1. MV-4920 Physical Modeling Remote Sensing Basics Mapping VR/Simulation Scientific Visualization/GIS Smart Weapons Physical Nomenclature Atmospherics Illumination Surface physics EO/IR

2. Radiometric Nomenclature “ivity” ending implies intrinsic surface measurement quantities absorbtivity  = power absorbed / power incident =  abs /  in reflectivity  = power reflected/power incident =  ref /  in transmissivity  = power transmitted/power incident =  trans /  in emissivity  = power emitted/power emitted from a blackbody =  bb /  in  in  ref  trans  abs  BB 

3. Light source Sensor Intrinsic Surface Reflectivity Is Characterized by BDRF Power into the surface J in cos(  i ) watts/cm 2 = E in (x s,y s  i,  i ) E in ( x s,y s  i,  i )  (  i,  i  r,  r, ) = N (x r,y r,  r,  r, ) Radiance Reflected from a surface. Measured perpendicular to the direction of travel N watts cm -2 cos -1 (  r ) AsAs

4. BDRF of a Lambertian Surface  =  o /  BDRF of a specular surface is  = R(  I )  (  -  I +  )  (  -  i ) cos(  r ) sin(  r ) where: R(  I ) is the Fresnel reflectance: R(  I ) = ½ [ sin 2 (  i -  t ) / sin 2 (  i +  t ) +tan 2 (  i -  t ) / tan 2 (  i +  t )] where:  t = the transmitted ray angle n = sin(  i )/sin(  I ) n = the index of refraction of the media All Surfaces can be considered as something between Lambertian and Specular, hence, two extremes form the first order starting points for all approximations. BDRF MODELS

5. Pure spectral Pure Lambertian Increasing roughness Foreward ScattererBackward Scatterer ii rr

6. TermIncident Surface Description Light Lambertian directional flat surface mat finish Specular, directional gloss surface, mirror Foreward Scattering Opposition effect, directional rough surface, shadowing Back scattering Ambient- diffuseany surface, all angles in all angles out approx.

7. Minneart BDRF  = (  o /  )[ cos(  r ) cos(  i ) ] k-1 Where: k = empirically derived limb darkening parameter  o = empirically derived reflectivity Originally derived to characterize the reflectance of the lunar disk. Extension of Lambertian surface As k>1 the limb darkens Used for Topographic Normalization to correct remote sensing reflectance. Does not handle forward or back scattering.

8. Microfacet Emperical models Basic correction to spectral reflection BDRF is incorporated in two factors D an G such that  = D · G ·  (,  i ) Where:  (,  i ) is the spectral reflection of an optically smooth surface. D is the fractional surface area oriented at an angle  to the surface normal N G is the self shadowing factor H N V L shadow 

9. Examples of BDRF Models B(  ) P(  ) D Description  exp  hcos(  i )/ ) 2  Davies h = average height (4m 2 cos 4 (  )) –1 exp(-tan 2 (  )/m 2 ) Beckman m= rms slope

10. H N V L  The sum of unit vectors L+V bisects the plane made by L and V. L+V makes an angle  with the facet normal H and  with the surface normal N. N and H make an angle . The angle  is the azimuth angle about the surface normal. Geometric Definitions Cos(  N H Cos(  H  L+V) /|L+V| Cos(  N  L+V) /|L+V| Cos(  )= -cos(  )cos(  ) + sin(  )sin(  )cos(  ) L+VL+V   

11. Assume each facet acts like a smooth finished surface whose reflection function is symmetric about H. Then  (,  i ) =>  0 B(  ) equal the fraction of the total reflected energy the facet reflects in the V direction. Let P(  ) equal the probability of the facets that make and angle alpha and phi with the surface normal so that it contributes B(  to the energy at the sensor. Calculating D · G ·  (,  i ) H N  L+VL+V    The fraction of energy contributed by the facet is then, D·G·  (,  i ) =     G(   0B(  ) P(  ) cos(  )d  d 

12. Calculating the Shadow effect G Shadowing depends strongly on the relative geometry of the surface. The two most critical parameters are the standard deviation of height  h and vertical spacing  v. We also use the slope m=  h /  v. hh vv ii rr Illuminated Visible Both The Probability that a particular facet will be illuminated is Pi = cos m (  i ) The probability that a particular facet will be visible is Pv = cos m (  v )

13. Calculating the Shadow effect G hh vv ii rr Illuminated Visible Both ii vv R=  v /  h The length “l” of the overlap between illuminated and visible portions is l = R (cos(  i ) +cos(  v )) The fraction of the whole is (  v - R (cos(  i ) +cos(  v ))) /  v

14. Calculating the Shadow effect G hh vv ii rr The probability that a particular facet is both illuminated and visible is Pi Pv = cos m (  i ) cos m (  v ) When the view and illumination angle are from the same direction these probabilities are in phase so we add a phase factor (1-cos(  i  v 

15. References: Alan Watt, Mark Watt, Advanced Animation and Rendering Techniques, Addison Wesley, ISBN 0-201-54412-1, Chap. 2 The theory and practice of light/object interaction