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Homogeneous representation

Homogeneous representation. Points Vectors Transformation representation. Lecture 4: Image formation. Image Formation. Vision infers world properties form images. How do images depend on these properties? Two key elements Geometry Radiometry We consider only simple models of these.

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Homogeneous representation

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  1. Homogeneous representation • Points • Vectors • Transformation • representation UCLA Vision Lab

  2. Lecture 4: Image formation UCLA Vision Lab

  3. Image Formation • Vision infers world properties form images. • How do images depend on these properties? • Two key elements • Geometry • Radiometry • We consider only simple models of these UCLA Vision Lab

  4. Image formation (Chapter 3) UCLA Vision Lab

  5. Representation of images UCLA Vision Lab

  6. Similar triangles <P’F’S’>,<ROF’> and <PSF><QOF>  UCLA Vision Lab

  7. Pinhole model UCLA Vision Lab

  8. Forward pinhole UCLA Vision Lab

  9. Distant objects are smaller UCLA Vision Lab (Forsyth & Ponce)

  10. Parallel lines meet Common to draw image plane in front of the focal point. Moving the image plane merely scales the image. UCLA Vision Lab (Forsyth & Ponce)

  11. Vanishing points • Each set of parallel lines meets at a different point • The vanishing point for this direction • Sets of parallel lines on the same plane lead to collinear vanishing points. • The line is called the horizon for that plane UCLA Vision Lab

  12. Properties of Projection • Points project to points • Lines project to lines • Planes project to the whole image or a half image • Angles are not preserved • Degenerate cases • Line through focal point projects to a point. • Plane through focal point projects to line • Plane perpendicular to image plane projects to part of the image (with horizon). UCLA Vision Lab

  13. Orthographic projection UCLA Vision Lab

  14. UCLA Vision Lab

  15. Cameras with Lenses UCLA Vision Lab (Forsyth & Ponce)

  16. UCLA Vision Lab

  17. Assumptions for thin lens equation • Lens surfaces are spherical • Incoming light rays make a small angle with the optical axis • The lens thickness is small compared to the radii of curvature • The refractive index is the same for the media on both sides of the lens UCLA Vision Lab

  18. UCLA Vision Lab

  19. Blur circle Points a t distance are brought into focus at distance is imaged at point A point at distance from the lens and so Thus points at distance will give rise to a blur circle of diameter with d the diameter of the lens UCLA Vision Lab

  20. Interaction of light with matter • Absorption • Scattering • Refraction • Reflection • Other effects: • Diffraction: deviation of straight propagation in the presence of obstacles • Fluorescence:absorbtion of light of a given wavelength by a fluorescent molecule causes reemission at another wavelength UCLA Vision Lab

  21. Refraction n1, n2: indexes of refraction UCLA Vision Lab

  22. steradian (sr) dw q w dw = df sinq dq f Solid Angle hemisphere radian dq q Sphere: 4p UCLA Vision Lab

  23. Radiometric Terms UCLA Vision Lab

  24. Irradiance and Radiance Irradiance Definition: power per unit area incident on a surface [W/m2 = lux] Radiance Definition: power per unit area and projected solid angle [W/m2sr] UCLA Vision Lab

  25. Radiant Intensity Definition: flux per unit solid angle : Radiant flux[W] [W/sr = cd (candela)]  [ ] UCLA Vision Lab

  26. Isotropic Point Source UCLA Vision Lab

  27. Isotropic Point Source : Radiant flux[W] All directions: solid angle 4p • Radiant flux per unit solid angle [W/sr] r : Radiant intensity • Note inverse square law fall off. UCLA Vision Lab

  28. Isotropic Point Source : Radiant flux[W] All directions: solid angle 4p • Radiant flux per unit solid angle [W/sr] r : Radiant intensity h • Note cosine dependency. UCLA Vision Lab

  29. Isotropic Point Source Point source at a finite distance r h • Note inverse square law fall off. • Note cosine dependency. UCLA Vision Lab

  30. Irradiance from Area Sources UCLA Vision Lab

  31. Hemispherical Source L UCLA Vision Lab

  32. Lr(x,w) Ei Reflectance The surface becomes a light source Li(x,wi) qi Ei dLr=fr dEi Reflectance: ratio of radiance to irradiance UCLA Vision Lab

  33. BRDF UCLA Vision Lab

  34. BRDF UCLA Vision Lab

  35. Reflection Equation UCLA Vision Lab

  36. qi  LB(qr, fr) is constant for all directions (qr, fr) Perfectly Diffuse Reflection • Perfectly Diffuse Surface • Appears equally bright from all viewing directions (qr, fr) • Reflects all incident light, i.e., UCLA Vision Lab

  37. qi Common Diffuse Reflection • Normal Diffuse Surface • Appears almost equally bright from most viewing • directions (qr, fr), qr << 90° • Reflects only a fraction of incident light, i.e., Reflectance : Albedo UCLA Vision Lab

  38. qi Perfectly Diffuse Reflection Distant point light source Lambertian cosine Law UCLA Vision Lab

  39. Law of Reflection UCLA Vision Lab

  40. Perfectly Specular Reflection From the definition of BRDF, the surface radiance is: To satisfy: UCLA Vision Lab

  41. Lambertian Examples Lambertian sphere as the light moves. (Steve Seitz) Scene (Oren and Nayar) UCLA Vision Lab

  42. Lambertian + Specular Model UCLA Vision Lab

  43. Lambertian + specular • Two parameters: how shiny, what kind of shiny. • Advantages • easy to manipulate • very often quite close true • Disadvantages • some surfaces are not • e.g. underside of CD’s, feathers of many birds, blue spots on many marine crustaceans and fish, most rough surfaces, oil films (skin!), wet surfaces • Generally, very little advantage in modelling behaviour of light at a surface in more detail -- it is quite difficult to understand behaviour of L+S surfaces (but in graphics???) UCLA Vision Lab

  44. Lambertian+Specular+Ambient (http://graphics.cs.ucdavis.edu/GraphicsNotes/Shading/Shading.html) UCLA Vision Lab

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