Human vision is determined based on information theory:

Transcription

1 Human vision is dtrmind basd on information thory: Supplmntary Information Alfonso Dlgado-Bonal,2 and F. Javir Martn Torrs,3 [] Instituto Andaluz d Cincias d la Tirra CSIC-UGR, Avda. d Las Palmras n 4, Armilla, 800, Granada, Spain [2] Univrsidad d Salamanca, Instituto d Física Fundamntal y Matmáticas, Pza d la Mrcd S/N, 37008, Salamanca, Spain [3] Division of Spac Tchnology, Dpartmnt of Computr Scinc, Elctrical and Spac Enginring, Lulå Univrsity of Tchnology, Kiruna, Swdn * alfonso dlgado@usal.s Supplmntary information Figur 3: Win s displacmnt law of th ntropy of radiation Using th statistical dfinition of ntropy proposd by Boltzmann, Planck dtrmind th analytical xprssion of th ntropy of radiation ntropy of bosons:

2 S λ = 2kc { } + λ5 L λ log + λ5 L λ λ5 L λ λ log λ5 L λ 2 and aftrwards drivd th law that charactrizs th bhavior of th radiation intnsity Planck s law: L λ = 22 λ 5 kλt 2 Onc th distribution of th radiation nrgy is known, is possibl to driv th law that dtrmins th maximum of nrgy for a givn blackbody tmpratur, which is calld Win s displacmnt law. This law stats that th spctral radianc of black body radiation pr unit wavlngth, paks at th wavlngth λ max that is invrsly proportional to tmpratur: λ max = b T 3 whr T is th absolut tmpratur of th body in Klvin and b is a constant of proportionality calld Win s constant qual to m K. Th constant b was dtrmind xprimntally, firstly by Lummr and Pringshim and latr by Paschn, and was thortically obtaind by Planck onc its distribution law was formulatd. It is worth noticing that Win proposd th law that carris his nam bfor Planck obtaind th spctral distribution. 2

3 Win s law dtrmins th wavlngth of th maximum mission of radiation by a blackbody, but it should not b confusd with th maximum information. Following th rasoning proposd by Planck in th Part II of this book [Planck, 94], th Win s displacmnt law should not b rstrictd to th spctral intnsity of radiation. Morovr, th spctral ntropy of radiation will follow a similar law, but th valu of th constant dos not hav to b th sam. Equation 20 can b rwrittn as: S λ = 2kc { + x log + x x log x} 4 λ4 whr x = L λ λ =. It can b convrtd by simpl arithmtic into: / S λ = 22 T λ + 5 T L λ + 2kc λ log 4 5 In th maximum of th function w hav th condition is ds λ dλ = 0, which lads to: ds λ dλ = 02 + λ 6 T T + 8kc λ 5 log λ 6 T + 22 λ 6 T 2h2 c3 λ 7 kt = Looking for common dnominator, λ 7 T 2 2: 3

4 2 ds 02 λt 0 2 λt λ dλ = λ 7 T 2 λ 7 T 2 2 2h 2 c 3 8kc log λ 2 T λ 7 T 2 λ 7 T λt + 2 = 0 λ 7 T 2 7 Th numrator must b zro in th prvious quation in ordr for th ntropy to b a maximum: ds λ dλ = 02 λt 8kc log λt 2 λ 2 T λt = 0 + 2h 2 c 3 8 Using th transformation x = λt = kx : ds λ dλ = 0h2 c 3 x 2 0h2 c 3 x + 2h2 c 3 x kx kx k h 2 c 2 8kc log x k 2 x 2 x x kx x 9 =0 Rmoving common factor 2h2 c 3 k : 4

5 ds λ dλ = 5 x 2 5 x + x x 2 4 log x x x x 2 + x x x = 0 0 and multiplying by x 2 : ds λ dλ = 5x x 2 5x x + x 2 x 4 log x 2 + x x x = 0 x w obtain a transcndntal quation which cannot b solvd analytically but that can b solvd numrically Figur 3. Th solution givs x = and rvrsing th x variabl: x = λt = k = m K 2 which is th Win s displacmnt law of th ntropy of radiation. Figur 2: Function of normalizd ratio of ntropy to nrgy qual to unity Th ntropy contnt in radiation is not uniformly distributd along th spctra. Th diffrnt location of th maxima and th shap of th spctra divids th spac in two ntropic rgions. Th curv that limits ths rgions is charactrizd for having a valu of th ratio of normalizd ntropy to nrgy qual to th unity. Th curv is vrifis th quation: 5

6 S λ S λ,max L λ = 3 L λ,max Whr S λ is givn by Equation 20, L λ by Equation 2, and S λ,max and L λ,max ar th maximum of th ntropy and th nrgy rspctivly. Th rlation is rwrittn as: S λ L λ,max = S λ,max S λ 4 As th wavlngth of th maximum of th ntropy and th nrgy ar dtrmind by th Win s displacmnt laws dscribd in this papr, w can us th rlations λ max,nrgy T = b max,nrgy and λ max,ntropy T = b max,ntropy in Equations 20 and 2 to dtrmin S λ,max and L λ,max. Using th xprssion of th ntropy of Equation 5, w hav: S λ,max = 22 T b 5 ntropy + T 2 2 b 5 ntropy + 2kc kbntropy b 4 ntropy T 4 log kbntropy 5 L λ,max = 22 b 5 nrgy 6 kbnrgy For simplicity, w call c = 2 2, c 2 = /k and c 3 = 2kc. Th rlation dscribd in Equation 4 lads to th quation: 6

7 c T λ 5 + c 3 λ 4 log c T b 5 ntropy c b 5 nrgy c 2/b nrgy + T c 2/λT c λ 5 c 2/λT + T + c 3 T 4 log b 4 ntropy c λ 5 c 2/λT c b nrgy c 2/b nrgy = c 2/b ntropy c b 5 ntropy c b 5 nrgy c 2/b nrgy c 2/b ntropy c λ 5 c 2/λT c λ 5 c 2/λT 7 It is asy to prov that rmoving T 4 λ 5, calling x = c 2 λt and multiplying both sids by x, th quation is rducd to: { x + x bnrgy 5 log = kbnrgy + x x b ntropy kbntropy + kb ntropy log } kbntropy 8 that is a transcndntal quation with numrical solution x = Undoing th chang of variabl, w hav: λt = k = m K 9 Figur : Optimal wavlngth: maximum product As xplaind in th main txt, th product of nrgy and ntropy is maximizd in a point btwn th maxima of both distributions. Th quation that dtrmins th optimal wavlngth is drivd as th maxima of th product of th two distributions. As sn bfor, th distributions of ntropy and nrgy ar rspctivly: 7

8 S λ = 2kc { } + λ5 L λ log + λ5 L λ λ5 L λ λ log λ5 L λ 2 20 L λ = 22 λ 5 kλt 2 and thir drivats: ds λ dλ = 02 + λ 6 T T + 8kc λ 5 log λ 6 T + 22 λ 6 T 2h2 c3 λ 7 kt 2 22 dl λ dλ = λ 6 2h2 c3 λ 7 kt 2 23 Th maximum product is accomplishd whn d S dλ λ L λ = 0, i.., ds λ L dλ λ + dl λ S dλ λ = 0. Using th prvious quations, rmoving th common factors λ 2 T 2 k λt 3, and 3, and doing th chang of variabl x =, th quation is transformd to: 0 x x 2 0 x x + 2 x 9 x 2 x 2 log x x x + x x + x x x log x x =

9 which numrically solvd, givs th valu x = Undoing th chang of variabl, w obtain λt = m K. In thos situations whr th qual wighting rul cannot b applid, th optimal function would hav a diffrnt valu. For xampl, in computr vision whr th nrgy is constant, th optimal wavlngth is dtrmind by d dλ S λ L λ = L λ d dλ S λ = 0, i.. it is rducd to th classical maximum ntropy rul ds λ dλ = 0. 9