1. Camera: optical system d  21 thin lens small angles: Y Z  2 1 curvature radius

2. Y Z incident light beam deviated beam deviation angle ?   ’’  lens refraction index: n

3. Thin lens rules a) Y=0   = 0 f Y parallel rays converge onto a focal plane  b) f  = Y  beams through lens center: undeviated independent of y

4. r f Y h Where do all rays starting from a scene point P converge ? Z Fresnel law P Obs. For Z  ∞, r  f O p ?

5. d f a Z if d ≠ r … focussed image:  blurring circle) <image resolution depth of field: range [Z1, Z2] where image is focussed image plane P p O r  (blurring circle)=a (d-r)/r image of a point = blurring circle

6. the image of a point P belongs to the line (P,O) p P O p = image of P = image plane ∩ line(O,P) interpretation line of p: line(O,p) = locus of the scene points projecting onto image point p image plane r  f Hp: Z >> a

7. Until here: where goes light ? But: how much light does reach an image point?

8. Image Formation: Reflectance Map Simplified model: light originates at a source light is reflected by an object light collected by camera lens and focused to image [Hemant D. Tagare, CV course notes, adapted from B.Horn]

9. The Reflectance Map dA at P receives flux of d  watt: Irradiance at P: (watt/meter 2, spatial density of flux at P)

10. The Reflectance Map ctd. d  infinitesimal solid angle, centered along incident direction infinitesimal flux d 2  passes through it, incident on dA ,  : zenith,azimuth dA cos  : fore- shortened area dA f Radiance of incident flux: Units: watt per m 2 per steradian

11. The Reflectance Map ctd. Object is a point (no area) flux d  incident on it from solid angle d  Radiant intensity of flux: units: watt per steradian

12. Computation of Solid Angle Solid angle d  : solid angle centered around direction ,  spherical geometry: dA = r 2 sin  d  d 

13. Computation of Flux Flux  : irradiance E and radiance L are derivatives of flux  flux comp. as integral L(P, ,  ): radiance along ,  at any point P of surface E(P): irradiance at P  : net flux received by object and,

14. Generalization: Reflected Light so far: radiance and radiant intensity defined in terms of incident light definitions also apply to reflected / emitted light  flux d  is assumed to have reverse direction (leaves surface) radiance of reflected light (dA through d  ) :

15. image intensity: proportional to irradiance E  whereis the radiance reflected towards the lens Zf

16. p P O Z Y X c y x perspective projection f -nonlinear -not shape-preserving -not length-ratio preserving

17. Point [x,y] T expanded to [u,v,w] T Any two sets of points [u 1,v 1,w 1 ] T and [u 2,v 2,w 2 ] T represent the same point if one is multiple of the other [u,v,w] T  [x,y] with x=u/w, and y=v/w [u,v,0] T is the point at the infinite along direction (u,v) In 2D: add a third coordinate, w Homogeneous coordinates

18. Transformations translation by vector [ d x,d y ] T scaling (by different factors in x and y) rotation by angle 

19. Homogeneous coordinates In 3D: add a fourth coordinate, t Point [X,Y,Z] T expanded to [x,y,z,t] T Any two sets of points [x 1,y 1,z 1,t 1 ] T and [x 2,y 2,z 2,t 2 ] T represent the same point if one is multiple of the other [x,y,z,t] T  [X,Y,Z] with X=x/t, Y=y/t, and Z=z/t [x,y,z,0] T is the point at the infinite along direction (x,y,z)

20. Transformations scaling translation rotation Obs: rotation matrix is an orthogonal matrix i.e.: R -1 = RTRT

21. with Scene->Image mapping: perspective transformation With “ad hoc” reference frames, for both image and scene

22. Let us recall them O Z Y X c y x f scene reference - centered on lens center - Z-axis orthogonal to image plane - X- and Y-axes opposite to image x- and y-axes image reference - centered on principal point - x- and y-axes parallel to the sensor rows and columns - Euclidean reference