2.2 Photometric Image Formation

Transcription

1 2.2 Photometrc Image Formaton mage plane n source sensor plane optcs!1

2 Illumnaton Computer son ory s ten deeloped wth assumpton a pont source at nfnty. But een sun has a fnte extent (about 0.5 deg sual angle) Typcal sual enronments hae more complex llumnaton!2

3 Measurng Lght Feld The feld at a pont can be measured by Takng calbrated photos a sphercal mrror Usng a sphercal camera e.g., Southampton-York Natural Scenes Dataset Spheron HDR Sphercal Camera!3

4 The BRDF The bal reflectance dstrbuton functon (BRDF) descrbes proporton comng from each ncdent that s redrected to each, as a functon waelength. BRDF s recprocal (can exchange ncdent and s). f r (,, r, r; ) θ = eleaton ncdent φ = azmuth ncdent θ r = eleaton φ r = azmuth n θ φ n θ r r φ r d y d x λ = waelength!4

5 The BRDF For sotropc s: f r (, r, r ; ) or f r (ˆ, ˆ r, ˆn; ), To calculate amount extng a pont p n ˆ r, ntegrate product ncomng L ˆ ;λ ( ) wth BRDF, takng nto account foreshortenng llumnant: Z L r (ˆ r ; )= L (ˆ ; )f r (ˆ, ˆ r, ˆn; ) cos + dˆ, where cos + = max(0, cos ). n θ n θ r r d y φ φ r d x!5

6 flecton, as well as darkenng n grooes and creases due to reduced arche.html.) nterreflectons. (Photo courtesy COMP Caltech 557 Vson Lab, - Lghtng, Materal, Shadng Whle s scattered unformly n all s,.e., 12BRDF s constant, rche.html.) scattered unformly n all s,.e., BRDF s constant, Dffuse (Lambertan, Matte) Reflecton The dffuse component We course where we wll be concerned fd (,now rbrdf, moe n ; )toscatters =thrd fd ( part ), (2.86) mostly wth w unformly, gng rse to Lambertan shadng. alues tobrdf put atseach pxel n an mage. We wll begn wth a few smple models o attered unformly n all s,.e., constant, fd (, r, n ; ) = fd ( ), (2.86) reflectance. I dscussed se qualtately, and n gae a more detaled angle between ncdent and descrpton model. he amount depends on f r,between n ; ) = fd ( ncdent ), (2.86) d ( angle t depends and amount becomes larger normal.on Ths s,because area exposed to a gen LIGHTING MATERIAL because area exposed to a gen amount becomes larger at oblque becomng self-shadowed as outgong normal ponts depends onangles, angle between completely ncdent and becomng completely self-shadowed as(thnk outgong normal ponts away from (Fgure 2.17a). about how you orent yourself towards sun or Colour greatly nfluenced by materal ecause area exposed to a gen amount becomes larger parallel ht (Fgure to 2.17a). (Thnk about how you orent towards sunoblquely or freplace get maxmum warmth how ayourself flash projected aganst a wall s comng completely self-shadowed asand outgong normal ponts axmum warmth and how a flash projected oblquely aganst a wall s ess brght one pontng drectly at t.) The shadng equaton (Fgure 2.17a). about how you orent yourself towards sun orfor dffuse reflecton can than The(Thnk amount stll depends upon ne pontng drectly at t.) The shadng equaton dffuse reflecton can ncdent angle due to oblquely for foreshortenng factor mum and a flash projected aganst a wall s hus bewarmth wrtten as howeleaton pontng drectly at t.) TheX shadng equaton for dffusex reflecton can + mage formaton X L ( ; ) = X)f ( ) cos+ = 2.2 Photometrc ambent L ( L ( )f ( )[ n ], (2.87) d r d d + + r; )= L ( )fd ( ) cos = L ( )fd ( )[ n ], X X )= L ( where )fd ( ) cos+ = L ( )fd ( )[ n ]+, where (2.87) (2.87) + [ n ] [ n ] = max(0, n ). = max(0, n ). + [ n ]+ = max(0, n ). Specular reflecton n (2.88) <1 0 < n dffuse ambent (OpenGL =1 n (2.88) pont (2.88) component a typcal BRDF sspecular or s hgh) The second major component a typcal(gloss BRDF specularreflecton, (gloss or hgh) reflecton, ongly on.consder reflectng f a reflectng f a whch depends strongly on outgong oroutgong Consder omponent a typcal BRDF s specular (gloss hgh). reflecton, Henrch Lambert Fgure 2.17b). Incdent rays are njohann arays that s( ) rotated mrrored (Fgure 2.17b). Incdent are ngly on outgong. Consder reflectngspot fnaa that s rotated (a) EECS 4422/5323 Computer Vson!6 e normal n. Usng same notaton as n Equatons ( ), by 180 around normal n. Usng same that notaton as n Equatons ( ), gure 2.17b). Incdent rays are n a s rotated mrror =0 n 0 n <glossy

7 to be a set The un-axal transforms, y can always be represented usng s reflecton: ncdent ray onto specular - - depresented transformatons. a rotaton, or equalently by a 3D nd bynormal n. axs n and an angle (Mrror) Specular Reflecton gure 2.5 shows how we can compute equalent rotaton. Frst, we n < n 0 < 0 : 180 deg rotaton ntal twst) Specular reflecton onto axs n to obtan he specular reflecton s as around normal. presented (a) by(a) a rotaton axs n and ant angle(b), or (b) equalently by a 3D T ), = n (n ) = (n n (2.29) k = s = (2n n I). (2.89) returned? caused gure 2.5 shows howk we can compute equalent rotaton. we he dmnuton by foreshortenng depends on Frst, n, 2.2 Photometrc mage formaton The dmnuton returned caused by foreshortenng depends on n, onto axs n to obtan ngle between ncdent normal n. (b) and onent that s not affected by rotaton. Next, we compute Recall: angle between ncdent and normal n. (b) eflecton: The ncdent ray s n onto specular =1 ual from n, T The ray0 depends s ontoangle specular <r 1thus reflecton: n on s = s n (n ) = (n n (2.29) <), n k =ncdent normal n.a gen nd n. r and specular tween ewnormal s. For example, n T? s=not affected k = (Iby n n rotaton. ). (2.30) nent that Next, we compute del uses a power cosne angle, =0 ealspecular reflecton s as n from n, sometmes referred to as gmbal lock. - he specular reflecton s as Thusf ( ; ) = k ( ) cos ke.s, (2.90) 180 s T T s = = ks s = (2n n I) (2.89)? = (I n n ). (2.30)? k T s = = (2n n I). (2.89) n < 0? k e and Sparrow (1967) mcro-facet model uses a Gaussan, 65 metmes referred to as gmbal lock. (a) (b) 1 n a gen rnthus depends on angle s = Amount r depends on angle θ = cos r s ). ( 2 2 s fs ( s ; ) =Fgure ks ( 2.17 ) exp( csdmnuton s ). (a) The caused foreshortenng depends on n, ween ew specular s returned. For example, by(2.91) r and n a gen rbetween thus depends on angle and = normal n. (b) s cosne angle ncdent e.g., model: el uses a power Phong cosne angle, Mrror reflecton: The ncdent ray s onto specular tween ew Gaussan r (specular) and c specular s specular. For example, ke (or nerse wdths to more s s ) correspond s normal n.wth ster gloss. karound e angle, s, whle exponents better model materals del uses a power cosne fsmaller ( ; ) = k ( ) cos, (2.90) s s s s ke = kespecular we can compute reflecton s as and Sparrow (1967) uses fs ( mcro-facet ( ) cos as,gaussan, (2.90) s ; ) = ksmodel Colour Sharpness f ( ; ) = k ( ) exp( c ). 2 2 s = 7! k? = (2n n T I). (2.91) (2.89)

8 Phong Shadng The full Phong model combnes dffuse and specular components contrbuted by man llumnant wth an ambent term that attempts to account for all or ncdent upon from or parts scene (sky, walls, etc.) L r (ˆ r ; )=k a ( )L a ( )+k d ( ) X L ( )[ˆ ˆn] + + k s ( ) X L ( )(ˆ r ŝ ) k e Ambent Dffuse Specular NB: I can t make sense Fg. 2.18: please gnore. Typcally: k a k s L a ( λ)! k d ( λ) (both due to sub- scatter). ( λ)! constant, thus specularty assumes colour llumnant. ( λ) L ( λ) Bu Tuong Phong ( )!8

9 Ray Tracng The Phong model assumes a fnte number dscrete sources. Lght emtted by se sources bounces f and nto camera. In realty, some se sources may be shadowed by or objects, and s generally also llumnated by nter-reflectons (multple bounces) Two approaches, dependng on nature scene: If mostly specular, use ray tracng: Follow each ray from camera across multple bounces toward sources If mostly matte, use radosty: Model nterchanged between all pars patches, and n sole as lnear system wth sources as forcng functon.!9

10 Optcs In Lecture 2.1, we treated projecton to mage usng a pnhole camera model. To account for focus, aperture, aberratons etc. we need to elaborate ths model.!10

11 Thn Lens Model Assume low-curature, symmetrc, conex sphercal lens f = 100 mm W=35mm c f.o.. P d Δz z =102 mm z o =5 m f = focal length W = sensor wdth z0 = dstance from optcal centre to object z = dstance from optcal centre to where focused mage object s formed d = aperture c = crcle confuson!11

12 Lens Equaton f = 100 mm W=35mm c f.o.. P d Δz z =102 mm z o =5 m 1 z o + 1 z = 1 f f-number (f-stop) = f / d. Note that lm z0 z = f. If sensor plane does not le at z, a pont on object wll be maged as a blurred dsk ( crcle confuson c). (a) Allowable depth araton that lmts ths blur to an acceptable leel called depth feld. Depth feld ncreases wth larger apertures and longer ewng dstances.!12

13 Chromatc Aberraton Index refracton glass ares sly as a functon waelength. As a result, dfferent waelengths focus at sly dfferent dstances. To reduce aberratons, most photographc lenses are compound lenses usng multple elements.!13