Examples. Epipoles. Epipolar geometry and the fundamental matrix


 Roy Wiggins
 1 years ago
 Views:
Transcription
1 Epipoar gomtry and th fundamnta matrix Epipoar ins Lt b a point in P 3. Lt x and x b its mapping in two imags through th camra cntrs C and C. Th point, th camra cntrs C and C and th (3D points corrspon to) th mappd points x and x wi i in th sam pan π. This pan is cad th pipoar pan for C, C and. Givn a point x in imag 1, th pipoar pan π is dfind by th ray through x and C and th basin through C and C. A corrsponding point x thus has to i on th intrscting in btwn th pipoar pan π and imag pan 2. Th in is th projction of th ray through x and in imag 2 and is cad th pipoar in to x. pipoar pan π x x x C C pipoar in for x p. 1 Epipos Examps Th intrsction points btwn th bas in and th imag pans ar cad pipos. Th pipo in imag 2 is th mapping of th camra cntr C. Th pipo in imag 1 is th mapping of th camra cntr C. Sinc a pipoar pans intrsct both camra cntrs, a pipoar ins wi intrsct th pipos. π basin basin p. 3
2 Th fundamnta matrixf Th fundamnta matrixf Th fundamnta matrix F is th agbraic rprsntation of th pipoar gomtry. It dscribs th mapping x btwn a point x in on imag and its pipoar in in anothr imag. Lt P and P b th camra matrics for imag 1 and 2. Th ray in P 3 that is projctd onto th point x in imag 1 is (λ) = P + x + λc, whr P + is th psudoinvrs to P, i.. PP + = I, and PC = 0. Th in (λ) intrscts th points P + x and C. Ths points ar mappd into th othr camra P at P P + x and P C. Th pipoar in intrscts ths projctd points, i.. = (P C) (P P + x). Th point P C is th pipo, i.. th projction of th camra cntr in th othr camra. Th pipoar in can thus b writtn as = (P P + x) or whr = [ ] (P P + )x = Fx, F = [ ] (P P + ). p. 5 Examp Corrspondnc Assum th camra matrics corrspond to a caibratd stro rig with th word origin in camra cntr 1. P = K[I 0], P = K [R t]. Thn P + = " K 1 0, C = " 0 1 and F = [P C] P P + = [K t] K RK 1 = K [t] RK 1 = K R[R t] K 1 = K RK [KR t] Not that th pipos ar W can thus writ = P " R t 1 = KR t, = P " 0 1 = K t. Givn two camras with diffrnt camra cntrs, th fundamnta matrix F is a 3 3 homognous matrix with rank 2. For ach corrsponding point pair x x it satisfis x Fx = 0, sinc if x and x ar corrsponding points, thn x is on th pipoar in = Fx corrsponding to x, i.. x = 0 = x Fx. Simiary, = F x is th pipoar in in imag 1 corrsponding to th point x in imag 2. F = [ ] K RK 1 =... = K RK [], F = = K R K [ ]. x pipoar in for x p. 7
3 Th pipos Th numbr of dgrs of frdom Th pipoar in = Fx to ach point x (xcpt ) intrscts th pipo. Thus satisfis (Fx) = ( F)x = 0 for a x. This impis that F = 0 or F = 0. Th pipo is thus a nu vctor to F (in th ft nuspac of F). Simiary, F = 0, i.. is a nuvctor to F (in th right nuspac of F). Th fundamnta matrix F has 7 dgrs of frdom: A 3 3 homognous matrix has 8 dgrs of frdom. Th constraint rank(f) = 2 or dt(f) = 0 rducs th numbr to 7. p. 9 Projktiv invarianc Th corrspondnc ration x Fx = 0 is invariant undr a homography in P 2. If ˆx = Hx and ˆx = H x thn x Fx = ˆx H FH 1ˆx = ˆx ˆFˆx, whr ˆF = H FH 1 is th fundamnta matrix corrsponding to ˆx ˆx. Th fundamnta matrix F is invariant undr a homography in P 3. Lt H b a 4 4 matrix corrsponding to a projctiv mapping of P 3. Thn th camra pairs (P,P ) and (PH,P H) hav th sam fundamnta matrix. Th points x = P = (PH)(H 1 ) and x = P = (P H)(H 1 ) ar corrsponding mappings of in th camras P and P and corrsponding mappings of H 1 in th camras PH and P H. Thus a homography H in P 3 dos affct th word points and camras P,P, but not F. This mans that th fundamnta matrix F dtrmins th camra matrics P,P up to a right mutipication by a 3D projctiv transformation. Canonica form Givn this ambiguity a canonica form for th camra pairs is dfind corrsponding to a fundamnta matrix whr th first camra P = [I 0] has cntr at th origin and word coordinat axs. If th scond camra is P = [M m] thn th fundamnta matrix F corrsponding to th canonica camras is F = [m] M. For finit camras P = K[I 0],P = K [R t] w hav F = [K t] K RK 1. p. 11
4 symmtry and th fundamnta matrix Canonica camra pairs givnf A nonzro matrix F is th fundamnta matrix corrsponding to th camra pair P,P iff P FP is skw symmtric. Th condition that P FP is skw symmtrica is quivant to that P FP = 0 for a. With x = P and x = P this bcoms x Fx = 0, which is th dfining quation for th fundamnta matrix. Lt F b a fundamnta matrix and S an arbitrary skw symmtric matrix. Dfin th camra matrics P = [I 0] and P = [SF ], whr is th ft pipo of F, F = 0 and assum that P is a vaid camra matrix (has rank 3). Thn F is th fundamnta matrix corrsponding to (P,P ). Chck by vrifying that P FP = [SF ] F[I 0] = [ F S F 0 F 0 ] = [ F S F ] is skw symmtric. p. 13 Canonica camra pairs givnf In ordr for th matrix P to hav rank 3 s has to b nonzro, whr s is a nuvctor of S = [s]. A working choic is s = ading to th camra pairs P = [I 0] and P = [[ ] F ]. Th most gnra formuation for a canonica camra pair is P = [I 0], P = [[ ] F + v λ ], whr v is an arbitrary 3vctor and λ is a scaar. Normaizd coordinats Study a camra matrix P = K[R t] and t x = P b an arbitrary point in th imag. If th camra caibration matrix K is known w may appy its invrs on th point x and gt ˆx = K 1 x. Thn ˆx = [R t] is th projction of xprssd in normaizd coordinats. Th camra matrix K 1 P = I[R t] is cad a normaizd camra matrix and has camra caibration matrix K = I. p. 15
5 Th ssntia matrixe Th numbr of dgrs of frdom for Study a normaizd camra pair P = [I 0], P = [R t]. Th fundamnta matrix corrsponding to normaizd camra pairs is cad th ssntia matrix and is on th form E = [t] R = R[R t]. Th dfining quation for th ssntia matrix is Th ssntia matrix E = [t] R has 5 dgrs of frdon; 3 rotation angs in R, 3 mnts in t, but arbitrary sca. Th fwr dgrs of frdom corrspond to on additiona constraint; a 3 3 matrix is an ssntia matrix if two of its singuar vaus ar qua and th ast is zro. ˆx Eˆx = 0, xprssd in normaizd imag coordinats for th corrsponding points x x. Substitution with ˆx and ˆx givs x K EK 1 x = 0 ading to F = K EK 1 or E = K FK. p. 17 h numbr of dgrs of frdom fore Study th factorization E = [t] R = SR, whr S is skw symmtric. W wi us th matrics W = and Z = that ar orthogona and skw symmtric, rspctivy. Not that Z = diag(1,1, 0)W Cacuation of th camra matrics from Lt th first camra matrix b P = [I 0]. In ordr to cacuat th othr camra matrix P it is ncssary to factoriz E into a product SR by a skw symmtric and a rotation matrix. Givn S = [t] and R, P is givn by P = [R t]. Lt E has th singuar vau dcomposition E = U diag(1, 1, 0)V. Ignoring sign, thr ar two possib factorizations E = SR: S = UZU, R = UWV or R = UW V A skw symmtric matrix S can b writtn as S = kuzu, whr U is orthogona. Thus up to sca and S = U diag(1,1, 0)WU E = SR = U diag(1,1,0)(wu R), which is a singuar vau dcomposition of E with two singuar vaus qua and th third qua to zro. p. 19 Th factorization givs th t part of th camra matrix P up to sca from S = [t]. If w choos t = 1 w gt a unit basin. Furthrmor St = 0 St = UZU t = U ( UZ ) t = U [ u 2 u 1 0 ] t = 0 givs that t = u 3, whr u i is th i:th coumn of U. Th sign of E and hnc t can howvr not b dtrmind which ads to 4 diffrnt possibiitis for th scond camra P.
6 Th 4 camra factorizations ofe Givn a singuar vau dcomposition of E = U diag(1, 1, 0)V and a canonica camra 1 P = [I 0], thr ar 4 atrnativ camra pairs: P = [UWV + u 3 ], P = [UWV u 3 ], P = [UW V + u 3 ], P = [UW V u 3 ]. Ths 4 options hav gomtric intrprtations; basin rvrsa and rotation by th scond camra 180 around th basin. A (a) B B (b) A A B B A (c) (d) p. 21
Epipolar Geometry and the Fundamental Matrix
9 Epipolar Gomtry and th Fundamntal Matrix Th pipolar gomtry is th intrinsic projctiv gomtry btwn two viws. It is indpndnt of scn structur, and only dpnds on th camras intrnal paramtrs and rlativ pos.
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) 92.222  Linar Algbra II  Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial
More informationModule 7: Discrete State Space Models Lecture Note 3
Modul 7: Discrt Stat Spac Modls Lctur Not 3 1 Charactristic Equation, ignvalus and ign vctors For a discrt stat spac modl, th charactristic quation is dfind as zi A 0 Th roots of th charactristic quation
More informationNonHomogeneous Systems, Euler s Method, and Exponential Matrix
NonHomognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous firstordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach
More informationQuestion 3: How do you find the relative extrema of a function?
ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating
More informationExponential Growth and Decay; Modeling Data
Exponntial Growth and Dcay; Modling Data In this sction, w will study som of th applications of xponntial and logarithmic functions. Logarithms wr invntd by John Napir. Originally, thy wr usd to liminat
More informationNew Basis Functions. Section 8. Complex Fourier Series
Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ralvalud Fourir sris is xplaind and formula ar givn for convrting
More informationProjections  3D Viewing. Overview Lecture 4. Projection  3D viewing. Projections. Projections Parallel Perspective
Ovrviw Lctur 4 Projctions  3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl  Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog
More informationSAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII
SAMPLE QUESTION PAPER MATHEMATICS (4) CLASS XII 67 Tim allowd : 3 hours Maimum Marks : Gnral Instructions: (i) All qustions ar compulsor. (ii) This qustion papr contains 9 qustions. (iii) Qustion  4
More informationCPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions
CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:
More information14.3 Area Between Curves
14. Ara Btwn Curvs Qustion 1: How is th ara btwn two functions calculatd? Qustion : What ar consumrs and producrs surplus? Earlir in this chaptr, w usd dfinit intgrals to find th ara undr a function and
More informationLecture 3: Diffusion: Fick s first law
Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th
More informatione = C / electron Q = Ne
Physics 0 Modul 01 Homwork 1. A glass rod that has bn chargd to +15.0 nc touchs a mtal sphr. Aftrword, th rod's charg is +8.00 nc. What kind of chargd particl was transfrrd btwn th rod and th sphr, and
More informationQUANTITATIVE METHODS CLASSES WEEK SEVEN
QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.
More informationThe example is taken from Sect. 1.2 of Vol. 1 of the CPN book.
Rsourc Allocation Abstract This is a small toy xampl which is wllsuitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of Cnts. Hnc, it can b rad by popl
More informationSUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* RostovonDon. Russia
SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* RostovonDon. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs
More informationII. Equipment. Magnetic compass, magnetic dip compass, Helmholtz coils, HP 6212 A power supply, Keithley model 169 multimeter
Magntic fild of th arth I. Objctiv: Masur th magntic fild of th arth II. Equipmnt. Magntic compass, magntic dip compass, Hlmholtz s, HP 6212 A powr supply, Kithly modl 169 multimtr III Introduction. IIIa.
More informationby John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia
Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs
More informationModelling and Solving TwoStep Equations: ax + b = c
Modlling and Solving ToStp Equations: a + b c Focus on Aftr this lsson, you ill b abl to φ modl problms φ ith tostp linar quations solv tostp linar quations and sho ho you ord out th ansr Cali borrod
More informationAdverse Selection and Moral Hazard in a Model With 2 States of the World
Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,
More informationMathematics. Mathematics 3. hsn.uk.net. Higher HSN23000
hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails
More informationSingleton Theorem Using Models
Singlton Thorm Using Modls Srivathsan B, Igor Walukiwicz LaBRI Paris, March 2010 Srivathsan B, Igor Walukiwicz (LaBRI) Singlton Thorm Using Modls Paris, March 2010 1 / 17 Introduction Singlton Thorm [Statman
More informationSharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means
Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s166001507411 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans WiMao Qian
More information10/06/08 1. Aside: The following is an online analytical system that portrays the thermodynamic properties of water vapor and many other gases.
10/06/08 1 5. Th watrair htrognous systm Asid: Th following is an onlin analytical systm that portrays th thrmodynamic proprtis of watr vapor and many othr gass. http://wbbook.nist.gov/chmistry/fluid/
More information5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power
Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim
More informationA Note on Approximating. the Normal Distribution Function
Applid Mathmatical Scincs, Vol, 00, no 9, 4549 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and
More informationEFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS
25 Vol. 3 () JanuaryMarch, pp.375/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut
More informationME 612 Metal Forming and Theory of Plasticity. 6. Strain
Mtal Forming and Thory of Plasticity mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.
More informationForeign Exchange Markets and Exchange Rates
Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls
More informationBasis risk. When speaking about forward or futures contracts, basis risk is the market
Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also
More informationLecture 20: Emitter Follower and Differential Amplifiers
Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.
More informationPrinciples of Humidity Dalton s law
Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid
More information7 Timetable test 1 The Combing Chart
7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing
More informationConstraintBased Analysis of Gene Deletion in a Metabolic Network
ConstraintBasd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFGRsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, 14195 Brlin, Grmany
More informationHANDOUT E.19  EXAMPLES ON FEEDBACK CONTROL SYSTEMS
MEEN 64 Parauram Lctur 9, Augut 5, HANDOUT E9  EXAMPLES ON FEEDBAC CONTOL SSTEMS Exampl Conidr th ytm hown blow Th opn loop tranfr function i givn by Th clod loop tranfr function i Exampl Conidr th ytm
More informationSection 55 Inverse of a Square Matrix
 Invrs of a Squar Matrix 9 (D) Rank th playrs from strongst to wakst. Explain th rasoning hind your ranking. 68. Dominan Rlation. Eah mmr of a hss tam plays on math with vry othr playr. Th rsults ar givn
More informationA Derivation of Bill James Pythagorean WonLoss Formula
A Drivation of Bill Jams Pythagoran WonLoss Formula Ths nots wr compild by John Paul Cook from a papr by Dr. Stphn J. Millr, an Assistant Profssor of Mathmatics at Williams Collg, for a talk givn to th
More informationEcon 371: Answer Key for Problem Set 1 (Chapter 1213)
con 37: Answr Ky for Problm St (Chaptr 23) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc
More informationFundamentals of Tensor Analysis
MCEN 503/ASEN 50 Chptr Fundmntls of Tnsor Anlysis Fll, 006 Fundmntls of Tnsor Anlysis Concpts of Sclr, Vctor, nd Tnsor Sclr α Vctor A physicl quntity tht cn compltly dscrid y rl numr. Exmpl: Tmprtur; Mss;
More informationWARING S PROBLEM RESTRICTED BY A SYSTEM OF SUM OF DIGITS CONGRUENCES
WARING S PROBLEM RESTRICTED BY A SYSTEM OF SUM OF DIGITS CONGRUENCES OLIVER PFEIFFER AND JÖRG M. THUSWALDNER Abstract. Th aim of th prsnt papr is to gnraiz arir work by Thuswadnr and Tichy on Waring s
More informationGenetic Drift and Gene Flow Illustration
Gntic Drift and Gn Flow Illustration This is a mor dtaild dscription of Activity Ida 4, Chaptr 3, If Not Rac, How do W Explain Biological Diffrncs? in: How Ral is Rac? A Sourcbook on Rac, Cultur, and Biology.
More informationSPECIAL VOWEL SOUNDS
SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)
More informationStatistical Machine Translation
Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm
More informationAP Calculus MultipleChoice Question Collection 1969 1998. connect to college success www.collegeboard.com
AP Calculus MultiplChoic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos
More informationFinite Element Vibration Analysis
Finit Elmnt Vibration Analysis Introduction In prvious topics w larnd how to modl th dynamic bhavior of multidof systms, as wll as systms possssing infinit numbrs of DOF. As th radr may raliz, our discussion
More informationDifferential Equations (MTH401) Lecture That a nonhomogeneous linear differential equation of order n is an equation of the form n
Diffrntial Equations (MTH40) Ltur 7 Mthod of Undtrmind CoffiintsSurosition Aroah Rall. That a nonhomognous linar diffrntial quation of ordr n is an quation of th form n n d d d an + a a a0 g( ) n n +
More informationCPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.
Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by
More informationChapter 10 Function of a Matrix
EE448/58 Vrsion. John Stnsby Chatr Function of a atrix t f(z) b a comlxvalud function of a comlx variabl z. t A b an n n comlxvalud matrix. In this chatr, w giv a dfinition for th n n matrix f(a). Also,
More informationCIRCUITS AND ELECTRONICS. Basic Circuit Analysis Method (KVL and KCL method)
6. CIRCUITS AND ELECTRONICS Basic Circuit Analysis Mthod (KVL and KCL mthod) Cit as: Anant Agarwal and Jffry Lang, cours matrials for 6. Circuits and Elctronics, Spring 7. MIT 6. Fall Lctur Rviw Lumpd
More informationCHAPTER 4c. ROOTS OF EQUATIONS
CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03  Computation Mthod in Civil Enginring II Dpartmnt o Civil
More informationTraffic Flow Analysis (2)
Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. GangLn Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,
More informationMaking and Using the Hertzsprung  Russell Diagram
Making and Using th Hrtzsprung  Russll Diagram Nam In astronomy th HrtzsprungRussll Diagram is on of th main ways that w organiz data dscribing how stars volv, ags of star clustrs, masss of stars tc.
More information5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:
.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This
More informationQuantum Graphs I. Some Basic Structures
Quantum Graphs I. Som Basic Structurs Ptr Kuchmnt Dpartmnt of Mathmatics Txas A& M Univrsity Collg Station, TX, USA 1 Introduction W us th nam quantum graph for a graph considrd as a ondimnsional singular
More informationSection 7.4: Exponential Growth and Decay
1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 117 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart
More informationGas Radiation. MEL 725 PowerPlant Steam Generators (300) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi
Gas Radiation ME 725 PowrPlant Stam Gnrators (300) Dr. Prabal Talukdar Assistant Profssor Dpartmnt of Mchanical Enginring T Dlhi Radiation in absorbingmitting mdia Whn a mdium is transparnt to radiation,
More informationParallel and Distributed Programming. Performance Metrics
Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:
More informationDeer: Predation or Starvation
: Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,
More informationA Theoretical Model of Public Response to the Homeland Security Advisory System
A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Amy (Wnxuan) Ding Dpartmnt of Information and Dcision Scincs Univrsity of Illinois Chicago, IL 60607 wxding@uicdu Using a diffrntial
More informationArchitecture of the proposed standard
Architctur of th proposd standard Introduction Th goal of th nw standardisation projct is th dvlopmnt of a standard dscribing building srvics (.g.hvac) product catalogus basd on th xprincs mad with th
More information811ISD Economic Considerations of Heat Transfer on Sheet Metal Duct
Air Handling Systms Enginring & chnical Bulltin 811ISD Economic Considrations of Hat ransfr on Sht Mtal Duct Othr bulltins hav dmonstratd th nd to add insulation to cooling/hating ducts in ordr to achiv
More informationSimulated Radioactive Decay Using Dice Nuclei
Purpos: In a radioactiv sourc containing a vry larg numbr of radioactiv nucli, it is not possibl to prdict whn any on of th nucli will dcay. Although th dcay tim for any on particular nuclus cannot b prdictd,
More informationVibrational Spectroscopy
Vibrational Spctroscopy armonic scillator Potntial Enrgy Slction Ruls V( ) = k = R R whr R quilibrium bond lngth Th dipol momnt of a molcul can b pandd as a function of = R R. µ ( ) =µ ( ) + + + + 6 3
More informationGround Fault Current Distribution on Overhead Transmission Lines
FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 19, April 2006, 7184 Ground Fault Currnt Distribution on Ovrhad Transmission Lins Maria Vintan and Adrian Buta Abstract: Whn a ground fault occurs on
More informationSolutions to Homework 8 chem 344 Sp 2014
1. Solutions to Homwork 8 chm 44 Sp 14 .. 4. All diffrnt orbitals mans thy could all b paralll spins 5. Sinc lctrons ar in diffrnt orbitals any combination is possibl paird or unpaird spins 6. Equivalnt
More information3. Yes. You can put 20 of the 6V lights in series, or you can put several of the 6V lights in series with a large resistance.
CHAPTE 6: DC Circuits sponss to Qustions. Evn though th bird s ft ar at high potntial with rspct to th ground, thr is vry littl potntial diffrnc btwn thm, bcaus thy ar clos togthr on th wir. Th rsistanc
More informationIntroduction to Finite Element Modeling
Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation
More information(Analytic Formula for the European Normal Black Scholes Formula)
(Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually
More informationAbstract. Introduction. Statistical Approach for Analyzing Cell Phone Handoff Behavior. Volume 3, Issue 1, 2009
Volum 3, Issu 1, 29 Statistical Approach for Analyzing Cll Phon Handoff Bhavior Shalini Saxna, Florida Atlantic Univrsity, Boca Raton, FL, shalinisaxna1@gmail.com Sad A. Rajput, Farquhar Collg of Arts
More informationSimulation of the electric field generated by a brown ghost knife fish
C H A P T R 2 7 Simulation of th lctric fild gnratd by a brown ghost knif fish lctric fild CONCPTS 27.1 Th fild modl 27.2 lctric fild diagrams 27.3 Suprposition of lctric filds 27.4 lctric filds and forcs
More informationAnalyzing the Economic Efficiency of ebaylike Online Reputation Reporting Mechanisms
A rsarch and ducation initiativ at th MIT Sloan School of Managmnt Analyzing th Economic Efficincy of Baylik Onlin Rputation Rporting Mchanisms Papr Chrysanthos Dllarocas July For mor information, plas
More informationElectronic Commerce. and. Competitive FirstDegree Price Discrimination
Elctronic Commrc and Comptitiv FirstDgr Pric Discrimination David Ulph* and Nir Vulkan ** Fbruary 000 * ESRC Cntr for Economic arning and Social Evolution (ESE), Dpartmnt of Economics, Univrsity Collg
More informationPlanar Graphs. More precisely: there is a 11 function f : V R 2 and for each e E there exists a 11 continuous g e : [0, 1] R 2 such that
Planar Graphs A graph G = (V, E) is planar i it can b drawn on th plan without dgs crossing xcpt at ndpoints a planar mbdding or plan graph. Mor prcisly: thr is a  unction : V R 2 and or ach E thr xists
More informationUpper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing
Uppr Bounding th Pric of Anarchy in Atomic Splittabl Slfish Routing Kamyar Khodamoradi 1, Mhrdad Mahdavi, and Mohammad Ghodsi 3 1 Sharif Univrsity of Tchnology, Thran, Iran, khodamoradi@c.sharif.du Sharif
More informationAccurate Doppler Prediction Scheme for Satellite Orbits
Accurat Dopplr Prdiction Schm for Satllit Orbits NASER AYAT, MOHAMAD MEHDIPOUR Computr nginring group Payam noor univrsity Lashgarak st., Nakhl st., Thran IRAN Abstract:  In satllit communications particular
More informationhttp://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force
ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd
More informationThe Normal Distribution: A derivation from basic principles
Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn
More informationNoise Power Ratio (NPR) A 65Year Old Telephone System Specification Finds New Life in Modern Wireless Applications.
TUTORIL ois Powr Ratio (PR) 65Yar Old Tlphon Systm Spcification Finds w Lif in Modrn Wirlss pplications ITRODUTIO by Walt Kstr Th concpt of ois Powr Ratio (PR) has bn around sinc th arly days of frquncy
More informationBudget Optimization in SearchBased Advertising Auctions
Budgt Optimization in SarchBasd Advrtising Auctions ABSTRACT Jon Fldman Googl, Inc. Nw York, NY jonfld@googl.com Martin Pál Googl, Inc. Nw York, NY mpal@googl.com Intrnt sarch companis sll advrtismnt
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scic March 15, 005 Srii Dvadas ad Eric Lhma Problm St 6 Solutios Du: Moday, March 8 at 9 PM Problm 1. Sammy th Shar is a fiacial srvic providr who offrs loas o th followig
More informationDept. of Materials Science and Engineering. Problem Set 8 Solutions
MSE 30/ECE 30 Elctrical Prortis Of Matrials Dt. of Matrials Scinc and Enginring Fall 0/Bill Knowlton Problm St 8 Solutions. Using th rlationshi of n i n i i that is a function of E g, rcrat th lot shown
More informationWORKLOAD STANDARD DEPARTMENT OF CIVIL ENGINEERING. for the. Workload Committee : P.N. Gaskin (Chair) J.W. Kamphuis K. Van Dalen. September 24, 1997
WORKLOAD STANDARD for th DEPARTMENT OF CIVIL ENGINEERING Sptmbr 24, 1997 Workload Committ : P.N. Gaskin (Chair) J.W. Kamphuis K. Van Daln 9 2 TABLE OF CONTENTS l. INTRODUCTION... 3 2. DEFINITION OF WORKLOAD
More informationEntityRelationship Model
EntityRlationship Modl Kuanghua Chn Dpartmnt of Library and Information Scinc National Taiwan Univrsity A Company Databas Kps track of a company s mploys, dpartmnts and projcts Aftr th rquirmnts collction
More informationSPECIFIC HEAT AND HEAT OF FUSION
PURPOSE This laboratory consists of to sparat xprimnts. Th purpos of th first xprimnt is to masur th spcific hat of to solids, coppr and rock, ith a tchniqu knon as th mthod of mixturs. Th purpos of th
More informationLAB 3: VELOCITY AND ACCELERATION GRAPHS
Goas: LAB 3: ELOCITY AND ACCELERATION GRAPHS Invstigat accration vs. tim graphs Prdict accration graphs from vocity graphs Invstigat accration as sop of vocity vs. tim graph Part 1  Making ocity Graphs
More informationSigmoid Functions and Their Usage in Artificial Neural Networks
Sigmoid Functions and Thir Usag in Artificial Nural Ntworks Taskin Kocak School of Elctrical Enginring and Computr Scinc Applications of Calculus II: Invrs Functions Eampl problm Calculus Topic: Invrs
More informationVan der Waals Forces Between Atoms
Van dr Waals Forcs twn tos Michal Fowlr /8/7 Introduction Th prfct gas quation of stat PV = NkT is anifstly incapabl of dscribing actual gass at low tpraturs, sinc thy undrgo a discontinuous chang of volu
More informationFactorials! Stirling s formula
Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 51 Orthonormal
More informationCamera calibration and epipolar geometry. Odilon Redon, Cyclops, 1914
Camera calibration and epipolar geometry Odilon Redon, Cyclops, 94 Review: Alignment What is the geometric relationship between pictures taken by cameras that share the same center? How many points do
More informationAP Calculus AB 2008 Scoring Guidelines
AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos mission is to connct studnts to collg succss and opportunity.
More informationIn the previous two chapters, we clarified what it means for a problem to be decidable or undecidable.
Chaptr 7 Computational Complxity 7.1 Th Class P In th prvious two chaptrs, w clarifid what it mans for a problm to b dcidabl or undcidabl. In principl, if a problm is dcidabl, thn thr is an algorithm (i..,
More informationInference by Variable Elimination
Chaptr 5 Infrnc by Variabl Elimination Our purpos in this chaptr is to prsnt on of th simplst mthods for gnral infrnc in Baysian ntworks, known as th mthod of Variabl Elimination. 5.1 Introduction Considr
More information7. Dry Lab III: Molecular Symmetry
0 7. Dry Lab III: Moecuar Symmetry Topics: 1. Motivation. Symmetry Eements and Operations. Symmetry Groups 4. Physica Impications of Symmetry 1. Motivation Finite symmetries are usefu in the study of moecues.
More informationSPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM
RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Ths paprs
More informationLABORATORY 1 IDENTIFICATION OF CIRCUIT IN A BLACKBOX
LABOATOY IDENTIFICATION OF CICUIT IN A BLACKBOX OBJECTIES. To idntify th configuration of an lctrical circuit nclosd in a twotrminal black box.. To dtrmin th valus of ach componnt in th black box circuit.
More informationAxial flow rate (per unit circumferential length) [m 2 /s] R B, R J =R Bearing Radius ~ Journal Radius [m] S
NOTE 4 TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING Lctur 4 introducs th fundamnts of journal baring analysis. Th long and short lngth baring modls ar introducd. Th prssur fild in a short lngth baring
More informationMAXIMAL CHAINS IN THE TURING DEGREES
MAXIMAL CHAINS IN THE TURING DEGREES C. T. CHONG AND LIANG YU Abstract. W study th problm of xistnc of maximal chains in th Turing dgrs. W show that:. ZF + DC+ Thr xists no maximal chain in th Turing dgrs
More informationOn the moments of the aggregate discounted claims with dependence introduced by a FGM copula
On th momnts of th aggrgat discountd claims with dpndnc introducd by a FGM copula  Mathiu BARGES Univrsité Lyon, Laboratoir SAF, Univrsité Laval  Hélèn COSSETTE Ecol Actuariat, Univrsité Laval, Québc,
More informationGraph Theory. 1 Graphs and Subgraphs
1 Graphs and Subgraphs Graph Thory Dfinition 1.1. A multigraph or just graph is an ordrd pair G = (V, E) consisting of a nonmpty vrtx st V of vrtics and an dg st E of dgs such that ach dg E is assignd
More information