Estimating Surface Normals in Noisy Point Cloud Data


 Ami Cain
 1 years ago
 Views:
Transcription
1 Estiatig Suface Noals i Noisy Poit Cloud Data Niloy J. Mita Stafod Gaphics Laboatoy Stafod Uivesity CA, A Nguye Stafod Gaphics Laboatoy Stafod Uivesity CA, ABSTRACT I this pape we descibe ad aalyze a ethod based o local least squae fittig fo estiatig the oals at all saple poits of a poit cloud data (PCD) set, i the pesece of oise. We study the effects of eighbohood size, cuvatue, saplig desity, ad oise o the oal estiatio whe the PCD is sapled fo a sooth cuve i 2 o a sooth suface i 3 ad oise is added. The aalysis allows us to fid the optial eighbohood size usig othe local ifoatio fo the PCD. Expeietal esults ae also povided. Categoies ad Subject Desciptos I.3.5 [ Coputig Methodologies ]: Copute Gaphics Coputatioal Geoety ad Object Modelig [Cuve, suface, solid, ad object epesetatios] Keywods oal estiatio, oisy data, eige aalysis, eighbohood size estiatio. INTRODUCTION Mode age sesig techology eables us to ae detailed scas of coplex objects geeatig poit cloud data (PCD) cosistig of illios of poits. The data acquied is usually distoted by oise aisig out of vaious physical easueet pocesses ad liitatios of the acquisitio techology. The taditioal way to use PCD is to ecostuct the udelyig suface odel epeseted by the PCD, fo exaple as a tiagle esh, ad the apply well ow ethods o that udelyig aifold odel. Howeve, whe the size of the PCD is lage, such ethods ay be expesive. To do suface ecostuctio o a PCD, oe would fist eed to filte out the oise fo the PCD, usually by soe soothig filte [2]. Such a pocess ay eove shap featues, Peissio to ae digital o had copies of all o pat of this wo fo pesoal o classoo use is gated without fee povided that copies ae ot ade o distibuted fo pofit o coecial advatage ad that copies bea this otice ad the full citatio o the fist page. To copy othewise, to epublish, to post o seves o to edistibute to lists, equies pio specific peissio ad/o a fee. SoCG 03, Jue 8 0, 2003, Sa Diego, Califoia, USA. Copyight 2003 ACM /03/ $5.00. howeve, which ay be udesiable. A ecostuctio algoith such as those i [2, 4, 8] the coputes a esh that appoxiates the oise fee PCD. Both the soothig ad the suface ecostuctio pocesses ay be coputatioally expesive. Fo cetai applicatios lie edeig o visualizatio, such a coputatio is ofte uecessay ad diect edeig of PCD has bee ivestigated by the gaphics couity [4, 6]. Alexa et al. [] ad Pauly et al. [6] have poposed to use PCD as a ew odelig piitive. Algoiths uig diectly o PCD ofte equie ifoatio about the oal at each of the poits. Fo exaple, oals ae used i edeig PCD, aig visibility coputatio, asweig isideoutside queies, etc. Also soe cuve (o suface) ecostuctio algoiths, as i [6], eed to have the oal estiates as a pat of the iput data. The oal estiatio poble has bee studied by vaious couities such as copute gaphics, iage pocessig, ad atheatics, but ostly i the case of aifold epesetatios of the suface. We would lie to estiate the oal at each poit i a PCD, give to us oly as a ustuctued set of poits sapled fo a sooth cuve i 2 o a sooth suface i 3 ad without ay additioal aifold stuctue. Hoppe et al. [] poposed a algoith whee the oal at each poit is estiated as the oal to the fittig plae obtaied by applyig the total least squae ethod to the eaest eighbos of the poit. This ethod is obust i the pesece of oise due to the iheet low pass filteig. I this algoith, the value of is a paaete ad is chose aually based o visual ispectio of the coputed estiates of the oals, ad diffeet tial values of ay be eeded befoe a good selectio of is foud. Futheoe, the sae value of is used fo oal estiatio at all poits i the PCD. We ote that the accuacy of the oal estiatio usig a total least squae ethod depeds o () the oise i the PCD, (2) the cuvatue of the udelyig aifold, (3) the desity ad the distibutio of the saples, ad (4) the eighbohood size used i the estiatio pocess. I this pape, we ae pecise such depedecies ad study the cotibutio of each of these factos o the oal estiatio pocess. This aalysis allows us to fid the optial eighbohood size to be used i the ethod. The eighbohood size ca be coputed adaptively at each poit based o its local ifoatio, give soe estiates about the oise, the local saplig desity, ad bouds o the local cuvatue. The coputatioal coplexity of estiatig all 322
2 oals of a PCD with poits is oly O( log ).. Related Wo I this sectio, we suaize soe of the pevious wos that ae elated to the coputatio of the oal vectos of a PCD. May cuet suface ecostuctio algoiths [2, 4, 8] ca eithe copute the oal as pat of the ecostuctio, o the oal ca be tivially coputed oce the suface has bee ecostucted. As the algoiths equie that the iput is oise fee, a aw PCD with oise eeds to go though a soothig pocess befoe these algoiths ca be applied. The wo of Hoppe et al. [] fo suface ecostuctio suggests a ethod to copute the oals fo the PCD. The oal estiate at each poit is doe by fittig a least squae plae to its eaest eighbos. The value of is selected expeietally. The sae appoach has also bee adopted by Pauly et al. [6] fo local suface estiatio. Highe ode sufaces have bee used by Welch et al. [5] fo local paaeteizatio. Howeve, as poited out by Aeta et al. [3] such algoiths ca fail eve i cases with abitaily dese set of saples. This poble ca be esolved by assuig uifoly distibuted saples which pevets eos esultig fo biased fits. As oted befoe, all these algoiths wo well eve i pesece of oise because of the iheet filteig effect. The success of these algoiths depeds lagely o selectig a suitable value fo, but usually little guidace is give o the selectio of this cucial paaete..2 Pape Oveview I sectio 2, we study the oal estiatio fo PCD which ae sapligs of cuves i 2, ad the effects of diffeet paaetes o the eo of that estiatio pocess. I sectio 3, we deive siila esults fo PCD which coe fo a suface i 3. I sectio 4, we povide soe siulatios to illustate the esults obtaied i sectios 2 ad 3. We also show how to use ou theoetical esult o pactical data. We coclude i sectio NORMAL ESTIMATION IN 2 I this sectio, we coside the poble of appoxiatig the oals to a poit cloud i 2. Give a set of poits, which ae oisy saples of a sooth cuve i 2, we ca use the followig ethod to estiate the oal to the cuve at each of the saple poits. Fo each poit O, we fid all the poits of the PCD iside a cicle of adius ceteed at O, the copute the total least squae lie fittig those poits. The oal to the fittig lie gives us a appoxiatio to the udiected oal of the cuve at O. Note that the oietatio of the oals is a global popety of the PCD ad thus caot be coputed locally. Oce all the udiected oals ae coputed, a stadad beadth fist seach algoith [] ca be applied to obtai all the oal diectios i a cosistet way. Though out this pape, we oly coside the coputatio of the udiected oals. We aalyze the eo of the appoxiatio whe the oise is sall ad the saplig desity is high eough aoud O. Ude these assuptios, which we will ae pecise late, the coputed oal appoxiates well the tue oal. We obseve that if is lage, the eighbohood of the poit caot be well appoxiated by a lie i the pesece of cuvatue i the data ad we ay icu lage eo. O the othe had, if is sall, the oise i the data ca esult i sigificat estiatio eo. We ai fo the optial that sties a balace betwee the eos caused by the oise ad the local cuvatue. 2. Modelig Without lost of geeality, we assue that O is the oigi, ad the yaxis is alog the oal to the cuve at O. We assue that the poits of the PCD i a dis of adius aoud O coe fo a seget of the cuve (a D topological dis). Ude this assuptio, the seget of the cuve ea O is locally a gaph of a sooth fuctio y = g(x) defied ove soe iteval R cotaiig the iteval [, ]. We assue that the cuve has a bouded cuvatue i R, ad thus thee is a costat κ > 0 such that g (x) < κ x R. Let p i = (, y i) fo i be the poits of the PCD that lie iside a cicle of adius ceteed at O. We assue the followig pobabilistic odel fo the poits p i. Assue that s ae istaces of a ado vaiable X taig values withi [, ], ad y i = g() + i, whee the oise tes i ae idepedet istaces of a ado vaiable N. X ad N ae assued to be idepedet. We assue that the oise N has zeo ea ad stadad deviatio σ, ad taes values i [, ]. Usig Taylo seies, thee ae ubes ψ i, i such that g() = g (ψ i)x 2 i /2 with ψ i. Let γ i = g (ψ i), the γ i κ. Note that if κ is lage, eve whe thee is o oise i the PCD, the oal to the best fit lie ay ot be a good appoxiatio to the taget as show i Figue. Siilaly, if σ / is lage ad the oise is biased, this oal ay ot be a good appoxiatio eve if the oigial cuve is a staight lie, see Figue 2. I ode to eep the oal appoxiatio eo low we assue a pioi that κ ad σ / ae sufficietly sall. κ 2 Figue : Cuvatue causes eo i the estiated oal Figue 2: Noise causes eo i the estiated oal We assue that the data is evely distibuted; thee is a adius 0 > 0 (possibly depedet o O) so that ay eighbohood of size 0 i R cotais at least 2 poits of the s but o oe tha soe sall costat ube of the. We obseve that the ube of poits iside ay dis of adius is bouded fo above by Θ()ρ, ad also is bouded fo below by aothe Θ()ρ, whee ρ is the saplig desity of the poit cloud. Hee we use Θ() to deote soe sall positive costat, ad fo otatioal siplicity, diffeet appeaaces of Θ() ay deote diffeet costats. We ote that distibutios satisfyig the (ɛ, δ) saplig coditio poposed by Dey et. al. [7] ae evely distibuted. 323
3 Ude the above assuptios, we would lie to boud the oal estiatio eo ad study the effects of diffeet paaetes. The aalysis ivolves pobabilistic aguets to accout fo the ado atue of the oise. 2.2 Total Least Squae Lie I this sectio, we biefly descibe the wellow total least squae ethod. Give a set of poits p i, i, we would lie to fid the lie a T x = c, with a T a = such that the su of squae distaces fo the poits p i s to i the lie is iiized. Let f(a, c) = 2 at 2 pipt a We would lie to fid a ad c iiizig f(a, c) ude the costait that a T a =. To solve this quadatic optiizatio poble, we eed to solve the followig syste of equatios: i f(a, c) = λa p ip T a c p = λa, a (at p i c) 2 = c p T a + 2 c2 whee p = pi. c f(a, c) = 0 pt a + c = 0, T whee λ is a Lagagia ultiplie. It follows that c = p T a, pipt i p p a = λa, ad f(a, c) = λ. Thus 2 λ is a eigevalue of M = pipt i p p T with a as the coespodig eigevecto. It is clea that to iiize f(a, c), λ has to be the iiu eigevalue of M. The coespodig eigevecto a is the oal to the total least squae lie ad is ou oal estiate. Note that this appoach ca be geealized to highe diesioal space. The oal to the total least squae fittig plae (o hypeplae) of a set of poits p i, i i d fo d 2 ca be obtaied by coputig the eigevecto coespodig to the sallest eigevalue of M = pipt i p p T. We obseve that M ca be witte as M = (pi p)(pi p)t ad thus it is always syetic positive seidefiite, ad has oegative eigevalues ad oegative diagoal. 2.3 Eigeaalysis of M We ca wite the 2 2 syetic atix M, as defied i the pevious sectio, as 2 Note that i absece of oise ad cuvatue, 2 = 22 = 0 which eas is the sallest eigevalue of M with [0 ] T as the coespodig eigevecto. Ude ou assuptio that the oise ad the cuvatue ae sall, y i s ae sall, ad thus 2 ad 22 ae sall. Let α = ( )/. We would lie to estiate the sallest eigevalue of M ad its coespodig eigevecto whe α is sall. Usig the Geshgoi Cicle Theoe [9], thee is a eigevalue λ such that λ 2, ad a eigevalue λ 2 such that 22 λ 2 2. Whe α /2, we have that λ λ 2. It follows that the two eigevalues ae distict, ad λ 2 is the sallest eigevalue of M. Let [v ] T be the eigevecto coespodig to λ 2, the Thus λ 2 2 v = v = λ 2 v, 2 22 λ 2. v = ( λ2)2 + 2(22 λ2), ( λ 2) () v 2 ( λ2 + 2 ), ( λ 2) 2 α( + α) ( α) 2. Thus, the estiatio eo, which is the agle betwee the estiated oal ad the tue oal (which is [0 ] T i this case), is less tha ta (α(+α)/( α) 2 ) α, fo sall α. Note that we could wite the eo explicitly i closed fo, the boud it. Ou appoach is oe coplicated, though as we will show late, it ca be exteded to obtai the eo boud fo the 3D case. To boud the estiatio eo, we eed to estiate α. 2.4 Estiatig Eties of M The assuptio that the saple poits ae evely distibuted i the iteval [, ] iplies that, give ay ube that iteval, the ube of poits p i s satisfyig x /4 is at least Θ(). It follows easily that = (xi x)2 Θ() 2. The costat Θ() depeds oly o the distibutio of the ado vaiable X. Fo the eties 2 ad 22, we use ad y i κ 2 /2 + to obtai the followig tivial boud: Thus, 2 = 22 y i 2 2(κ 2 /2 + ), y 2 i 2((κ 2 /2) ). α Θ() κ κ Θ() κ +. (2) This boud illustates the effects of, κ ad o the eo. Fo lage values of, the eo caused by the cuvatue κ doiates, while fo a sall eighbohood the te / is doiatig. Nevetheless, the expessio depeds o the absolute boud of the oise N. This boud ca be uecessaily lage o ubouded fo ay distibutio odels of N. We would lie to use ou assuptio o the distibutio of the oise N to ipove ou boud o α futhe. y i 324
4 ɛρ M Note that 2 = + 2 y i 2 i i) y (γ ix 3 i /2 + i) (γ ix 2 i i /2 + i Θ()κ 3 + +Θ() κ 2 +. Futheoe, ude the assuptio that X ad N ae idepedet, we have E[ i] = E[]E[ i] = 0 sice E[ i] = 0 ad Va( i) = Θ() 2 σ 2 sice Va( i) = σ. 2 Let ɛ > 0 be soe sall costat. Usig the Chebyshev Iequality [3], we ca show that the followig boud o 2 holds with pobability at least ɛ: 2 Θ()κ 3 + Θ() 2 σ 2 ɛ + Θ() σ 2 ɛ = Θ()κ 3 + Θ() 2 σ 2 ɛρ + Θ() σ 2 ɛρ Θ()κ 3 + Θ()σ ɛρ. (3) Fo easoable oise odels, we also have that 22 2(γ 2 i x 4 i /4 + 2 i ) Θ()κ Θ()σ Eo Boud fo the Estiated Noal Fo the estiatios of the eties of M, we obtai the followig boud o α, with pobability at least ɛ: α Θ()κ + Θ() σ + Θ() σ2 3. (4) 2 Note that this boud depeds o the stadad deviatio σ of the oise N athe tha its agitude boud. Fo a give set of paaetes κ, σ, ρ, ad ɛ, we ca fid the optial that iiizes the ight had side of iequality 4. As this optial value of is ot easily expessed i closed fo, let us coside a few extee cases. Whe thee is o cuvatue (κ = 0) we ca ae the boud o α abitaily sall by iceasig. Fo sufficietly lage, the boud is liea i σ ad it deceases as 3/2. Whe thee is o oise, we ca ae the eo boud sall by choosig as sall as possible, say = 0. Whe both oise ad cuvatue ae peset, the eo boud caot be abitaily educed. Whe the desity ρ of the PCD is sufficietly high, α Θ()κ + Θ()σ/ 2 2. The eo boud is iiized whe = Θ()σ 2/3 κ /3, i which case α Θ()κ 2/3 σ 2/3. The sufficietly high desity coditio o ρ ca be show to be ρ > Θ()ɛ σ 4/3 κ /3. / Whe thee ae both oise ad cuvatue, ad the desity ρ is sufficietly low, α Θ()κ +Θ()σ ɛρ 3. The boud is sallest whe = Θ()(σ/(ɛρκ 2 2 )) /5, i which case, α Θ()(κ 3 σ/(ɛρ)) 2 /5. The sufficietly low coditio o ρ ca be expessed oe specifically as ρ < Θ()ɛ σ 4/3 κ /3. We would lie to poit out that the costat hidde i the Θ() otatio i the sufficietly low coditio is 3/4 of that i the sufficietly high coditio. 3. NORMAL ESTIMATION IN 3 We ca exted the esults obtaied fo cuves i 2 to sufaces i 3. Give a poit cloud obtaied fo a sooth 2aifold i 3 ad a poit O o the suface, we ca estiate the oal to the suface at O as follows: fid all the poits of the PCD iside a sphee of adius ceteed at O, the copute the total least squae plae fittig those poits. The oal vecto to the fittig plae is ou estiate of the udiected oal at O. Give a set of poits p i, i, let M = p p T, whee p = pi. As poited out i subsectio 2.2, the oal to the total least squae plae fo this set of poits is the eigevecto coespodig to the iiu eigevalue of the M. We would lie to boud the agle betwee this eigevecto ad the tue oal to the suface. 3. Modelig pipt i We odel the PCD i a siila fashio as i the 2 case. We assue that O is the oigi, the zaxis is the oal to the suface at O, ad that the poits of the PCD i the sphee of adius aoud O ae saples of a topological dis o the suface. Ude these assuptios, we ca epeset the suface as the gaph of a fuctio z = g(x) whee x = [x, y] T. Usig Taylo Theoe, we ca wite g(x) = 2 xt Hx whee H is the Hessia of f at soe poit ψ such that ψ x. We assue that the suface has bouded cuvatue i soe eighbohood aoud O so that thee is a κ > 0 such that the Hessia H of g satisfies H 2 κ i that eighbohood. Wite the poits p i as p i = (, y i, z i) = (, z i). We assue that z i = g( ) + i, whee the i s ae idepedet istaces of soe ado vaiable N with zeo ea ad stadad deviatio σ. We siilaly assue that the poits ae evely distibuted i the xyplae o a dis D of adius ceteed at O, i.e. thee is a adius 0 such that ay dis of size 0 iside D cotais at least 3 poits aog the s but o oe tha soe sall costat ube of the. We also assue that the oise ad the suface cuvatue ae both sall. 3.2 Eigeaalysis i 3 = We wite the aalogous atix M = M 3 M3 T As poited out i subsectio 2.2, M is sy
5 etic ad positive seidefiite. Ude the assuptios that the oise ad the cuvatue ae sall, ad that the poits ae evely distibuted, ad 22 ae the two doiat eties i M. We assue, without lost of geeality, that 22. Let α = ( )/( 2 ). As i the 2 case, we would lie to boud the agle betwee the coputed oal ad the tue oal to the poit cloud i te of α. Deote by λ λ 2 the eigevalues of the 2 2 syetic atix M. Usig agai the Geshgoi Cicle Theoe, it is easy to see that 2 λ, λ Let λ be the sallest eigevalue of M. Fo the Geshgoi Cicle Theoe we have λ = α( 2) αλ. Let [v T ] T be the eigevecto of M coespodig with λ. The, as with Equatio, we have that: v = (M λi) 2 + M 3M T 3 ((M λi)m 3 + M 3( 33 λ)) = (M λi) 2 I + (M λi) 2 M 3M T 3 ((M λi)m 3 + M 3( 33 λ)), v 2 (M λi) 2 2 Note that Thus It follows that v 2 I + (M λi) 2 M 3M3 2 T ( (M λi) 2 M M λ ). (M λi) 2 M 3M T 3 2 (M λi) 2 2 M 3 2 M T 3 2 (λ λ) 2 ( ) ( α) 2 α 2. I + (M λi) 2 M 3M3 2 T ( α)2 ( α) 2 α2 2α. ( α) 2 λ 2 α( + α) 2α ( α) 2 2α λ 2 λ. (λ2αλ + αλαλ) Whe α is sall, the ight had side is appoxiately (λ 2/λ )α, ad thus the agle betwee the coputed oal ad the tue oal, ta v 2, is appoxiately bouded by (λ 2/λ )α (( )/( 2 ))α, 3.3 Estiatio of the eties of M As i the 2 case, fo the assuptio that the saples ae evely distibuted, we ca show that Θ() 2, We ca also show that 33 Θ()κ Θ()σ. 2 Let ρ be the saplig desity of the PCD at O, the = Θ()ρ 2. Agai, let ɛ > 0 be soe sall positive ube. Usig the Chebyshev iequality, we ca show that 3, 23 Θ()κ 3 + Θ()σ / ɛ Θ()κ 3 + Θ()σ / ɛρ with pobability at least ɛ. Fo the te 2, we ote that E[y i] = 0 ad V a(y i) = Θ() 4, ad so, by the Chebyshev iequality, 2 Θ()/ ɛρ with pobability at least ɛ. 3.4 Eo Boud fo the Estiated Noal Let β = 2/. We estict ou aalysis to the cases whe β is sufficietly less tha, say β < /2. This estictio siply eas that the poits s ae ot degeeate, i.e. ot all of the poits s ae lyig o o ea ay give lie o the xyplae. With this estictio, it is clea that (λ 2/λ )α ( 22/ )(( + β)/( β))α = Θ()α. Fo the estiatios of the eties of M, we obtai the followig boud with pobability at least ɛ: λ 2 σ α Θ()κ + Θ() λ 2 ɛρ Θ()κ Θ() σ2 2 σ Θ()κ + Θ() 2 ɛρ + Θ() σ2 2 This is a appoxiate boud o the agle betwee the estiated oal ad the tue oal. To iiize this eo boud, it is = clea that we should pic σ + c 2σ κ c 2, (5) ɛρ /3 fo soe costats c, c 2. The costats c ad c 2 ae sall ad they deped oly o the distibutio of the PCD. We otice that fo the above esult, whe thee is o oise, we should pic the adius to be as sall as possible, say = 0. Whe thee is o cuvatue, the adius should be as lage as possible. Whe the saplig desity is high, the optial value of that iiizes the eo boud is appoxiately = Θ()(σ/κ) 2 /3. This esult is siila to that fo cuves i 2, ad it is ot at all ituitive. 4. EXPERIMENTS I this sectio, we discuss soe siulatios to validate ou theoetical esults. We the show how to use the esults i obtaiig a good eighbohood size fo the oal coputatio with the least squae ethod. 4. Validatio We cosideed a PCD whose poits wee oisy saples of the cuves (x, κ sg(x) x 2 /2), fo x [, ] fo diffeet values of κ. We estiated the oals to the cuves at the oigi by applyig the least squae ethod o thei coespodig PCD. As the yaxis is ow to be the tue oal to the cuves, the agles betwee the coputed oals ad the yaxis gives the estiatio eos. To obtai the PCD i ou expeiets, we let the saplig desity ρ be 00 poits pe uit legth, ad let x be uifoly distibuted i the iteval [, ]. The y copoets of the data wee polluted with uifoly ado oise i the iteval [, ], fo soe value. The stadad deviatio σ of this oise is / 3. Figue 3 shows the eo as a fuctio of the eighbohood size whe = 0.05 fo 3 diffeet values of κ, κ =,, ad.2. As pedicted by Equatio 4 fo lage value of, the eo iceases as iceases. I the expeiets, it ca be see that the eo is popotioal to κ fo > 0.2. Note 326
6 that the PCD we chose geeates the wost case behavio of the eo Eo Agle 0.5 Eo Agle Radius Radius Figue 5: The aveage eo ove 50 us exhibits a clea tedecy to decease as iceases fo sall. Figue 3: The oal estiatio eo iceases as iceases fo > 0.2. Figue 4 shows the estiatio eo as a fuctio of the eighbohood size fo sall whe κ =.2 fo 3 diffeet values of, = 0.07, 0.033, ad We obseve that the eo teds to decease as iceases fo < This is expected as fo Equatio 4, the boud o the eo is a deceasig fuctio of whe is sall. Figue 6: Taget plaes o the oigial buy Eo Agle Radius Figue 4: The oal estiatio eo deceases as deceases ad iceases fo < The depedecy of the eo o fo sall values of ca be see oe easily i Figue 5, which shows the aveage of the estiatio eos ove 50 us fo each. 4.2 Estiatig Neighbohood Size fo the Noal Coputatio I this pat, we used the esults obtaied i Sectio 3 to estiate the oals of a PCD. The data poits i the PCD wee assued to be oisy saples of a sooth suface i 3. This is the case, fo exaple, fo PCD obtaied by age scaes. To obtai the eighbohood size fo the oal coputatio usig the least squae ethod, we would lie to use Equatio 5. We assued that the stadad deviatio σ of the oise was give to us as pat of the iput. We estiated the othe local paaetes i Equatio 5, the coputed. Note that this value of iiizes the boud of the oal coputatio eo, ad thee is o guaatee that this would iiize the eo itself. The costats c ad c 2 deped o the saplig distibutio of the PCD. While we could attept to copute the exact values of c ad c 2, we siply guessed the value c ad c 2. The value of ɛ was fixed at 0.. Give a PCD, we estiated the local saplig desity as follows. Fo a give poit p i the PCD, we used the appoxiate eaest eighbo libay ANN [5] to fid the distace s fo p to its th eaest eighbo fo soe sall ube, = 5 i ou expeiets. The local saplig desity at p was the appoxiated as ρ = /(πs 2 ) saples pe uit aea. To estiate the local cuvatue, we used the ethod poposed by Guhold et al. [0]. Let p j, j be the eaest saple poits aoud p, ad let µ be the aveage distace fo p to all the poits p j. We coputed the best fit least squae plae fo those poits, ad let d be the distace fo p to that best fit plae. The local cuvatue at p ca the be estiated as κ = 2d/µ 2. Oce all the paaetes wee obtaied, we coputed the eighbohood size usig Equatio 5. Note that the estiated value of could be used to obtai a good value fo, which ca to be used to eestiate the local desity ad the local cuvatue. This suggests a iteative schee i which we epeatedly estiate the local desity, the local cuvatue, ad the eighbohood size. I ou expeiets, we foud that 3 iteatios wee eough to obtai good values fo all the quatities. We still have pobles with obtaiig good estiates fo the costats c ad c 2. Fotuately, we oly have to estiate the costats oce fo a give PCD, ad we ca use the sae costats fo ay PCD with a siila poit distibutio. I ou expeiets, we used the sae value fo both c ad c 2. This value was chose so that the coputed oals o a sall egio of the PCD wee visually satisfactoy. Figue 6 shows the coputed taget plaes fo the oig 327
7 We also wat to tha the ueous efeees of the pevious vesios of this pape fo thei exteely useful suggestios. Figue 7: Noal estiatio eos fo the buy PCD with oise added. The subfigues show the poits of the PCD usig the pi colo wheeve the eos ae above 0, 8, ad 5 espectively. ial Stafod buy. The plaes ae daw as sall fixed size squae patches. We oted that ou coputed oals ae siila to those obtaied usig the cocoe ethod by Aeta et al. [4]. Noisy PCD used i ou expeiets wee obtaied by addig oise to the oigial buy. The x, y, ad z copoets of the oise wee chose idepedetly ad uifoly ado i the age [ , ]. The aplitude of this oise is copaable to the aveage distace betwee the saple poits ad thei eaest saples. We coputed the oals of the oisy PCD, ad used the agles betwee those oals ad the oals of the oigial PCD as estiates of the oal coputatio eos. I Figue 7, we colo coded the estiatio eos usig a covetio i which the colo of the squae patch at a poit of the PCD showed the eo at that poit. The colo of a patch is blue whe thee is o eo, ad it gets dae as the eo iceases. Whe the eo is lage tha a cetai theshold, the patch becoes pi. Figue 7 shows the taget plaes whee the thesholds ae 0, 8, ad 5 espectively. We a the least squae oal estiatio algoith o the buy with diffeet aouts of oise added to it ad obseved that the algoith woed well. We also oted that the oal estiatio ethod based o cocoe pefoed pooly i the pesece of oise. 5. CONCLUSIONS We have aalyzed the ethod of least squae i estiatig the oals to a poit cloud data deived eithe fo a sooth cuve i 2 o a sooth suface i 3, with oise added. I both cases, we povided theoetical boud o the axiu agle betwee the estiated oal ad the tue oal of the udelyig aifold. This theoetical study allowed us to fid a optial eighbohood size to be used i the least squae ethod. 6. ACKNOWLEDGMENTS We would lie to tha Leoidas Guibas fo his suggestios, coets ad ecouageet. We ae gateful to Taal K. Dey fo useful discussios ad also fo povidig the softwae fo evaluatig oals usig cocoe. We also tha Mac Levoy, Ro Fediw fo helpful discussios. We acowledge the geeous suppot of the Stafod Gaduate Fellowship poga ad of NSF CARGO gat The sall holes i the buy ae obsevable due to the fact that the patches do ot cove the buy etiely. 7. REFERENCES [] M. Alexa, J. Beh, D. CoheO, S. Fleisha, D. Levi, ad C. T. Silva. Poit set sufaces. IEEE Visualizatio 200, pages 2 28, Octobe 200. ISBN x. [2] N. Aeta, M. Be, ad M. Kavysselis. A ew Voooibased suface ecostuctio algoith. Copute Gaphics, 32(Aual Cofeece Seies):45 42, 998. [3] N. Aeta ad M. W. Be. Suface ecostuctio by voooi filteig. I Syposiu o Coputatioal Geoety, pages 39 48, 998. [4] N. Aeta, S. Choi, T. K. Dey, ad N. Leeha. A siple algoith fo hoeoophic suface ecostuctio. Iteatioal Joual of Coputatioal Geoety ad Applicatios, 2(2):25 4, [5] S. Aya, D. M. Mout, N. S. Netayahu, R. Silvea, ad A. Y. Wu. A optial algoith fo appoxiate eaest eighbo seachig fixed diesios. Joual of the ACM, 45(6):89 923, 998. [6] J.D. Boissoat ad F. Cazals. Sooth suface ecostuctio via atual eighbou itepolatio of distace fuctios. I Syposiu o Coputatioal Geoety, pages , [7] T. K. Dey, J. Giese, S. Goswai, ad W. Zhao. Shape diesio ad appoxiatio fo saples. I Poc. 3 th ACMSIAM Sypos, Discete Algoiths, pages , [8] S. Fue ad E. Raos. Soothsuface ecostuctio i ealiea tie, [9] G. Golub ad C. V. Loa. Matix Coputatios. The Joh Hopis Uivesity Pess, Baltioe, 996. [0] S. Guhold, X. Wag, ad R. MacLeod. Featue extactio fo poit clouds. I 0 th Iteatioal Meshig Roudtable, Sadia Natioal Laboatoies, pages , Octobe 200. [] H. Hoppe, T. DeRose, T. Duchap, J. McDoald, ad W. Stuetzle. Suface ecostuctio fo uogaized poits. Copute Gaphics, 26(2):7 78, 992. [2] I. Lee. Cuve ecostuctio fo uogaized poits. Copute Aided Geoetic Desig, 7:6 77, [3] A. LeoGacia. Pobability ad Rado Pocesses fo Electical Egieeig. Addiso Wesley, 994. [4] S. Rusiiewicz ad M. Levoy. QSplat: A ultiesolutio poit edeig syste fo lage eshes. I K. Aeley, edito, Siggaph 2000, Copute Gaphics Poceedigs, pages ACM Pess / ACM SIGGRAPH / Addiso Wesley Loga, [5] W. Welch ad A. Witi. Feefo shape desig usig tiagulated sufaces. Copute Gaphics, 28(Aual Cofeece Seies): , 994. [6] M. Zwice, M. Pauly, O. Koll, ad M. Goss. Poitshop 3d: A iteactive syste fo poitbased suface editig. I Poc. ACM SIGGRAPH 02, Copute Gaphics Poceedigs, Aual Cofeece Seies,
STATISTICS: MODULE 12122. Chapter 3  Bivariate or joint probability distributions
STATISTICS: MODULE Chapte  Bivaiate o joit pobabilit distibutios I this chapte we coside the distibutio of two adom vaiables whee both adom vaiables ae discete (cosideed fist) ad pobabl moe impotatl whee
More informationPeriodic Review Probabilistic MultiItem Inventory System with Zero Lead Time under Constraints and Varying Order Cost
Ameica Joual of Applied Scieces (8: 37, 005 ISS 546939 005 Sciece Publicatios Peiodic Review Pobabilistic MultiItem Ivetoy System with Zeo Lead Time ude Costaits ad Vayig Ode Cost Hala A. Fegay Lectue
More informationChisquared goodnessoffit test.
Sectio 1 Chisquaed goodessoffit test. Example. Let us stat with a Matlab example. Let us geeate a vecto X of 1 i.i.d. uifom adom vaiables o [, 1] : X=ad(1,1). Paametes (1, 1) hee mea that we geeate
More informationTwo degree of freedom systems. Equations of motion for forced vibration Free vibration analysis of an undamped system
wo degee of feedom systems Equatios of motio fo foced vibatio Fee vibatio aalysis of a udamped system Itoductio Systems that equie two idepedet d coodiates to descibe thei motio ae called two degee of
More informationUnderstanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions
Udestadig Fiacial Maagemet: A Pactical Guide Guidelie Aswes to the Cocept Check Questios Chapte 4 The Time Value of Moey Cocept Check 4.. What is the meaig of the tems isketu tadeoff ad time value of
More informationLearning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV)
Leaig Objectives Chapte 2 Picig of Bods time value of moey Calculate the pice of a bod estimate the expected cash flows detemie the yield to discout Bod pice chages evesely with the yield 21 22 Leaig
More informationOn the Optimality and Interconnection of Valiant LoadBalancing Networks
O the Optimality ad Itecoectio of Valiat LoadBalacig Netwoks Moshe Babaioff ad Joh Chuag School of Ifomatio Uivesity of Califoia at Bekeley Bekeley, Califoia 94720 4600 {moshe,chuag}@sims.bekeley.edu
More informationMoney Math for Teens. Introduction to Earning Interest: 11th and 12th Grades Version
Moey Math fo Tees Itoductio to Eaig Iteest: 11th ad 12th Gades Vesio This Moey Math fo Tees lesso is pat of a seies ceated by Geeatio Moey, a multimedia fiacial liteacy iitiative of the FINRA Ivesto Educatio
More informationFinance Practice Problems
Iteest Fiace Pactice Poblems Iteest is the cost of boowig moey. A iteest ate is the cost stated as a pecet of the amout boowed pe peiod of time, usually oe yea. The pevailig maket ate is composed of: 1.
More informationAnnuities and loan. repayments. Syllabus reference Financial mathematics 5 Annuities and loan. repayments
8 8A Futue value of a auity 8B Peset value of a auity 8C Futue ad peset value tables 8D Loa epaymets Auities ad loa epaymets Syllabus efeece Fiacial mathematics 5 Auities ad loa epaymets Supeauatio (othewise
More informationCHAPTER 4: NET PRESENT VALUE
EMBA 807 Corporate Fiace Dr. Rodey Boehe CHAPTER 4: NET PRESENT VALUE (Assiged probles are, 2, 7, 8,, 6, 23, 25, 28, 29, 3, 33, 36, 4, 42, 46, 50, ad 52) The title of this chapter ay be Net Preset Value,
More informationDerivation of Annuity and Perpetuity Formulae. A. Present Value of an Annuity (Deferred Payment or Ordinary Annuity)
Aity Deivatios 4/4/ Deivatio of Aity ad Pepetity Fomlae A. Peset Vale of a Aity (Defeed Paymet o Odiay Aity 3 4 We have i the show i the lecte otes ad i ompodi ad Discoti that the peset vale of a set of
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationECONOMICS. Calculating loan interest no. 3.758
F A M & A N H S E E S EONOMS alculatig loa iterest o. 3.758 y Nora L. Dalsted ad Paul H. Gutierrez Quick Facts... The aual percetage rate provides a coo basis to copare iterest charges associated with
More informationTHE PRINCIPLE OF THE ACTIVE JMC SCATTERER. Seppo Uosukainen
THE PRINCIPLE OF THE ACTIVE JC SCATTERER Seppo Uoukaie VTT Buildig ad Tapot Ai Hadlig Techology ad Acoutic P. O. Bo 1803, FIN 02044 VTT, Filad Seppo.Uoukaie@vtt.fi ABSTRACT The piciple of fomulatig the
More informationThe Binomial Multi Section Transformer
4/15/21 The Bioial Multisectio Matchig Trasforer.doc 1/17 The Bioial Multi Sectio Trasforer Recall that a ultisectio atchig etwork ca be described usig the theory of sall reflectios as: where: Γ ( ω
More informationBreakeven Holding Periods for Tax Advantaged Savings Accounts with Early Withdrawal Penalties
Beakeve Holdig Peiods fo Tax Advataged Savigs Accouts with Ealy Withdawal Pealties Stephe M. Hoa Depatmet of Fiace St. Boavetue Uivesity St. Boavetue, New Yok 4778 Phoe: 76375209 Fax: 7637529 email:
More informationLogistic Regression, AdaBoost and Bregman Distances
A exteded abstact of this joual submissio appeaed ipoceedigs of the Thiteeth Aual Cofeece o ComputatioalLeaig Theoy, 2000 Logistic Regessio, Adaoost ad egma istaces Michael Collis AT&T Labs Reseach Shao
More informationA Comparison of Hypothesis Testing Methods for the Mean of a LogNormal Distribution
World Applied Scieces Joural (6): 845849 ISS 88495 IDOSI Publicatios A Copariso of Hypothesis Testig ethods for the ea of a ogoral Distributio 3 F. egahdari K. Abdollahezhad ad A.A. Jafari Islaic Azad
More information580.439 Course Notes: Nonlinear Dynamics and HodgkinHuxley Equations
58.439 Couse Notes: Noliea Dyamics ad HodgkiHuxley Equatios Readig: Hille (3 d ed.), chapts 2,3; Koch ad Segev (2 d ed.), chapt 7 (by Rizel ad Emetout). Fo uthe eadig, S.H. Stogatz, Noliea Dyamics ad
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationbetween Modern Degree Model Logistics Industry in Gansu Province 2. Measurement Model 1. Introduction 2.1 Synergetic Degree
www.ijcsi.og 385 Calculatio adaalysis alysis of the Syegetic Degee Model betwee Mode Logistics ad Taspotatio Idusty i Gasu Povice Ya Ya 1, Yogsheg Qia, Yogzhog Yag 3,Juwei Zeg 4 ad Mi Wag 5 1 School of
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationMaximum Entropy, Parallel Computation and Lotteries
Maximum Etopy, Paallel Computatio ad Lotteies S.J. Cox Depatmet of Electoics ad Compute Sciece, Uivesity of Southampto, UK. G.J. Daiell Depatmet of Physics ad Astoomy, Uivesity of Southampto, UK. D.A.
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationVolume 1: Distribution and Recovery of Petroleum Hydrocarbon Liquids in Porous Media
LNAPL Distibutio ad Recovey Model (LDRM) Volume 1: Distibutio ad Recovey of Petoleum Hydocabo Liquids i Poous Media Regulatoy ad Scietific Affais Depatmet API PUBLICATION 4760 JANUARY 007 3 Satuatio, Relative
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationLongTerm Trend Analysis of Online Trading A Stochastic Order Switching Model
Asia Pacific Maagemet Review (24) 9(5), 893924 LogTem Ted Aalysis of Olie Tadig A Stochastic Ode Switchig Model Shalig Li * ad Zili Ouyag ** Abstact Olie bokeages ae eplacig bokes ad telephoes with
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More information9.5 Amortization. Objectives
9.5 Aotization Objectives 1. Calculate the payent to pay off an aotized loan. 2. Constuct an aotization schedule. 3. Find the pesent value of an annuity. 4. Calculate the unpaid balance on a loan. Congatulations!
More informationStrategic Remanufacturing Decision in a Supply Chain with an External Local Remanufacturer
Assoiatio fo Ifomatio Systems AIS Eletoi Libay (AISeL) WHICEB 013 Poeedigs Wuha Iteatioal Cofeee o ebusiess 55013 Stategi Remaufatuig Deisio i a Supply Chai with a Exteal Loal Remaufatue Xu Tiatia Shool
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More informationPortfolio Performance Attribution
Potfolio Pefoance Attibution Alia Biglova, Svetloa Rachev 2 Abstact In this pape, we povide futhe insight into the pefoance attibution by developent of statistical odels based on iniiing ETL pefoance isk
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationGraphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.
Gaphs of Equations CHAT PeCalculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such
More informationThe dinner table problem: the rectangular case
The ie table poblem: the ectagula case axiv:math/009v [mathco] Jul 00 Itouctio Robeto Tauaso Dipatimeto i Matematica Uivesità i Roma To Vegata 00 Roma, Italy tauaso@matuiomait Decembe, 0 Assume that people
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More informationA Faster ClauseShortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 4960 A Faster ClauseShorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
More informationAnt Colony Algorithm Based Scheduling for Handling Software Project Delay
At Coloy Algorith Based Schedulig for Hadlig Software Project Delay Wei Zhag 1,2, Yu Yag 3, Juchao Xiao 4, Xiao Liu 5, Muhaad Ali Babar 6 1 School of Coputer Sciece ad Techology, Ahui Uiversity, Hefei,
More informationThroughput and Delay Analysis of Hybrid Wireless Networks with MultiHop Uplinks
This paper was preseted as part of the ai techical progra at IEEE INFOCOM 0 Throughput ad Delay Aalysis of Hybrid Wireless Networks with MultiHop Upliks Devu Maikata Shila, Yu Cheg ad Tricha Ajali Dept.
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More information9.4 Annuities. Objectives. 1. Calculate the future value of an ordinary annuity. 2. Perform calculations regarding sinking funds.
9.4 Annuities Objectives 1. Calculate the futue value of an odinay annuity. 2. Pefo calculations egading sinking funds. Soewhee ove the ainbow... skies ae blue,... and the deas that you dae to dea...eally
More informationEquation of a line. Line in coordinate geometry. Slopeintercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Pointslope form ( 點 斜 式 )
Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationSupplementary Material for EpiDiff
Supplementay Mateial fo EpiDiff Supplementay Text S1. Pocessing of aw chomatin modification data In ode to obtain the chomatin modification levels in each of the egions submitted by the use QDCMR module
More informationTracking/Fusion and Deghosting with Doppler Frequency from Two Passive Acoustic Sensors
Tacking/Fusion and Deghosting with Dopple Fequency fom Two Passive Acoustic Sensos Rong Yang, Gee Wah Ng DSO National Laboatoies 2 Science Pak Dive Singapoe 11823 Emails: yong@dso.og.sg, ngeewah@dso.og.sg
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More information9.8: THE POWER OF A TEST
9.8: The Power of a Test CD91 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationController Area Network (CAN) Schedulability Analysis with FIFO queues
Cotroller Area Network (CAN) Schedulability Aalysis with FIFO queues Robert I. Davis RealTie Systes Research Group, Departet of Coputer Sciece, Uiversity of York, YO10 5DD, York, UK rob.davis@cs.york.ac.uk
More informationConvex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells
Caad. J. Math. Vol. 60 (1), 2008 pp. 3 32 Covex Bodies of Miimal Volume, Surface Area ad Mea Width with Respect to Thi Shells Károly Böröczky, Károly J. Böröczky, Carste Schütt, ad Gergely Witsche Abstract.
More informationANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS... 1 ANNUITIES SOFTWARE ASSIGNMENT... 2 WHAT IS AN ANNUITY?... 2 EXAMPLE 1... 2 QUESTIONS...
ANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS ANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS... 1 ANNUITIES SOFTWARE ASSIGNMENT... WHAT IS AN ANNUITY?... EXAMPLE 1... QUESTIONS... EXAMPLE BRANDON S
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationHadoop Performance Modeling for Job Estimation and Resource Provisioning
his aticle has been accepted fo publication in a futue issue of this jounal, but has not been fully edited. Content ay change pio to final publication. Citation infoation: DOI 0.09/PDS.05.40555, IEEE ansactions
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More information4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then
SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or pseries (the Compariso Test), but of
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationReview for 1 sample CI Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review for 1 sample CI Name MULTIPLE CHOICE. Choose the oe alterative that best completes the statemet or aswers the questio. Fid the margi of error for the give cofidece iterval. 1) A survey foud that
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationOn Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.
C.Candan EE3/53METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof
More informationFM4 CREDIT AND BORROWING
FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationOptimal Pricing Decision and Assessing Factors in. ClosedLoop Supply Chain
Applied Matheatical Sciences, Vol. 5, 2011, no. 80, 40154031 Optial Picing Decision and Assessing Factos in ClosedLoop Supply Chain Yang Tan Picing Science and Engineeing Depatent, FedEx Expess Wold
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationPerformance Analysis of an Inverse Notch Filter and Its Application to F 0 Estimation
Cicuits and Systems, 013, 4, 1171 http://dx.doi.og/10.436/cs.013.41017 Published Online Januay 013 (http://www.scip.og/jounal/cs) Pefomance Analysis of an Invese Notch Filte and Its Application to F 0
More informationLesson 7 Gauss s Law and Electric Fields
Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationA ConstantFactor Approximation Algorithm for the Link Building Problem
A CostatFactor Approximatio Algorithm for the Lik Buildig Problem Marti Olse 1, Aastasios Viglas 2, ad Ilia Zvedeiouk 2 1 Ceter for Iovatio ad Busiess Developmet, Istitute of Busiess ad Techology, Aarhus
More informationPower and Sample Size Calculations for the 2Sample ZStatistic
Powe and Sample Size Calculations fo the Sample ZStatistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.
More informationOn Efficiently Updating Singular Value Decomposition Based Reduced Order Models
On Efficiently dating Singula alue Decoosition Based Reduced Ode Models Ralf Zieann GAMM oksho Alied and Nueical Linea Algeba with Secial Ehasis on Model Reduction Been Se..3. he PODbased ROM aoach.
More informationDistributed Storage Allocations for Optimal Delay
Distributed Storage Allocatios for Optial Delay Derek Leog Departet of Electrical Egieerig Califoria Istitute of echology Pasadea, Califoria 925, USA derekleog@caltechedu Alexadros G Diakis Departet of
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationPearson Physics Level 30 Unit VI Forces and Fields: Chapter 10 Solutions
Peason Physics Level 30 Unit VI Foces and Fields: hapte 10 Solutions Student Book page 518 oncept heck 1. It is easie fo ebonite to eove electons fo fu than fo silk.. Ebonite acquies a negative chage when
More informationOptimizing Result Prefetching in Web Search Engines. with Segmented Indices. Extended Abstract. Department of Computer Science.
Optiizig Result Prefetchig i Web Search Egies with Segeted Idices Exteded Abstract Roy Lepel Shloo Mora Departet of Coputer Sciece The Techio, Haifa 32000, Israel eail: frlepel,orag@cs.techio.ac.il Abstract
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationCDAS: A Crowdsourcing Data Analytics System
CDAS: A Crowdsourcig Data Aalytics Syste Xua Liu,MeiyuLu, Beg Chi Ooi, Yaya She,SaiWu, Meihui Zhag School of Coputig, Natioal Uiversity of Sigapore, Sigapore College of Coputer Sciece, Zhejiag Uiversity,
More informationarxiv:0903.5136v2 [math.pr] 13 Oct 2009
First passage percolatio o rado graphs with fiite ea degrees Shakar Bhaidi Reco va der Hofstad Gerard Hooghiestra October 3, 2009 arxiv:0903.536v2 [ath.pr 3 Oct 2009 Abstract We study first passage percolatio
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More information