The Light FieldPowerPoint PresentationField RadianceSlide 5Properties of Radiance1st Law: Conservation of RadianceSlide 8Spherical Gantry 4D Light FieldTwo-Plane Light FieldMulti-Camera Array Light FieldThroughput Counts RaysConservation of ThroughputConservation of RadianceQuizSlide 16Radiance: 2nd LawSlide 18Slide 19Slide 20Parameterizing RaysSlide 22Slide 23Projected Solid AngleParameterizing Rays: S2 × R2Parameterizing Rays: M2 × S2Incident Surface RadianceExitant Surface RadianceSlide 29Irradiance from the EnvironmentUniform Area SourceUniform Disk SourceSpherical SourceThe SunPolygonal SourceSlide 36Slide 37Slide 40Types of ThroughputProbability of Ray IntersectionAnother FormulationForm FactorRadiosityForm Factors and ThroughputThe Light FieldLight field = radiance function on raysConservation of radianceMeasurement equationThroughput and counting raysConservation of throughputArea sources and irradianceForm factors and radiosityFrom London and UptonLight Field = Radiance(Ray)Definition: The field radiance (lu m inance) at a point in space in a given direction is the power per unit solid angle per unit area perpendicular to the directionRadiance is the quantity associated with a rayField RadiancedAdω( , )L x ωProperties of RadianceProperties of RadianceFundamental field quantity that characterizes the distribution of light in an environment.Radiance is a function on raysAll other field quantities are derived from itRadiance invariant along a ray.5D ray space reduces to 4DResponse of a sensor proportional to radiance.1st Law: Conservation of RadianceThe radiance in the direction of a light ray remains constant as the ray propagates1dω2dω21d Φ22d ΦΦ = Φ2 21 2d d1dA2dAr1L2L1st Law: Conservation of RadianceThe radiance in the direction of a light ray remains constant as the ray propagates21 1 1 1d L d dAωΦ =1 21 1 2 22dA dAd dA d dArω ω= =1 2L L∴ =1dω2dω21d Φ22d ΦΦ = Φ2 21 2d d1dA2dAr22 2 2 2d L d dAωΦ =1L2LSpherical Gantry 4D Light FieldCapture all the light leavingan object - like a hologram( , , , )L x y θ ϕ( , )θ ϕTwo-Plane Light Field2D Array of Cameras 2D Array of Images( , , , )L u v s tMulti-Camera Array Light FieldThroughput Counts Rays•Define an infinitesimal beam as the set of rays intersecting two infinitesimal surface elements•T measures/count the number of rays in the beam21 221 2dA dAd Tx x=−1 1 1( , )dA u v2 2 2( , )dA u v1 1 2 2( , , , )r u v u vConservation of ThroughputThroughput conserved during propagationNumber of rays conservedAssuming no attenuation or scatteringn2 (index of refraction) times throughput invariant under the laws of geometric opticsReflection at an interfaceRefraction at an interfaceCauses rays to bend (kink)Continuously varying index of refractionCauses rays to curve; miragesConservation of RadianceRadiance is the ratio of two quantities:1. Power2. ThroughputSince power and throughput are conserved,∴ Radiance conservedΔ →ΔΦ Δ Φ= =Δ0( )( ) limTT dL rT dTQuizDoes radiance increase under a magnifying glass?QuizDoes radiance increase under a magnifying glass?No!!Radiance: 2nd LawThe response of a sensor is proportional to the radiance of the surface visible to the sensor.L is what should be computed and displayed.T quantifies the gathering power of the device; the higher the throughput the greater the amount of light gatheredAR Ld dA LTωΩ= =∫∫AT d dAωΩ=∫∫ApertureSensorΩAQuizDoes the brightness that a wall appears to the sensor depend on the distance?Measuring Rays = ThroughputThroughput Counts Rays•Define an infinitesimal beam as the set of rays intersecting two infinitesimal surface elements•Measure/count the number of rays in the beam21 221 2dA dAd Tx x=−1 1 1( , )dA u v2 2 2( , )dA u v1 1 2 2( , , , )r u v u vParameterizing Rays•Parameterize rays wrt to receiver2 2 2( , )dA u vω θ φ2 2 2( , )d212 2 221 2dAd T dA d d Ax xω= =−2 2 2 2( , , , )r u v θ φParameterizing Rays•Parameterize rays wrt to source1 1 1( , )dA u vω θ φ1 1 1( , )d221 1 121 2dAd T dA dA dx xω= =−1 1 1 1( , , , )r u v θ φParameterizing RaysTilting the surfaces reparameterizes the rays All these throughputs must be equal.vvgvvg21 21 221 21 12 2cos cosd T dA d Ax xd dAd dAθ θωω=−==r1 1 1( , )dA u vr2 2 2( , )dA u v1 1 2 2( , , , )r u v u vProjected Solid Angleco s dθ ωθ2c osHdθ ω π=∫dωParameterizing Rays: S2 × R2Parameterize rays byMeasuring the number or rays that hit a shape( , , , )r x y θ φω θ ϕθ ϕ ω θ ϕπ===∫ ∫∫%%2 22( , ) ( , )( , ) ( , )4S RST d dA x yA dA( )A ωr%ωrSphere:%2 24 4T A Rπ π= =Projected areaParameterizing Rays: M2 × S2Parameterize rays by( , , , )r u v θ φ2 2( )( , ) cos ( , )M HT dA u v dSθ ω θ ϕπ⎡ ⎤⎡ ⎤=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦∫ ∫Nr1 4 2 4 31 4 4 42 4 4 43Sphere:2 24T S Rπ π= =Crofton’s Theorem:44SA S Aπ π= ⇒ =% %( , )u v( , )θ φNrDefinition: The incoming surface radiance (luminance) is the power per unit solid angle per unit projected area arriving at a receiving surfaceIncident Surface RadianceωωωΦ≡rrg2( , )( , )iid xL xd dAdArdωrExitant Surface RadianceDefinition: The outgoing surface radiance (luminance) is the power per unit solid angle per unit projected area leaving at surface Alternatively: the intensity per unit projected area leaving a surfaceωωωΦ≡rrg2( , )( , )ood xL xd dAdArdωrIrradiance from a Uniform Area SourceIrradiance from the Environment2( , ) ( , )cosi id x L x dA dω ω θ ωΦ =2( ) ( , ) cosiHE x L x dω θ ω=∫ω( , )iL xdAθdω( , ) ( , ) cosidE x L x dω ω θ ω=Uniform Area Source2( ) coscosHE x L dL dLθ ωθ ωΩ=== Ω∫∫%AΩ%ΩUniform Disk Sourcerhα%cos 21 0cos21222 2cos coscos22sind drr hα παθ φ θθππ απΩ ====+∫ ∫Geometric DerivationAlgebraic Derivation2sinπ αΩ =%sinαSpherical Source222c ossindrRθ ωπ απΩ ===∫%rRαGeometric DerivationAlgebraic Derivation2sinπ αΩ =%The SunSolar constant (normal incidence at zenith)Irradiance 1353 W/m2Illuminance 127,500 lm/m2 = 127.5 kiloluxSolar angle= .25 degrees = .004 radians (half angle) = 6 x 10-5 steradiansSolar radiance3 275 21.353 10 / W2.25 106 10E W mLsr m sr−×= = = ×Ω × ⋅%%2 2sinπ α παΩ = ≈αPolygonal SourcePolygonal SourcePolygonal SourceForm FactorsTypes of Throughput1. Infinitesimal beam of rays (radiance) 2. Infinitesimal-finite beam (irradiance calc.)3. Finite-finite beam (radiosity calc.)22cos cos( , ) (
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