Photon Differentials

Transcription

1 Photon Differentials Adaptive Anisotropic Density Estimation in Photon Mapping Jeppe Revall Frisvad Technical University of Denmark Trade-off problem in photon mapping Effect of changing bandwidth (no. of photons in estimates): Low High The trade-off is between noise and blur.

2 Why photon differentials? Using the same number of photons in the map: Standard PM Photon Differentials Ray differentials improve teture filtering. Photon differentials improve photon flu density estimation. Ray differentials image plane v ω D r u D r v r(s) ray r(s ) surface u A ray is modelled by the parametrisation of a straight line: r(s) = + s ω, s [0, [, ω = 1. Suppose we let u and v parameterise the image plane s be the distance to the first intersection along the ray then r(s ) r(u, v), and the ray differential [Igehy 1999] Dr = [ D u r D v r ] = [ ] r r u v tells where a ray would end up if slightly offset in uv-space. References - Igehy, H. Tracing ray differentials. In Proceedings of ACM SIGGRAPH 1999, A. Rockwood, Ed., ACM/Addison-Wesley, pp

3 First-order ray differentials surface D v r eye D = 0 A r = 0 D v ω r(1) D u ω image plane r(s ) D u r A r ray footprint In the first order Taylor approimation, a ray differential is given by two pairs of differential vectors. Positional differential vectors: D = [ D u D v ] Directional differential vectors: D ω = [ Du ω D v ω ]. The differential vectors span parallelograms which define ray footprint (D) and beam spread (D ω). Photon differentials Dφ ω Dθ ω D v Dv D u ω = r(s ) D u No camera: we need different local coordinate systems. u and v parameterise the light source surface. θ and φ parameterise the emission solid angle. Now r(s ) r(u, v; θ, φ) = (u, v) + s (u, v; θ, φ) ω(θ, φ). Photon differential: Dr = ( [ ] [ ] D u D v + Dθ D φ )r. Photon differential vectors: [ Positional differential vectors: D = Du D v ] Directional differential vectors: D ω = [ D θ ω D φ ω ] define light ray footprint (D) and beam spread (D ω).

4 Photon footprint The parallelogram spanned by the positional differential vectors is the ray footprint. Du ray footprint A r D u p photon footprint p A p Dv D v p The ma area ellipse inscribed in the parallelogram with centre in the photon position p is the photon footprint. The area of the photon footprint is then A p = π 4 A r = π 4 D u p D v p, A ṕ and, by analogy, the photon solid angle is ω p ω p = π 4 D θ ω p D φ ω p. A = 0 p Emitting photon differentials A light source emits photons from points e across an area A e and in directions ω e within a solid angle ω e. The initial differential vectors of an emitted photon are [ ] Du e D v [ e ] an orthogonal basis of the tangent plane at e. Dθ ω e D φ ω e an orthogonal basis of the plane normal to ωe. To ensure p A p = A e and p ω p = ω e, we set the initial lengths of the vectors to D u e = D v e = 2 D θ ω e = D φ ω e = 2 Ae πn e ωe πn e, where n e is the number of photons emitted from the source. Point lights emit photons with D u e = D v e = 0. Collimated lights emit photons with D θ ω e = D φ ω e = 0.

5 Photon tracing Emitted flu is confined by the solid angle of the ray. Flu carried by a ray changes like radiance upon reflection and refraction. Tracing photons is like tracing ordinary rays. Whenever the photon is traced to a non-specular surface: It is stored in a kd-tree. Position is stored. Direction from where it came is stored. Flu (Φ p ) is stored. Russian roulette is used to stop the recursive tracing. Tracing photon differentials Emitted flu is confined by the cone which is spanned by the photon differential. Photon differentials change like ray differentials upon reflection and refraction. Tracing photon differentials is like tracing ordinary ray differentials. Whenever the photon is traced to a non-specular surface: It is stored in a kd-tree. Position is stored. Direction from where it came is stored. Irradiance (Ep = Φ p /A p) is stored (instead of flu). Positional differential vectors Du and D v are stored. Russian roulette is used to stop the recursive tracing.

6 Radiance estimation using photon differentials Irradiance of a projected photon differential E p = Φ p /A p Reflected radiance L r (, ω) = 2π f r (, ω, ω) de(, ω) Radiance estimate L r (, ω) L r (, ω) = n f r (, ω p, ω) E p (, ω p ) p=1 To ensure that no energy is lost in the estimate, we must find all the n photons with footprints that overlap a surface point. We can induce smoothing by scaling all photon footprints. Adaptive anisotropic kernel density estimation Transform by M p = [ 1 2 D u p 1 2 D v p n p ] 1, where n p = D u p D v p D u p D v p is the surface normal at p. D u p D ^ u p M p p D v p p D ^ v p Geometry space Filter space Radiance estimate with filtering Lr (, ω) = n πk ( M p ( p ) 2) f r (, ω p, ω)e p p=1

7 Case studies Refraction Reflection Photon distribution in the map Rendered reference images Optimal bandwidth - knn photon mapping Finding the optimal bandwidth using image quality measures: RMSE: root mean square error. SSIM: structural similarity inde RMSE SSIM inde Bandwidth [k] Bandwidth [k]

8 Optimal bandwidth - photon differentials Finding the optimal bandwidth using image quality measures: RMSE: root mean square error. SSIM: structural similarity inde RMSE SSIM inde Bandwidth [s] Bandwidth [s] Refraction - equal number of photons comparison Method RMSE-optimal bandwidth SSIM-optimal bandwidth knn pd RMSE = SSIM = RMSE = SSIM = Using 20,000 photons in the map. Comparing knn k-nearest neighbours photon mapping. pd photon differentials.

9 Refraction - equal quality comparison Method RMSE-optimal bandwidth SSIM-optimal bandwidth knn n = 200,000, RMSE = n = 200,000, SSIM = n = 500,000, RMSE = n = 500,000, SSIM = pd n = 20,000, RMSE = n = 20,000, SSIM = Reflection - comparison Method RMSE-optimal bandwidth SSIM-optimal bandwidth knn n = 20,000, RMSE = n = 20,000, SSIM = n = 75,000, RMSE = n = 75,000, SSIM = n = 420,000, RMSE = n = 420,000, SSIM = pd n = 20,000, RMSE = n = 20,000, SSIM =

10 The gold ring cardioid caustic - equal time comparison RMSE=0.085 path traced reference (20 h) SSIM=0.79 path tracing 20 h) ( 250 RMSE=0.044 RMSE=0.030 SSIM=0.95 SSIM=0.96 standard photon mapping photon differentials References on photon differentials and more applications I Photon differentials - Schjøth, L., Frisvad, J. R., Erleben, K., and Sporring, J. Photon differentials. In Proceedings of GRAPHITE 2007, pp , ACM, I Photon differentials for diffuse interreflections. - Fabianowski, B., and Dingliana, J. Interactive global photon mapping. Computer Graphics Forum (Proceedings of EGSR 2009) 28, 4 (June-July), pp , I Photon differentials for temporal blur. - Schjøth, L., Frisvad, J. R., Erleben, K., and Sporring, J. Photon differentials in space and time. In Computer Vision, Imaging and Computer Graphics: Theory and Applications, P. Richard and J. Braz, eds., Communications in Computer and Information Science 229, pp , December I Photon differentials for participating media. - Schjøth, L. Anisotropic Density Estimation in Global Illumination, PhD thesis, University of Copenhagen, Faculty of Science, Jarosz, W., Nowrouzezahrai, D., Sadeghi, I., and Jensen, H. W. A comprehensive theory of volumetric radiance estimation using photon points and beams. ACM Transactions on Graphics 30(1), pp. 5:1 5:19, January 2011.