Computational Photography. Lecture #2. Today. Photometric Stereo. Shape from X. Francesc Moreno Noguer

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1 elected oc Comute Vo Comutatoal Photogah Lectue # oda hae Fom X Radomet ad Reflectace Photometc teeo Facec Moeo ogue fmoeogue@gmal.com hae fom X hae fom X Method fo obtag the 3D hae fom mage. X ca be ma dffeet cue: teeo o moe vew Moto movg camea o object hadg Chagg Lghtg Photometc teeo etue vaato Focu/Defocu tuctued lght

2 hae fom X hae Fom teeo ad Moto hae fom X hae Fom hadg -agulate the ame ot o two o moe mage -mage acued multaeoul F teeo -mage fom a movg camea o object F Moto hadg: Vaato of bghte due to chage oetato hadg eveal 3D geomet hae-fom-hadg: Jut oe mage to ecove hae Reue ma cotat hae fom X Photometc teeo hae fom X hae Fom Focu/Defocu gle vewot, ad multle mage ude dffeet lghtg. g Camea Focu/Defocu aothe cue to etmate deth.

3 hae fom X hae Fom etue hae fom X tuctue Lght ad Lae Rage Fde etue: Reetto of a elemet o the aeaace of a ecfc temlate ove a hae-fom-etue: gue the hae of a fom the defomato of t eel Xtue Lemet Actve agulato Pojecto-lae / mage-lae tagulato hae fom X adtoal Aoache teeo Moto hadg Photometc teeo Lectue # etue vaato Camea Focu/Defocu Lectue #3 tuctued Lght Radomet ad Reflectace ovel echue Helmholtz teeo Lectue #3 Pojecto Defocu Lectue #4 Coded ad Mult Aetue Lectue #5

4 Dgtal mage Fomato Dgtal mage Fomato Camea Lghtg g Lght ouce mage Plae: Comute Phcal Model Lght ouce uface uface uface Le mage adace Radace adace L cee mage Plae Meaued Pel: We eed to udetad the elato betwee the lghtg, eflectace ad medum ad the mage of the cee. mage adace Camea lectoc Meaued Pel Value, Dgtal mage Fomato Dgtal mage Fomato Lght ouce mage Plae: Lght ouce uface uface uface Le mage adace Radace adace L z omal decto θ φ ouce cdet decto θ, φ θ, φ omal elemet vewg decto mage Plae Meaued Pel: mage adace Camea lectoc Meaued Pel Value, L θ, φ θ, φ BRDF : adace at uface decto θ, φ Radace of uface decto θ, φ f θ, φ ; θ, φ L θ, φ θ, φ Bdectoal Reflectace Dtbuto Fucto

5 Dgtal mage Fomato Dgtal mage Fomato mage lae Lght ouce mage Plae: atch da θ Lght ouce uface uface uface Le mage adace Radace adace L α α mage atch da mage Plae Meaued Pel: mage adace Camea lectoc Meaued Pel Value, z t ca be how that: L f π d 4 co α 4 f mage adace PROPOROAL to uface Radace! mall feld of vew ffect of 4 th owe of coe ae mall d/f the vee of the f-umbe. Dgtal mage Fomato Dgtal mage Fomato Lght ouce mage Plae: he camea eoe fucto elate mage adace at the mage lae to the meaued el tet value. Lght ouce uface uface uface Le mage adace Radace adace L g : mage Plae Meaued Pel: Pel value the fal mage mage adace Camea lectoc Meaued Pel Value, Gobeg ad aa, CVPR 003 Uuall th a o-lea Mag!

6 Dgtal mage Fomato Mecham of Reflecto Radometc Leazato of the Camea motat eoceg te fo ma vo ad gahc algothm uch a hotometc t teeo, vaat, de-weatheg, vee edeg, mage baed edeg, etc. g : Ue a colo Magbeth chat wth ecel kow eflectace. 90% 59.% 36.% 9.8% 9.0% 3.% Pel Value 55 0 g g? 0? adace cot * Reflectace Method aume cotat lghtg o all atche ad wok bet whe ouce fa awa eamle ulght. Bod Reflecto: ouce bod eflecto cdet decto Dffue Reflecto Matte Aeaace o-homogeeou Medum Cla, ae, etc eflecto uface Reflecto: ecula Reflecto Glo Aeaace Hghlght Domat fo Metal Uue vee et becaue g mootoc ad mooth fo all camea. mage tet Bod Reflecto uface Reflecto amle uface Reflectace Model How to model the mecham of eflectace? Dffue Reflecto Dffue ecula Reflecto ecula Reflecto Dffue Reflecto: Lambeta Model v uface aea euall bght fom ALL decto! deedet of ouce tet K cdet decto omal v vewg decto θ elemet Lambeta BRDF ml a cotat : f θ, φ ; θ, φ ρd π albedo

7 Reflectace Model Dffue Reflecto: Lambeta Model uface Radace : ρd L K co θ π ρd π K ouce tet Commol ued Vo ad Gahc! h the Lambet coe law of eflecto fom matte. Reflectace Model ecula Reflecto Mo BRDF Vald fo ve mooth. All cdet lght eeg eflected a GL decto ouce tet K ecula/mo decto θ, φ cdet cde decto θ, φ elemet omal vewg decto v θ v, φ v Mo BRDF ml a double-delta delta fucto : f θ, φ; θv, φv ρ δ θ θv δ φ π φv ecula albedo Reflectace Model ecula Reflecto Mo BRDF uface Radace : L K ρ δ θ θ δ φ π φ K ρ L 0 f v v θ θ ad φ φ π v othewe v θ, φ z θ ecula/mo decto, φ θ φ π θ, Reflectace Model Dffue ecula Reflecto Ma eal combe dffue ad glo comoet of eflecto ode to aomate thee Model combg Dffue ad ecula eflecto Phog, Dchomatc, Obeved mage Colo α Bod Colo β ecula Reflecto Colo φ Vewe eceve lght ol whe v dffue ecula dffueecula

8 Reflectace Model Phog Model Ambet Dffue ecula α kaa kd d k v lght a,d,: d : tete of the ambet ecula ad dffue comoet of the lght ouce. ka,kd,k: mateal comoet α : he how evel the lght eflected Model a agula falloff of the hghlght v Reflectace Model Homewok: mlemet Phog Model fo a hee fo a hee ea to comute the omal,,, R R Reflectace Model Reult Photometc teeo Phog Model Paamete a:0.9 d:0.44 :0.59 ka:0.03 kd:0.47 k:0.50 alha:8.94 Phog Model Paamete a:0.7 d:0.88 :0.4 ka:0.3 kd:0.47 k:0.53 alha:7.09 Phog Model Paamete a:0.6 d:0.43 :0.45 ka:0.4 kd:0.49 k:0.43 alha:8.63

9 uface Oetato Gadet ace z f, ' mage Plae omal ' uface :,,,f, aget vecto :, f,0,, f 0,, omal :,,,, z z θ z Ut omal vecto ouce vecto,,,, lae called the Gadet ace lae t comoet ad, ae the loe the - ad - decto ve ot o t coeod to a atcula oetato Reflectace Ma Relate mage adace, to oetato, fo GV ouce decto ad eflectace ool ued develog method fo ecoveg hae fom mage. Let code a eamle: Lambeta cae:, k : ouce bghte ρ : albedo eflectace c : cotat otcal tem mage adace: ρ ρ kc coθ kc π π ρ Let kc the co π θ θ v Reflectace Ma Lambeta Cae coθ R, Reflectace Ma Lambeta Cotou of cotat bghte ae coc ecto the -lae: c R, c

10 Reflectace Ma Reflectace Ma Lambeta Cae c o-bghte cotou R, c Lambeta Cae o-bghte cotou 0.9.0, 0.8 R, 0. 7 coe of cotat θ o θ R,,, ote: mamum whe 0.0 Reflectace Ma Dffue ecula Reflectace ma wth two eak: oetato that mamze each of the two dffeet te of eflecto., Dffue eak ecula eak hae fom a gle mage? Gve a gle mage of a object wth kow eflectace take ude a kow lght ouce, ca we ecove the hae of the object? Gve R,, ad eflectace ca we deteme, uuel fo each mage ot? All the ot o the le, ae oluto fo, R, 0. 5, O

11 oluto Photometc teeo ake moe mage Photometc teeo Add moe cotat hae-fom-hadg g oo etctve to be ueful,, 3 3 3, Photometc teeo 3 v We ca wte th mat fom: Lambeta cae: ρ kc coθ ρ π mage adace: ρρ ρ ρ ρ kc π olvg the uato 3 ρ ~ ~ vee ρ ~ ~ ~ ~ ρ ce a ut vecto, the om of wll be ρ

12 Moe tha hee Lght ouce Get bette eult b ug moe lght M Leat uae oluto: ~ ~ M ρ ~ olve fo ρ,a befoe ~ ~ 3 3 Mooe-Peoe eudo vee Colo mage he cae of RGB mage get thee et of euato, oe e colo chael: R ρr G ρg ρ B B mle oluto: ft olve fo ug oe chael he ubttute kow to above euato to get ρ, ρ, ρ R G O combe thee chael ad olve fo B R G B ρ ck fo Hadlg hadow Wegh each euato b the el bghte: ρ Gve weghted leat-uae mat euato: M olve fo ρ, a befoe M ρ Deth Fom omal Photometc teeo jut ecove the omal We eed to comute the deth!!! a Method: hae B tegato D Patal Devatve: Gve the chage heght wth a mall te. We ca get the b ummg the chage. tegated oetato 0 f f f f d 0 f

13 Deth Fom omal, 00 00, hae B tegato D 0 0, j j Deth Fom omal hae B tegato D Lmtato: tegato good ol f object hae cotuou t etve to oe: oluto: Aveage the eult ove ma dffeet ath. f j f j f d d f Aged a o j 0 j X X X X Hgh cot ad tll accuate!!!!!! Deth Fom omal Deth Fom omal Fakot-Chellaa Algothm Bet algothm foud to comute deth fom omal. Ve obut to oe. Fat ad ve mle mlemetato. Fomulato of the oblem:,, fˆ, Let be the to be ecotucted he bac dea to oject the etmated loe, ad, oto a et of tegable loe ˆ, ad ˆ, uch that:. tegatblt Cotat:. he dtace eo mmzed: f, f, ˆ ˆ ˆ ˆ dd Fakot-Chellaa Algothm oluto: f ˆ, ad b Foue: f ˆ, Cˆ ω φ,, ω ω Ω Foue Coeffcet C ˆ ω h eao eue tegablt codto befoe, becaue Foue ba fucto ae tegable We eed to fd the coeffcet that mmze the dtace eo of codto. he ae how that thee otmal coeffcet ae: jωc ω jω C ω Cˆ w ω ω C ω FF hee ae jut the Foue C whee: C ω FF afom of the ogal etmated loe ad!!!!!

14 Deth Fom omal Fakot-Chellaa Algothm Deth Fom omal Fakot-Chellaa Algothm fft ffthft ffthft