Stefano Soatto (c) UCLA Vision Lab 1 Homogeneous representation Points Vectors Transformation representation

Stefano Soatto (c) UCLA Vision Lab 2 Lecture 4: Image formation

Stefano Soatto (c) 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

Stefano Soatto (c)UCLA Vision Lab4 Image formation (Chapter 3)

Stefano Soatto (c) UCLA Vision Lab 5 Representation of images

Stefano Soatto (c) UCLA Vision Lab 6 Similar triangles, and 

Stefano Soatto (c) UCLA Vision Lab 7 Pinhole model

Stefano Soatto (c) UCLA Vision Lab 8 Forward pinhole

Stefano Soatto (c) UCLA Vision Lab 9 Distant objects are smaller (Forsyth & Ponce)

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

Stefano Soatto (c) UCLA Vision Lab 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

Stefano Soatto (c) 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).

Stefano Soatto (c) UCLA Vision Lab 13 Orthographic projection

Stefano Soatto (c) UCLA Vision Lab 14

Stefano Soatto (c) UCLA Vision Lab 15 Cameras with Lenses (Forsyth & Ponce)

Stefano Soatto (c) UCLA Vision Lab 16

Stefano Soatto (c) 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

Stefano Soatto (c) UCLA Vision Lab 18

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

Stefano Soatto (c) 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

Stefano Soatto (c) UCLA Vision Lab 21 Refraction n1, n2: indexes of refraction

Stefano Soatto (c) UCLA Vision Lab 22 Solid Angle  radian  steradian (sr)     d  d  sin  d  hemisphere Sphere: 4 

Stefano Soatto (c) UCLA Vision Lab 23 Radiometric Terms

Stefano Soatto (c) UCLA Vision Lab 24 Irradiance and Radiance Irradiance Definition: power per unit area incident on a surface  W/m 2 = lux  Radiance Definition: power per unit area and projected solid angle  W/m 2 sr 

Stefano Soatto (c) UCLA Vision Lab 25 Radiant Intensity  Radiant flux  W   Definition: flux per unit solid angle  W/sr = cd (candela)  [ ]

Stefano Soatto (c) UCLA Vision Lab 26 Isotropic Point Source

Stefano Soatto (c) UCLA Vision Lab 27 Isotropic Point Source  Radiant flux  W  All directions: solid angle 4   Radiant flux per unit solid angle  W/sr   Radiant intensity r Note inverse square law fall off.

Stefano Soatto (c) UCLA Vision Lab 28 Isotropic Point Source  Radiant flux  W  All directions: solid angle 4   Radiant flux per unit solid angle  W/sr   Radiant intensity r Note cosine dependency. h

Stefano Soatto (c) UCLA Vision Lab 29 Isotropic Point Source Point source at a finite distance r Note cosine dependency. h Note inverse square law fall off.

Stefano Soatto (c) UCLA Vision Lab 30 Irradiance from Area Sources

Stefano Soatto (c) UCLA Vision Lab 31 Hemispherical Source L

Stefano Soatto (c) UCLA Vision Lab 32 Reflectance Reflectance: ratio of radiance to irradiance dL r =f r dE i EiEi ii EiEi L r (x,  ) L i (x,  i ) The surface becomes a light source

Stefano Soatto (c) UCLA Vision Lab 33 BRDF

Stefano Soatto (c) UCLA Vision Lab 34 BRDF

Stefano Soatto (c) UCLA Vision Lab 35 Reflection Equation

Stefano Soatto (c) UCLA Vision Lab 36 Perfectly Diffuse Reflection ii Perfectly Diffuse Surface Appears equally bright from all viewing directions (  r,  r ) Reflects all incident light, i.e.,  L B (  r,  r ) is constant for all directions (  r,  r )

Stefano Soatto (c) UCLA Vision Lab 37 Common Diffuse Reflection ii Normal Diffuse Surface Appears almost equally bright from most viewing directions (  r,  r ),  r << 90° Reflects only a fraction of incident light, i.e., Reflectance : Albedo

Stefano Soatto (c) UCLA Vision Lab 38 Perfectly Diffuse Reflection ii Lambertian cosine Law Distant point light source

Stefano Soatto (c) UCLA Vision Lab 39 Law of Reflection

Stefano Soatto (c) UCLA Vision Lab 40 Perfectly Specular Reflection From the definition of BRDF, the surface radiance is: To satisfy:

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

Stefano Soatto (c) UCLA Vision Lab 42 Lambertian + Specular Model

Stefano Soatto (c) 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???)

Stefano Soatto (c) UCLA Vision Lab 44 Lambertian+Specular+Ambient (