Optical Models Jian Huang CS 594 Spring 2002 This set of slides are modified from slides used by Prof Torsten Moeller at Simon Fraser University BC Canada Optical Models Nelson Max Optical Models IEEE Transactions on Visualization and Computer Graphics Vol 1 No 2 1995 Jim Blinn s 1982 SIGGRAPH paper on scattering The mathematical framework for light transport in volume rendering Transport of Light Determination of Intensity Local Diffuse and Specular Global Radiosity Ray Tracing Ultimate Physcial Mechanisms in Ultimate Model Observer Emittance Absorption Scattering single vs multiple Light Transport of Light Typically based on S Chandrasekhar Radiative Transfer Oxford Universtiy Press 1950 Mathematically challenging Observer Approximate models required Blinn et al to the rescue Over Operator only emission and absorption Light Max 1995 Several cases Completely opaque or transparent voxels Variable opacity correction Self emitting glow Self emitting glow with opacity along viewing ray Single scattering of external illumination Multiple scattering Blinn Assumptions N Assumptions N surface normal E eye vector L light vector T surface thickness e angle btw E and N a angle btw E and L aka phase angle I angle btw N and L E e i a T Particles L Blinn Assumptions N Assumptions contd particles are little spheres with radius p n number density number of particles per unit volume cosine of angle e i e N E D proportional volume of the cloud occupied by particles 4 3 D n p 3 E e i a T Particles L Blinn transparency 1 Expected particles in a volume will be nV Probability that there are no particles in the way can be modeled as a Poisson process P 0 V e nV Cylinders must be empty E t Hence the probability that the light is making it through those tubes is P 0 V e L E Top Lit 2 2 p T n p T 0 n e t Cylinders of Integration L Bottom Lit Blinn transparency 2 Transparency through the cloud Tr e E T is called the optical depth n p 2T E Max absorption only I s intensity at distance s along a ray s extinction coefficient dI s I s ds s I s I 0 exp t dt 0 I 0T s T s transparency between 0 and s Max absorption only Linear variation of D D 0 T s exp t dt 0 0 D exp D 2 D t Max absorption only On the opacity D 1 T s 1 exp t dt 0 1 exp D 2 D D 2 assuming to be constant in the interval Max cloud model Using fractal structure Perlin Max self emitting glow Without extinction dI g s ds s I s I 0 g t dt 0 Max self emitting glow Without extinction Max self emitting glow With extinction dI g s s I s ds D D D I D I 0 exp t dt g s exp t dt ds 0 0 s D I D I 0T D g s T s ds 0 Volume Ray Integral 1 The continuous form D D D I D I 0 exp t dt g s exp t dt ds 0 0 s In general case can not compute analytically Volume Ray Integration 2 Practical Computation Method D D D I D I 0 exp t dt g s exp t dt ds 0 0 s ti exp i x x 1 i x x note x can be xi different everywhere n I D I 0 ti t j g i i 1 j i 1 i 1 which leads to the familiar BTF or FTB compositing g n t n g n 1 t n 1 g n 2 g1 t1 I 0 n n g s g s could be Self emitting particle glow Reflected color obtained via illumination The color is usually the sum of emitted color E and reflected color R Max self emitting glow Identical glowing spherical particles projected area a r2 surface glow color C number per unit volume N occluded area aNAdl total area A extinction coefficient a N added glow intensity per unit length g CaN C dl A Max self emitting glow Special Case g C and C constant D D D D g s exp t dt ds C s exp t dt ds 0 0 s s D C 1 exp t dt 0 I D I 0T D C 1 T D This is compositing color C on top of background I0 Max self emitting glow For I0 0 and varying according to f Volume Ray Integration Equation Self emitting glow none constant color D D D D g s exp t dt ds C s s exp t dt ds 0 0 s s Max reflection O X g x r x i x i x illumination reaching point x unit reflection direction vector unit illumination direction vector r x BRDF f x for conventional surface shading effects Max reflection For particle densities O X r x w x x p w x albedo Blinn assuming that the primary effect is from interaction of light with one single particle albedo proportion of light reflected from a particle in the range of 0 1 p phase function still unrealistic external reflection of outside illumination Blinn Phase Function how we see the particles depends on the angle of eye E and light vector L smooth drop off Top View a 0 EyeView L E L a 90 E a L 0 a 180 a 180 E Blinn Phase Function Many different models possible a 1 Constant function size of particles much less then wavelength of the light Anisotropic a 1 x cos a more light forward then backward essentially our diffuse shading 8 sin a a cos a a Lambert surfaces 3 spheres reflect according to Lamberts law physically based Blinn Phase Function a 3 1 cos 2 a 4 diffraction effects dominate Rayleigh Scattering Henyey Greenstein a 1 g 2 1 g 2 2 g cos a general model with good fit to empirical data Empirical Measurments tabulated phase function sums of functions weighted sum of functions model different effects in parallel 3 2 Max shadows Should account for transparency of volume between light source and point x i x L exp x t dt 0 those are the shadow feelers e g Kajiya s two pass algorithm O X Max shadows Max multiple scattering I x intensity at position x in direction g x source term glow at x in direction g x r x I x d 4 I x I 0 x D T D D r x s I x d T s ds 0 4 I I 0 KI Max multiple scattering use radiosity ideas for the solution simplified model scattering is isotropic i e g x does not depend on then we can compute the total contribution of all voxels to the iso tropic scattering S xi a xi Fij g x j where Fij are the form factors and ai are the albedo j coefficients do a first pass like Kayija and collect external illumination in E …
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