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Irradiance Gradients in the Presence of Media & Occlusions Wojciech Jarosz in collaboration with Matthias Zwicker and Henrik Wann Jensen University of California, San Diego June 23, 2008 Thursday, 6 September 12 Wojciech Jarosz 2007 2

  1. Volume Rendering Equation object T r ( x ↔ x s )                                                                                      L ( x s , ⇥ � ) x x s surface radiance 20 Thursday, 6 September 12 * the right-hand term incorporates lighting arriving from a surface * before reaching the eye, this radiance must travel through the medium and so is attenuated by a transmission term

  2. Volume Rendering Equation object � s 0                                                                                      x x s media radiance 21 Thursday, 6 September 12 * the left-hand term integrates the scattering of light from the medium along the whole length of the ray

  3. Volume Rendering Equation object � s 0                                                                                      x x s media radiance 21 Thursday, 6 September 12 * the left-hand term integrates the scattering of light from the medium along the whole length of the ray

  4. Volume Rendering Equation object L i ( x t , ⇥ � ) x x t media radiance 22 Thursday, 6 September 12 * the main quantity that is integrated, Li, is inscattered radiance * This represents the amount of light that reaches some point in the volume (from any other location in the scene), and then subsequently scatterers towards the eye

  5. Volume Rendering Equation object L i ( x t , ⇥ � ) x x t                                    T r ( x ↔ x t ) σ s ( x t ) media radiance 23 Thursday, 6 September 12 * as this scattered light travels towards the eye it is also dissipated by extinction through the medium * this computation is very expensive and there has been a lot of work on how to solve this problem e ffj ciently

  6. Contribution • Compute translational gradients of irradiance in the presence of media 24 Thursday, 6 September 12

  7. Contribution • Compute translational gradients of irradiance in the presence of media • Absorbing media • Emissive / scattering media 25 Thursday, 6 September 12

  8. Contribution • Compute translational gradients of irradiance in the presence of media • Absorbing media • Emissive / scattering media • Higher quality irradiance interpolation 26 Thursday, 6 September 12

  9. Irradiance in Part. Med. 27 Thursday, 6 September 12 * Irradiance is simply the integral of the cosine weighted radiance over the hemisphere * Since we decomposed the definition of radiance as radiance coming from surfaces and radiance coming from the media, we can perform the same decomposition on the hemispherical integral.

  10. Irradiance in Part. Med. 27 Thursday, 6 September 12 * Irradiance is simply the integral of the cosine weighted radiance over the hemisphere * Since we decomposed the definition of radiance as radiance coming from surfaces and radiance coming from the media, we can perform the same decomposition on the hemispherical integral.

  11. Irradiance in Part. Med. 27 Thursday, 6 September 12 * Irradiance is simply the integral of the cosine weighted radiance over the hemisphere * Since we decomposed the definition of radiance as radiance coming from surfaces and radiance coming from the media, we can perform the same decomposition on the hemispherical integral.

  12. Irradiance in Part. Med. 27 Thursday, 6 September 12 * Irradiance is simply the integral of the cosine weighted radiance over the hemisphere * Since we decomposed the definition of radiance as radiance coming from surfaces and radiance coming from the media, we can perform the same decomposition on the hemispherical integral.

  13. Irradiance Gradient 28 Thursday, 6 September 12 * Since the total irradiance is the sum of two terms, the total irradiance gradient is just the sum of two gradient terms. * The right hand term is the gradient due to surface irradiance and the left is the gradient due to media irradiance. * In the remainder of the talk I will describe how we compute the two irradiance values and their corresponding gradients.

  14. Irradiance Gradient Gradient from Surfaces 28 Thursday, 6 September 12 * Since the total irradiance is the sum of two terms, the total irradiance gradient is just the sum of two gradient terms. * The right hand term is the gradient due to surface irradiance and the left is the gradient due to media irradiance. * In the remainder of the talk I will describe how we compute the two irradiance values and their corresponding gradients.

  15. Irradiance Gradient Gradient from Media Gradient from Surfaces 28 Thursday, 6 September 12 * Since the total irradiance is the sum of two terms, the total irradiance gradient is just the sum of two gradient terms. * The right hand term is the gradient due to surface irradiance and the left is the gradient due to media irradiance. * In the remainder of the talk I will describe how we compute the two irradiance values and their corresponding gradients.

  16. Irradiance From Surfaces 29 Thursday, 6 September 12 * Given the definition of the surface irradiance, we can estimate it by performing a stratified Monte Carlo integration. * This involves subdividing the hemisphere of directions into a number of strata, or cells, and sampling the radiance using a jittered sample within each cell. * The irradiance is just the sum of all the radiance samples weighted by their cell area and the cosine term. * This is exactly the approach used by standard irradiance caching techniques.

  17. Irradiance From Surfaces 29 Thursday, 6 September 12 * Given the definition of the surface irradiance, we can estimate it by performing a stratified Monte Carlo integration. * This involves subdividing the hemisphere of directions into a number of strata, or cells, and sampling the radiance using a jittered sample within each cell. * The irradiance is just the sum of all the radiance samples weighted by their cell area and the cosine term. * This is exactly the approach used by standard irradiance caching techniques.

  18. Irradiance From Surfaces 30 Thursday, 6 September 12

  19. Irradiance From Surfaces 30 Thursday, 6 September 12

  20. Irradiance Gradient From Surfaces 31 Thursday, 6 September 12 * We can compute the translational gradient of this estimate by using the product rule within the summation.

  21. Irradiance Gradient From Surfaces 32 Thursday, 6 September 12

  22. Irradiance Gradient From Surfaces 32 Thursday, 6 September 12

  23. Irradiance Gradient From Surfaces 33 Thursday, 6 September 12 * Computing the gradient therefore involves estimating how the cell areas change due to a translation * This term is what Ward and Heckbert derived * Our contribution is additionally taking into account a gradient of the cell radiance.

  24. Irradiance Gradient From Surfaces gradient of cell area (Ward and Heckbert) 33 Thursday, 6 September 12 * Computing the gradient therefore involves estimating how the cell areas change due to a translation * This term is what Ward and Heckbert derived * Our contribution is additionally taking into account a gradient of the cell radiance.

  25. Irradiance Gradient From Surfaces gradient of gradient of cell area cell radiance (Ward and Heckbert) (our contribution) 33 Thursday, 6 September 12 * Computing the gradient therefore involves estimating how the cell areas change due to a translation * This term is what Ward and Heckbert derived * Our contribution is additionally taking into account a gradient of the cell radiance.

  26. Gradient of Cell Radiance 34 Thursday, 6 September 12 * In participating media, the surface radiance is the product of two terms, so its gradient can be computed using the product rule. * We recently published a method at TOG which derives the necessary expressions for computing the gradient of the transmittance. * The gradient of cell radiance was ignored by all previous methods. This implies that all these methods (including radiance caching for glossy surfaces) assumed that all surfaces visible during final gather are Lambertian surfaces in a vacuum. * By incorporating this term we can not only account for participating media, but also get the added benefit of being able to handle glossy indirect reflectors.

  27. Gradient of Cell Radiance 34 Thursday, 6 September 12 * In participating media, the surface radiance is the product of two terms, so its gradient can be computed using the product rule. * We recently published a method at TOG which derives the necessary expressions for computing the gradient of the transmittance. * The gradient of cell radiance was ignored by all previous methods. This implies that all these methods (including radiance caching for glossy surfaces) assumed that all surfaces visible during final gather are Lambertian surfaces in a vacuum. * By incorporating this term we can not only account for participating media, but also get the added benefit of being able to handle glossy indirect reflectors.

  28. Gradient of Cell Radiance 34 Thursday, 6 September 12 * In participating media, the surface radiance is the product of two terms, so its gradient can be computed using the product rule. * We recently published a method at TOG which derives the necessary expressions for computing the gradient of the transmittance. * The gradient of cell radiance was ignored by all previous methods. This implies that all these methods (including radiance caching for glossy surfaces) assumed that all surfaces visible during final gather are Lambertian surfaces in a vacuum. * By incorporating this term we can not only account for participating media, but also get the added benefit of being able to handle glossy indirect reflectors.

  29. Gradient of Cell Radiance Attenuation due to media Jarosz et al. ACM TOG ’08. 34 Thursday, 6 September 12 * In participating media, the surface radiance is the product of two terms, so its gradient can be computed using the product rule. * We recently published a method at TOG which derives the necessary expressions for computing the gradient of the transmittance. * The gradient of cell radiance was ignored by all previous methods. This implies that all these methods (including radiance caching for glossy surfaces) assumed that all surfaces visible during final gather are Lambertian surfaces in a vacuum. * By incorporating this term we can not only account for participating media, but also get the added benefit of being able to handle glossy indirect reflectors.

  30. Gradient of Cell Radiance Glossy indirect reflectors 34 Thursday, 6 September 12 * In participating media, the surface radiance is the product of two terms, so its gradient can be computed using the product rule. * We recently published a method at TOG which derives the necessary expressions for computing the gradient of the transmittance. * The gradient of cell radiance was ignored by all previous methods. This implies that all these methods (including radiance caching for glossy surfaces) assumed that all surfaces visible during final gather are Lambertian surfaces in a vacuum. * By incorporating this term we can not only account for participating media, but also get the added benefit of being able to handle glossy indirect reflectors.

  31. Hemispherical Sampling 35 Thursday, 6 September 12 * The way this derivation can be interpreted visually, is that we start with a hemispherical sampling around some point x

  32. Hemispherical Sampling 36 Thursday, 6 September 12 * We now know the radiance coming from each cell, and the distance to the surface within each cell, which results in a discretization of the visible environment. * In order to compute the gradient of irradiance, we consider how the contribution from each cell will change as we move the point x along the tangent plane.

  33. Hemispherical Sampling 36 Thursday, 6 September 12 * We now know the radiance coming from each cell, and the distance to the surface within each cell, which results in a discretization of the visible environment. * In order to compute the gradient of irradiance, we consider how the contribution from each cell will change as we move the point x along the tangent plane.

  34. Surface Irradiance Gradient    37 Thursday, 6 September 12 * Moving the point will result in the cell areas changing due to occlusions from neighboring surfaces (shown in grey). * Additionally, the radiance coming from each cell may change due to changes in extinction (shown in red).

  35. Absorbing Medium y x 38 Thursday, 6 September 12 * To validate this gradient formulation we visualized the gradients within this simple synthetic scene, which contains a ground plane, an occluding block, and a polygon reflecting indirect light. The whole scene is embedded within an absorbing medium.

  36. Absorbing Medium Irradiance on floor 39 Thursday, 6 September 12 * We can visualize the irradiance on the ground plane.

  37. Per-Pixel Irradiance Gradient (Finite Differences) 40 Thursday, 6 September 12 * We can also compute a ground truth solution to the gradient by performing finite di fg erences along the ground plane. * In these visualizations the absolute value of the x component of the gradient is shown in red and the y component is shown in blue. And we compute the gradient per-pixel * This unfortunately su fg ers from significant noise.

  38. Per-Pixel Irradiance Gradient (Finite Differences 10X) 41 Thursday, 6 September 12 * We can improve the quality by taking 10 times as many samples, and this starts to reveal the structure of the true gradient, however it is not a practical approach since it is very expensive

  39. Per-Pixel Irradiance Gradient (Our Method) 42 Thursday, 6 September 12 * Using our approach, we can match the behavior of this gradient, with less noise, and using only 1/10th of the number of samples.

  40. Per-Pixel Irradiance Gradient (Ward and Heckbert) 43 Thursday, 6 September 12 * If we were to compute the gradient using the original Ward and Heckbert formulation, the results are significantly di fg erent than the finite di fg erence gradients.

  41. Gradient Comparison Our Method Ward and Heckbert Why is the Ward & Heckbert gradient darker? 44 Thursday, 6 September 12 * If we look at these side by side we can immediately see that the Ward and Heckbert version is darker. * For Ward & Heckbert, the radiance has an inverse squared fallo fg * In participating media, which our gradients take into account, the radiance has a sharper fallo fg since it is also attenuated by transmittance. * This leads to a higher gradient value.

  42. Gradient Comparison Our Method Ward and Heckbert Why is the Ward & Heckbert gradient darker? 44 Thursday, 6 September 12 * If we look at these side by side we can immediately see that the Ward and Heckbert version is darker. * For Ward & Heckbert, the radiance has an inverse squared fallo fg * In participating media, which our gradients take into account, the radiance has a sharper fallo fg since it is also attenuated by transmittance. * This leads to a higher gradient value.

  43. Gradient Comparison || ∇ E || - Finite Difference 0.12 || ∇ E || - Our Method || ∇ E || - Ward and Heckbert ‘92 Gradient Magnitude 0.10 0.08 0.06 0.04 0.02 50 100 150 200 250 Pixel 45 Thursday, 6 September 12

  44. Extrapolated Irradiance Our Method Ward and Heckbert Same cache points 46 Thursday, 6 September 12

  45. Visual Break 47 Thursday, 6 September 12 This is frame 352 from the Patterson film taken on October 20 , 1967. It is the most famous picture of bigfoot ever taken.

  46. Irradiance From Media 48 Thursday, 6 September 12

  47. Irradiance From Media where: 48 Thursday, 6 September 12

  48. Ray Marching 49 Thursday, 6 September 12

  49. Ray Marching object x 49 Thursday, 6 September 12

  50. Irradiance From Media where: 50 Thursday, 6 September 12

  51. Irradiance From Media where: - Does not have an associated “distance” 50 Thursday, 6 September 12

  52. Irradiance From Media where: - Does not have an associated “distance” - Cannot use the same gradient formulation 50 Thursday, 6 September 12

  53. Hemispherical Sampling 51 Thursday, 6 September 12 * I’ll describe the process of computing the media irradiance gradient at a high level using this 2D example * the details of this process are in the paper.

  54. Ray Marching 52 Thursday, 6 September 12 * An individual cell in this case samples the medium at multiple steps using ray marching.

  55. Ray Marching 52 Thursday, 6 September 12 * An individual cell in this case samples the medium at multiple steps using ray marching.

  56. Ray Marching Medium radiance interpreted as coming from “shells” 53 Thursday, 6 September 12 * In order to compute the contribution to the gradient for each cell, we interpret these samples as radiance come from multiple shells of di fg erent radius

  57. Ray Marching shells Medium radiance interpreted as coming from “shells” 53 Thursday, 6 September 12 * In order to compute the contribution to the gradient for each cell, we interpret these samples as radiance come from multiple shells of di fg erent radius

  58. Ray Marching surfaces surfaces 54 Thursday, 6 September 12 * We also want to handle occlusions from neighboring surfaces, like in the Ward and Heckbert formulation

  59. Media Irradiance Gradient Each shell is occluded by surfaces at a different rate as we move x .    55 Thursday, 6 September 12 * To compute the gradient contribution of each cell, we determine how each shell may be occluded by surfaces. * The rate of occlusion depends on the distance to the “shell,” and the distance to the neighboring surface causing the occlusion (shown in blue) * This means that shells in front of neighboring surfaces do not get occluded with translation, and shells past surfaces get occluded faster with increased distance. * The media irradiance gradient can be thought of as applying the Ward and Heckbert gradient formulation to estimate the change in occlusion individually for each of these shells of increasing radius.

  60. Media Irradiance Gradient Each shell is Shells do occluded by not surfaces at a occlude different rate each other as we move x . Determined by distance to shell and to occluder.    55 Thursday, 6 September 12 * To compute the gradient contribution of each cell, we determine how each shell may be occluded by surfaces. * The rate of occlusion depends on the distance to the “shell,” and the distance to the neighboring surface causing the occlusion (shown in blue) * This means that shells in front of neighboring surfaces do not get occluded with translation, and shells past surfaces get occluded faster with increased distance. * The media irradiance gradient can be thought of as applying the Ward and Heckbert gradient formulation to estimate the change in occlusion individually for each of these shells of increasing radius.

  61. Scattering Medium y x 56 Thursday, 6 September 12 * Using a modification of the previous scene, we can validate the correctness of our media irradiance gradients. * In this case, we use a scattering media, and a point light source. * The scene is constructed in a way where all lighting on the ground plane has first scattered within the medium.

  62. Per-Pixel Irradiance Gradient (Finite Differences 10X) 57 Thursday, 6 September 12 * We compute a ground truth gradient using finite di fg erences * Even with a very large number of samples the finite di fg erence gradient su fg ers from significant noise

  63. Per-Pixel Irradiance Gradient (Our Method) 58 Thursday, 6 September 12 * The gradients estimated using our method match the behavior of the true gradient but have significantly less noise using only 1/10th of the number of samples

  64. Per-Pixel Irradiance Gradient (Ward and Heckbert) 59 Thursday, 6 September 12 * The Ward and Heckbert gradient formulation again significantly di fg ers from the true gradient since it does not take into account media scattering.

  65. Per-Pixel Irradiance Gradient Our Method Ward and Heckbert Why is the Ward & Heckbert gradient darker? 60 Thursday, 6 September 12 * In addition to having a di fg erent structure, it is also overall darker. * This is because for Ward & Heckbert, all radiance is assumed to come from the surface past the medium. * and since the gradient is inversely proportional to the distance, this underestimates the gradient. * By comparing the gradients along a single scanline

  66. Per-Pixel Irradiance Gradient Our Method Ward and Heckbert Why is the Ward & Heckbert gradient darker? 60 Thursday, 6 September 12 * In addition to having a di fg erent structure, it is also overall darker. * This is because for Ward & Heckbert, all radiance is assumed to come from the surface past the medium. * and since the gradient is inversely proportional to the distance, this underestimates the gradient. * By comparing the gradients along a single scanline

  67. Gradient Comparison || ∇ E || - Finite Difference 0.07 || ∇ E || - Our Method || ∇ E || - Ward and Heckbert ‘92 0.06 Gradient Magnitude 0.05 0.04 0.03 0.02 0.01 0 0 50 100 150 200 250 Pixel 61 Thursday, 6 September 12 * we see that our method (shown in blue) matches the ground truth, whereas Ward and Heckbert gradients (shown in red) significantly di fg er from this

  68. Extrapolated Irradiance Our Method Ward and Heckbert Same cache points 62 Thursday, 6 September 12

  69. Gradient Comparison • In a scene with no walls, Ward & Heckbert would estimate 0 gradients! y y x x 63 Thursday, 6 September 12 * This test scene is actually constructed to give the original Ward & Heckbert gradient a helping hand. * If we removed the box and the walls then Ward & Heckbert’s formulation would incorrectly estimate a 0 gradient everywhere, which would be of no benefit for interpolation.

  70. Convergence Irradiance Gradient Magnitude Convergence 1.0 0.8 Relative Error 0.6 || ∇ E || - Our Method 0.4 || ∇ E || - Ward and Heckbert ‘92 0.2 0.0 1 16 32 48 64 Number of Samples 64 Thursday, 6 September 12

  71. Results • Rendered at 1K horizontal resolution • On an Intel Core 2 Duo 2.4 GHz PC 65 Thursday, 6 September 12

  72. Smoky Cornell Box (8:14) 66 Thursday, 6 September 12

  73. Smoky Cornell Box Our Method Ward and Heckbert (8:14) (8:10) 67 Thursday, 6 September 12

  74. Beam through Window Our Method (3:25) Ward and Heckbert (3:17) 68 Thursday, 6 September 12 * still based on stochastic sampling, so there may still be errors (e.g., on the floor), not as bad as previous methods

  75. Disco Ball (10:33) 69 Thursday, 6 September 12 * All illumination on the ground plane has first scattered in the medium.

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