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Image Synthesis

Image Synthesis. Pre-Computed Radiance Transfer PRT. Radiometric quantities. Strahlungsenergie: radiant energy Q in Joule [J] Strahlungsleistung oder -fluss: radiant flux in Watt [W=J/s] Einfallende Flussdichte: irradiance (incident) power per area in [W/m 2 ]

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Image Synthesis

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  1. Image Synthesis Pre-ComputedRadiance Transfer PRT

  2. Radiometric quantities Strahlungsenergie: radiant energy Qin Joule [J] Strahlungsleistung oder -fluss: radiant fluxin Watt [W=J/s] Einfallende Flussdichte: irradiance (incident) power per area in [W/m2] ausgehende Flussdichte: radiosity (radiant exitance)power per area in [W/m2]

  3. d N  dA dN dA Strahldichte: Radiance Combination of flux and intensity Strahldichte = Radiance Central quantity in physics based images synthesis Units: [W/(m2sr)] Power per unit solid angle per projected unit area

  4. Precomputedradiancetransfer • Precomputeirradiance on a grid • Based on irradiancevolumes(Greger98) • Assumethescenedoesn´tchange • Precomputeirradiancesamplesatgridpoints • Interpolatelightatpointsin-between

  5. Precomputed radiance transfer • A cube-mapcapturesradiancedistributionat a pointfor all directions • Entrieshavetobescaledbyprojected solid angle • For diffuse surfaces, radianceisdefined in termsoftheirradianceE

  6. Precomputed radiance transfer • Idea: compute irradiance at a point for all possible „orientations“ • Gives irradiance distribution function • Looks like a radiance distribution function but it is convolved (blurred) with a cosine kernel • It is continuous over directions • Can be stored at a point using a “Diffuse Cube Map”, which is indexed with surface normal

  7. Radiance vs. Irradiance

  8. Precomputed radiance transfer • Volume subdivisionforirradiancecaching • Uniform, but someareashaveslowlychangingirradiancewhile in otherstheirradianceisquicklychanging • Idea: use adaptive subdivision

  9. Precomputed radiance transfer • Sampling the irradiance grid • Trilinear between grid vertices • At level transitions use vertices on either side

  10. Precomputed radiance transfer • Idea: project irradiance into a particular function space and use only „a few“ coefficients to reconstruct the original function • As irradiance is captured on a sphere, use basis functions on a sphere

  11. Diffuse PRT • http://people.csail.mit.edu/kautz/PRTCourse • Goal: to shade a diffuse object usingPrecomputedRadiance Transfer (PRT) • Diffuse: • Reflectedlightisview-independent • Simplifiesequations

  12. RememberRadiosity? B = E + TB = E + T(E+TB) = E + TE + T2B = ... = T0E + T1E + T2E + T3E + ... = B(0) + B(1) + B(2) + B(3) + ...

  13. Diffuse PRT To derive PRT for the diffuse case we are going to start with just the direct term from the Neumann expansion of the rendering equation and make several simplifying assumptions.

  14. Diffuse PRT The bottom equation is the “simplified” form. For diffuse objects light is reflected equally in all directions, so outgoing radiance is independent of view direction.

  15. Diffuse PRT This also means the BRDF is just a constant (and independent of direction) so it can be pulled out of the integral.

  16. Diffuse PRT „Visually”

  17. Diffuse PRT Visually, we integrate the product of three functions (light, visibility, and cosine).

  18. Diffuse PRT The main trick is to use for pre-computed radiance transfer (PRT) is to combine the visibility and the cosine into one function (cosine-weighted visibility or transfer function), which we integrate against the lighting.

  19. Problems • How to encode the spherical functions?We still need to encode the two spherical functions (lighting, cosine-weighted visibility/transfer function) • How to quickly integrate over the sphere?we need to perform the integration of the product of the two functions quickly-  PRT

  20. Diffuse PRT Now we are going to approximate the source radiance function with its projection into a set of basis functions on the sphere (denoted yi() in this equation.)

  21. Diffuse PRT Manipulating this expression exploiting the fact that integration is a linear operator (sum of integrals = integral of sums), we can generate the following equivalent expression. The important thing to note about the highlighted integral is that it is independent of the actual lighting environment being used, so it can be pre-computed.

  22. Diffuse PRT We call the precomputed integrals “transfer coefficients” Outgoing radiance: just a dot-product!

  23. Diffuse PRT A similar process can be used to model the other bounces, so that a final vector can be computed and used to map source radiance to outgoing radiance at every point on the object. Outgoing radiance is then just the dot product of the lights projection coefficients with the transfer vector.

  24. Puttingit all together We project the lighting into thebasis (integral against basis functions). If the object is rotatedwrt. to the lighting, we need toapply the inverse rotation to thelighting vector. At run-time, we need to lookupthe transfer vector at every pixel. A pixel-shader then computes the dotproduct between the coefficient vectors. The result of this computation is the outgoing radiance at that point.

  25. Results

  26. Results

  27. Results

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