CSE 272 Assignment 1

Transcription

1 CSE 7 Assignmnt 1 Kui-Chun Hsu Task 1: Comput th irradianc at A analytically (point light) For point light, first th nrgy rachd A was calculatd, thn th nrgy was rducd by a factor according to th angl btwn A s surfac normal and th light incidnt dirction. Φ Φ E= cosθ = cosθ da 4πr 50 (5,5,0) (0,1,0) 50 5 = = 4π (5,5,0) (0,0,0) (5,5,0) (0,1,0) 4π = Task : Comput th irradianc at A using progrssiv photon mapping (point light) Th point light tst scn was valuatd with th progrssiv photon mapping, with initial configuration radius R = 0. 7, alphaα = 0. 5, and photons mittd pr i itration K = In ach itration, th irradianc, radius ( R i ), and local photon count ( N ) wr valuatd. Aftr 100 million photons ar mittd, th thr valus i convrgd to , , and rspctivly. To gt an ovrall undrstanding of th convrgnc procss during progrssiv photon mapping, th thr valus wr also plottd as a function of mittd photons as follows: Irradianc at A Irradianc (W/(m*m)) Numbr of photons ( photons)

2 Maximum irradianc stimatd: 1647 at itration 5 Minimum irradianc stimatd: at itration 18 Radius Ri Numbr of photons ( photons) Local Photon Count Ni Numbr of photons ( photons)

3 Hackr points 1: Comput th irradianc at A analytically (squar light) 1. First th radianc mittanc (M) of th point light is calculatd: M = L θdω = cos Ω 0 M Φ L = = π πa π π 0 L cosθ sinθdθdϕ = πl. Th irradianc is thn computd by plugging in L : E = = Ω A ( ω' n') da L ( ω n) = L r L cosθdω = A cosθ ' da L cosθ r x y ( x y + y + z ) dxdy Evaluat th intgral with Matlab w gt E = Task 3: Comput th irradianc at A using progrssiv photon mapping (squar light) Th squar light tst scn was valuatd with th sam initial configuration as in Task. Aftr 100 million photons ar mittd, th thr valus convrgd to , , and rspctivly. Th valus of irradianc, radius, and local photon count during th progrssiv photon mapping wr also plottd: Irradianc at A Irradianc (W/(m*m)) Numbr of photons ( photons) Maximum irradianc stimatd: at itration 4 Minimum irradianc stimatd: 5999 at itration 106

4 Radius Ri Numbr of photons ( photons) Local Photon Count Ni Numbr of photons ( photons) Task 4: Bst initial configuration for radius (R), mittd photons pr itration (K), and α Th objct for this task is to achiv accurat irradianc stimation within 10 million photons mittd. In othr words, it is to find a configuration to achiv fast and accurat convrgnc. Whn th nw itration has littl contribution to th accumulatd flux, th irradianc is said to b convrgd. Th contribution is from th nw photons within th radius, which is α M. If w assum all photons ar mittd vnly in all dirctions, and th tst scn is simpl nough that no objcts ar occludd by othrs, thn a masur point has α M = K ω α nw photons in ach itration, whr ω is th solid angl of th masurd ara of th masur point. A trivial configuration to achiv fast convrgnc is to mak th thr componnts in M small, but it is asy to convrg to incorrct rsults if thy ar too small. Th tradoffs ar listd as blow:

5 Paramtrs Too larg Too small R ( ω ) 1. Tak too many far- Can t gt nough nw photons (low away photons into possibility of photon hits within th radius) flux valuation. convrg slowly K 1. convrg slowly Can t gt nough nw photons α 1. wast rsourcs. convrg slowly Estimatd ara of th first itration Th scond itration If α is too small, thn R shrinks fast. For simplicity, assum w convrg at th scond itration, as th figur shown abov, thn as w can s th two masurd ara hav a dramatic diffrnc; th blu ara covrs lots of far-away photons. Howvr, th contribution of th photons in both itration hav th sam wight in accumulatd flux, which mans at last half of th contribution coms from th far-away photons, and this lads to inaccurat convrgnc. To sarch for th bst configuration, two assumptions wr mad: 1. Assum th sarch won t fall into local minimum in th givn sarch rang. Onc a local minimum of th rror is rachd, th rsult configuration is th bst on.. Assum th thr paramtrs ar indpndnt with ach othr; thrfor th paramtrs can b sarchd on at a tim, whil all othr paramtrs stay fixd. Basd on th assumptions, a binary sarch was prformd to sarch for th bst configuration. Sinc α M = K ω α indicats th smallr paramtrs givs fastr convrgnc, th basic ida was to start with stting ach paramtr to a sufficintly larg numbr, and thn dcrasing it whn th smallr valu givs th bst accuracy so far, or incrasing it othrwis. Th sarchs for R and α stop whn a local

6 minimum is rachd. Th stop critrion for sarching K is a littl diffrnt sinc th rror is always small with largr K. Th sarch for K stops whn largr K givs no contribution to rduc th rror. K is also th last paramtr to b sarchd; aftr R and α ar dtrmind, th sarch for K is startd. Th rsult configuration and rror wr listd blow: Initial valu Point light Bst configuration Squar light R ( ω ) R = 1.00, α =0.90, K= K R = bst, α = bst, K= α R = bst, α =0.90, K= rror Th abov sarch startd from R, α, and thn K. Th bst mans th initial valus for th sarch ar changd to th bst configuration so far. Howvr, unfortunatly local minimum problm did occur. For th initial sarch valu abov, th bst K for point light should b 50000, which only gav rror = and usd half as much mmory as K=96777 did. Morovr, an intrsting obsrvation was th small K gav rasonably good rsult. For xampl (R = bst, α =bst, K=10000) gav rror quals to for squar light.