angled light spreads wider across a surface than perpendicular light

dA1 = dA2 cosq2

  • Foreshortening scales the flux density measured on the hemisphere to the flux density on the surface
  • E.g. irradiance is the foreshortened (cosqi) incident flux (Iidwi)

Ei = Iidwi cosqi

  • Hemispherical projection automatically foreshortens source (but not destination)

dq2

dA1

dA2

Unequal irradiances

A

q

(A/d2) cos q

d

1

Source foreshortening

radiance
Radiance
  • Radiance (L) – Power per unit solid angle per unit area (W/(srm2)), foreshortened
  • Radiance allows flux to be integrated across surface area and solid angle
  • Integration across all four degrees of freedom for a line
    • Two DOF for its position
    • Two DOF for its orientation

w

dw

L(x,w)

x

dA

radiance derivation i
Radiance Derivation I

dS

F

  • Consider the flux from source dS to receiver dR, and assume receiver is perpendicular to centerline between dS and dR
  • Radiance L(x,w) is the power received at point x from direction w
  • Flux from dS to dR is the power received by each point in each direction L(x,w) times the total number of points dR times the directions dw

dF = L(x,w) dR dw

  • Unforshortened, the radiance is

L(x,w) = dF/(dR dw)

dw

x

dR

dS

dR

radiance derivation ii
Radiance Derivation II

dS

  • Now consider different receiver dR’
  • No longer perpendicular to centerline
  • Flux between dS and dR’ identical to flux between dS and dR
  • But area of dR’ is larger than dR

dR = dR’ cos q

  • General flux equation is thus

dF = L(x,w) dR dw

= L(x,w) dR’cos qdw

  • Solve for the radiance

L(x,w) = dF/(dR’cos qdw)

dw

x

dR’

radiance v x
Radiance v. X
  • Intensity is radiance across solid angle

dI = L cos qdA

  • Irradiance is radiance across an area

dE = L cos qdw

1/cosqforeshortening

dF/dwFlux densitywrtspherical angle(Intensity)

dF/dAFlux density wrt area(Irradiance)

the brdf
The BRDF

wr

wi

dwr

  • Bidirectional Reflectance Distribution Function fr(wi, wr)
  • Measures the portion of incident irradiance (Ei) from wi that is reflected as radiance (Lr) toward wr

fr(wi, wr) = dLr(wr)/dEi(wi)

  • Or the ratio between incident radiance and reflected radiance

fr(wi, wr) = dLr(wr)/(Li(wi) cosqi dw)

  • BRDF can range from 0 to , especially when light comes in at grazing angles (cosqi 0)

dwi

wr

wi

Incident irradiance Ei is the surface power density of light incoming from direction wi

illumination via the brdf
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

wr

Incident irradiance Ei is the surface power density of light incoming from entire hemisphere W

parameterizations
6-D BRDF fr(wi, wr, x)

Incident direction L

Reflected direction V

Surface position x

Textured reflection (BTDF)

4-D BRDF fr(wi, wr)

Homogeneous material

Anisotropic, depends on incoming azimuth

e.g. hair, brushed metal, ornaments

3-D BRDF fr(qi, qr, fi – fr)

Isotropic, independent of incoming azimuth

e.g. Phong highlight

1-D BRDF fr(qi)

Perfectly diffuse

e.g. Lambertian

Parameterizations
brdf attributes
BRDF Attributes
  • Helmholtz Reciprocity

fr(wi, wr) = fr(wr, wi)

    • 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
diffuse reflection
Diffuse Reflection
  • Uniform
    • Reflects power equally in all dirs. Lr(w1) = Lr(w2)
    • BRDF constant
  • Perfect
    • all incoming light is reflected Br = Ei

q

Why 1/p ? Because the incident light from a single direction is distributed as reflected light across all directions of the hemisphere, which leaves only a 1/p portion for each direction

modeling brdf s
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
dirac d function
Dirac d Function

d(x)

x

  • Defined by its behavior when integrated
    • Zero almost everywhere
    • Except at zero, where its infinite
  • Used to represent the illumination from a point light source S

Li(wi) = LSd(wi – wS)

  • So…

0

S

wr

wS

Reflected radiance Lr is just the incident radiance LS from the point source S foreshortened and scaled by the BRDF fr

the reflectivity
The Reflectivity

Dwr

Dwi

  • Reflected flux across a solid angle as a ratio of the incident flux received across a solid angle

r(Dwi,Dwr) = dFr(Dwr)/dFi(Dwi)

  • Ratio of reflected intensity to incident intensity
  • Incident flux across a solid angle is the sum of differential
illumination via reflectivity
Illumination via Reflectivity

Ir = kara Ia (NL) +S (kdrd Id +ksrs Is) (NL) dw

  • Constants kd + ks = 1
  • Intensities
    • Ia = average color of reflected light in scene
    • Id = color of material
    • Is = color of light source
  • Reflectivities
    • Ambient: ra(V,L) = 1/ (NL) – constant color regardless of surface orientation
    • Lambertian: rd(V,L) = 1 – reflects light uniformly in all directions
    • (Phong) Specular: rs(V,L) = (VR)n/ (NL) – focuses light in reflected direction