PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Computer VisionJanuary 2002 L1.1© 2002 by Davi GeigerImage Formation Light can change the image (and appearances). What is the relation between pixel brightness and scene reflectance ?Computer VisionJanuary 2002 L1.2© 2002 by Davi GeigerImage Formation nˆR2cosRAsolid angle subtended by a small patch of area A.AL - radiance is the amount of light radiated from a surface per solid angle (power per unit area per unit solid angle emitted from a surface. )E - irradiance is the amount of light falling in a surface (power per unit area incident in a surface. )12 srmW2mWComputer VisionJanuary 2002 L1.3© 2002 by Davi GeigernˆAIzfSurface Radiance and Image Irradiance22)cos/(cos)cos/(cosfIzA2coscosfzIASame solid anglePinhole Camera ModelComputer VisionJanuary 2002 L1.4© 2002 by Davi GeigernˆAIzfdSurface Radiance and Image Irradiance3222cos4)cos/(cos4zdzd Solid angle subtended by the lens, as seen by the patch Acoscos4cos32zdALALPPower from patch A through the lens4232cos4coscos4fdLzdIALIPEThus, we concludeComputer VisionJanuary 2002 L1.5© 2002 by Davi GeigerConclusions 42cos4fdLE•The irradiance at the image pixel is converted into the brightness of the pixel•Image Irradiance is proportional to Scene Radiance•Scene distance, z, does not affect/reduce image brightness (the model is too simplified, since in practice it does.)•The angle of the scene patch with respect to the view ( reduces the brightness by the . In practice the effect is even stronger.4cosComputer VisionJanuary 2002 L1.6© 2002 by Davi GeigerThe Bidirectional Reflectance Distribution Function (BRDF)),(),(),,,(iieeeeiiELfBRDF - How bright a surface appears when viewed from one direction while light falls on it from another.),(ii),(eenˆnˆiiUsually f depends only on ,  true for matte surfaces and specularlyreflecting surfaces.ieei,Computer VisionJanuary 2002 L1.7© 2002 by Davi GeigerExtended Light Sources and BRDFiiisiniiAiiiiiiiEEsin),(),( Light source radiance arriving through solid angle iiiiiiiiiEAEAPsincos),(),(cos Power arriving at patch A from thus the irradiance arriving at patch A is 200sincos),(iiiiiiEAPEThe radiance of a patch A at direction is thus, given by 20sincos),(),,,(),(iiiiiieeiieeEfLForeshortening),(eeComputer VisionJanuary 2002 L1.8© 2002 by Davi GeigerSpecial Cases of BRDF1. Lambertian Surfaces (matte)- appears equally bright from all viewing directions and reflects all incident light, absorbing none, i.e. the BRDF is constant and . What constant f ? 20sincoseeeeThus, the total “reflected power” from patch A becomesUsing thatwe finally obtain1f 200sincos),(),( EfEfLiiiiiieeAEfALpeeeeee 020sincos),(  sinceForeshortening.0EL ,0EL 0EfApL and for Lambertian surfacesComputer VisionJanuary 2002 L1.9© 2002 by Davi GeigerSpecial Cases of BRDF1. S2. Specular Surfaces (mirrors) – reflects all light arriving from the direction into the direction . The BRDF is in this case proportional to the product of two impulses, and .What is the factor of proportionality ?),(ii),(ii)(ie)(ie),(iik 200sincos),(iiiiiiEE 20sincos),(eeeeeeLLeeeeeeiiiiiiieieiieeEkEkLsincos),(),(sincos),()()(),(),(20 we finally obtainiiieieeeiifsincos)()(),,,(0ELand for specular surfacesiiiiksincos1),( Computer VisionJanuary 2002 L1.10© 2002 by Davi GeigerLambertian Surface Brightness),(ii),(eenˆHow bright will a Lambertian surface bewhen it is illuminated by a point sourceof radiance E? and by a “sky” of uniformradiance E?For a point source the irradiance at the surface is and the radiance must then beieeEEfLcos1),(0iEEcos0Familiar cosine or “Lambert’s law” of reflection from matte surfaces(surfaces covered with finely powdered transparent materials such as barium sulfate or magnesium carbonate), and can approximate paper, snow and matte paint.Finally, for a “sky” of uniform radiance E we obtain!sincos1),(20EELiiiiee