1. Multiple Scattering in Vision and Graphics Lecture #21 Thanks to Henrik Wann Jensen

5. MistFog Glows of Light Sources

7. Properties of Scattering Media Scattering Coefficient: Fractional loss in intensity due to scattering per unit cross section Absorption Coefficient: Fractional loss in intensity due to absorption per unit cross section Extinction Coefficient: Scattering Coefficient + Absorption Coefficient Scattering Albedo: Scat. Coeff. / Ext. Coeff.

8. Phase Function Incident Direction Exiting Direction Probability of light getting scattered in a single direction Phase function integrates to 1 Light Scattered in any direction :

9. Recap Different Orders of Scattering

10. Particle Scattering Mechanisms ( Mie 1908 ) Incident Beam Size: 0.01Size: 0.1 Size: 1 Single Scattering: Independent Scattering: Incident Beam Distance of Separation >> Size of Particles

11. Attenuation Model – Zero th Order Scattering Scattering Medium Unit Cross Section X = 0 dx X = d Incident Light Attenuated Exiting Light Scattering Coefficient Brightness at Distance d : ( Bouguer’s Law, 1729 )

12. Airlight Model – First Order (Single) Scattering Sunlight Diffuse Skylight Diffuse Ground Light Object Observer d dV Brightness due to a Path of Length d : Horizon Brightness ( Koschmeider, 1924 )

13. Mountains Distant objects appear Bright !

14. Combining 0 th and 1 st orders: Useful for Vision Object Observer d Attenuation Sunlight Diffuse Skylight Diffuse Ground Light Airlight Intensity Distance Intensity

15. Multiple Scattering : Higher orders of scattering Incident Beam Particle Phase Function

16. Radiative Transfer Mathematical study of transport of radiation (in particular light). Finite Difference method used to model the rate of change of radiation along any direction in an infinitesimal volume. Can model multiple scattering elegantly. Solution to light transport gives the Light Field in the medium. But, hard to solve analytically. Why? Depends on medium geometry and location of sources. Only few special cases are known to have analytic solutions. (Plane Parallel, Spherical)

17. Plane Parallel and Spherical Radiative Transfer Isotropic Source Homogeneous Medium Scattered Light Field Plane Parallel Medium Scattered Light Field Distant Source Sun

18. Plane Parallel Medium Radiative Transfer in Plane Parallel Media [ Chandrasekhar 1960, Ishimaru 1997 ] Scattered Light Field Distant Source Sun Collimated Source Outside Medium Widely used in Atmospheric Optics, Remote Sensing Popular configuration for Subsurface Scattering in Graphics

19. Radiative Transfer in Plane Parallel Medium Infinitesimal Scattering Volume : Extinction Radiative Transfer Equation : Radiance Rate of Change Source Function Phase Function Optical Thickness Incident Beam Radiance Exiting Beam Radiance dR Direction

20. BSSRDFs Bidirectional Surface Scattering Reflectance Distribution Function The BSSRDF relates the outgoing radiance to the incident flux The BRDF is an approximation of the BSSRDF for which it is assumes that light enters and leaves at the same point The outgoing radiance is computed by integrating the incident radiance over incoming directions and area, A

21. Symbol Reference

22. Diffusion Approximation for Multiple Scattering An incoming ray is transformed into a dipole source for the diffusion approximation

23. The Diffusion Approximation The diffusion approximation is based on the observation that the light distribution in highly scattering media tends to become isotropic The volumetric source distribution can be approximated using the dipole method The dipole method consists of positioning two point sources near the surface in such a way as to satisfy the required boundary condition The diffuse reflectance due to the dipole source can be computed as Taking into account the Fresnel reflection at the boundary for both the incoming light and the outgoing radiance Where S d is the diffusion term of the BSSRDF, which represents multiple scattering

24. Single Scattering Term The total outgoing radiance, due to single scattering is computed by integrating the incident radiance along the refracted outgoing ray The single scattering BSSRDF is defined implicitly by the second line of this equation Single scattering occurs only when the refracted incoming and outgoing rays intersect, and is computed as an integral over path length s along the refracted outgoing ray

25. The BSSRDF Model The complete BSSRDF model is a sum of the diffusion approximation and the single scattering term This model accounts for light transport between different locations on the surface, and it simulates both the directional component (due to single scattering) as well as the diffuse component (due to multiple scattering)

26. Rendering Using the BSSRDF The BSSRDF model derived only applies to semi-infinite homogeneous media, for a practical model we must consider –Efficient integration of the BSSRDF (importance sampling) –Single scattering evaluation for arbitrary geometry –Diffusion approximation for arbitrary geometry –Texture (spatial variation on the object surface)

27. BRDF vs BSSRDF

30. Diffusion Approximation for Multiple Layers Donner, Jensen, Siggraph 05

37. Plane Parallel and Spherical Radiative Transfer Isotropic Source Homogeneous Medium Scattered Light Field Plane Parallel Medium Scattered Light Field Distant Source Sun

38. MistFog Glows of Light Sources (Narasimhan & Nayar, CVPR 2003)

39. Multiple Scattering in the Atmosphere Incident Beam Particle Light Source A T M O S P H E R E Phase Function Imaging Plane Glow Pinhole

40. Light Source in a Spherical Medium Isotropic Source Homogeneous Medium Spherical Radiative Transfer Equation: Phase Function Light Field Cosine of Angle Optical Thickness [ Chandrasekhar 1960 ] Scattered Light Field

41. Axially Symmetric Phase Functions Legendre Polynomial Expansion: [ Ishimaru 1997 ] [ Henyey et al., 1941 ] Legendre Polynomial Forward Scattering Parameter Incident Direction Exiting Direction

42. Light Source in a Spherical Medium Isotropic Source Homogeneous Medium Spherical Radiative Transfer Equation: Phase Function Light Field Cosine of Angle Optical Thickness [ Chandrasekhar 1960 ] Scattered Light Field

43. Phase Function Parameter Optical Thickness Exponential Coefficients : Radiant Intensity of Source Legendre Polynomial Analytic Multiple Scattering Solution Scattered Light Field :

44. Highlights of the Model 1.02 1.2 1.4 1.6 1.8 T m 160 120 60 30 10 Small Number of Coefficients (m) : Absorbing and Purely Scattering Media Single and Multiple Scattering Isotropic and Anisotropic Phase Functions

45. Scattered Light Field vs. Weather Condition Mild Weather (T = 1.2) Dense Weather (T = 4) Angular PSF : Scattered Light Field at a Point

46. Validation: Multiple Scattering in Milk Original Milk Images Increasing Milk Concentrations Rendered Milk Images Image acquired With No Milk

47. Number of Milk Concentrations : 15 Model Fitting Error : [ 1 % to 3 % ] Diffusion Fitting Error : [ 20 % to 50 % ] Model Fit Accuracy Low Milk ConcentrationHigh Milk Concentration

48. Model Fit Accuracy: Monte Carlo Simulations

49. Effect of Source Visibility 3152401809030 ooooo Increasing Milk Concentrations Observed Milk Images

50. Original Image Rendering Glows using Convolution Increasing Fog Rendered Images Joint work with Ramamoorthi

51. Original Image Single Scattering Multiple Scattering (Mild Condition) Multiple Scattering (Dense Condition) Single versus Multiple Scattering Joint work with Ramamoorthi

52. Inverse RTE : Weather from APSF Measured APSF : Meteorological Visibility: [ Middleton 1952] Weather Condition: [ Van de Hulst 1957] 01 Pure Air Small Aerosols HazeMistFogRain 0.10.40.70.90.8 Objective Function :

53. Computed Atmospheric Visibilities A Camera-based Weather Station 45 images of a light source (WILD Database ECCV 02) Computed Weather Conditions Ground TruthEstimated Ground Truth Estimated

54. Volume Rendering as Convolution Analytic Multiple Scattering Shedding Light on the Weather Model Validation using Milk Summary

55. Next Class: Fluids Lectures #22