Image formation

Camera: optical system da=a -a 2 1 curvature radius r r Z 1 2 thin lens small angles:

Y incident light beam lens refraction index: n deviated beam Z deviation angle ? Dq = q’’-q

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

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

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

Hp: Z >> a r  f the image of a point P belongs to the line (P,O) P image plane O p 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

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

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 [adapted from Hemant D. Tagare]

Reflectance Map dA at P receives flux of d watt: Irradiance at P: (watt/meter2, spatial density of flux at P)

reflectance map ctd. d infinitesimal solid angle, centered along incident direction infinitesimal flux d2 passes through it, incident on dA ,: zenith,azimuth dA cos : fore-shortened area dAf Radiance of incident flux: Units: watt per m2 per steradian

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

Solid Angle Solid angle d: solid angle centered around direction , spherical geometry: dA = r2 sin dd

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,

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) :

a f Z image intensity: proportional to irradiance E where is the radiance reflected towards the lens