Mean Value Coordinates for Closed Triangular Meshes


 Jessica Greene
 3 years ago
 Views:
Transcription
1 Mean Value Coordnates for Closed Trangular Meshes Tao Ju, Scott Schaefer, Joe Warren Rce Unversty (a) (b) (c) (d) Fgure : Orgnal horse model wth enclosng trangle control mesh shown n black (a). Several deformatons generated usng our 3D mean value coordnates appled to a modfed control mesh (b,c,d). Abstract Constructng a functon that nterpolates a set of values defned at vertces of a mesh s a fundamental operaton n computer graphcs. Such an nterpolant has many uses n applcatons such as shadng, parameterzaton and deformaton. For closed polygons, mean value coordnates have been proven to be an excellent method for constructng such an nterpolant. In ths paper, we generalze mean value coordnates from closed D polygons to closed trangular meshes. Gven such a mesh P, we show that these coordnates are contnuous everywhere and smooth on the nteror of P. The coordnates are lnear on the trangles of P and can reproduce lnear functons on the nteror of P. To llustrate ther usefulness, we conclude by consderng several nterestng applcatons ncludng constructng volumetrc textures and surface deformaton. CR Categores: I.3.5 [Computer Graphcs]: Computatonal Geometry and Object Modelng Boundary representatons; Curve, surface, sold, and object representatons; Geometrc algorthms, languages, and systems Keywords: barycentrc coordnates, mean value coordnates, volumetrc textures, surface deformaton Introducton Gven a closed mesh, a common problem n computer graphcs s to extend a functon defned at the vertces of the mesh to ts nteror. For example, Gouraud shadng computes ntenstes at the vertces of a trangle and extends these ntenstes to the nteror usng lnear nterpolaton. Gven a trangle wth vertces {p, p, p 3 } and assocated ntenstes { f, f, f 3 }, the ntensty at pont v on the nteror of the trangle can be expressed n the form ˆf[v] = f j () where w j s the area of the trangle {v, p j, p j+ }. In ths formula, note that each weght w j s normalzed by the sum of the weghts, w to form an assocated coordnate j. The nterpolant ˆf[v] s then smply the sum of the f j tmes ther correspondng coordnate. Mesh parameterzaton methods [Hormann and Grener 000; Desbrun et al. 00; Khodakovsky et al. 003; Schrener et al. 004; Floater and Hormann 005] and freeform deformaton methods [Sederberg and Parry 986; Coqullart 990; MacCracken and Joy 996; Kobayash and Ootsubo 003] also make heavy use of nterpolants of ths type. Both applcatons requre that a pont v be represented as an affne combnaton of the vertces on an enclosng shape. To generate ths combnaton, we smply set the data values f j to be ther assocated vertex postons p j. If the nterpolant reproduces lnear functons,.e.; v = p j, w j the coordnate functons are the desred affne combnaton. For convex polygons n D, a sequence of papers, [Wachspress 975], [Loop and DeRose 989] and [Meyer et al. 00], have proposed and refned an nterpolant that s lnear on ts boundares and only nvolves convex combnatons of data values at the vertces of the polygons. Ths nterpolant has a smple, local defnton as a ratonal functon and reproduces lnear functons. [Warren 996; Warren et al. 004] also generalzed ths nterpolant to convex shapes n hgher dmensons. Unfortunately, Wachspress s nterpolant does not generalze to nonconvex polygons. Applyng
2 (a) (b) such a generalzaton for arbtrary closed surfaces and show that the resultng nterpolants are wellbehaved and have lnear precson. Appled to closed polygons, our constructon reproduces D mean value coordnates. We then apply our method to closed trangular meshes and construct 3D mean value coordnates. (In ndependent contemporaneous work, [Floater et al. 005] have proposed an extenson of mean value coordnates from D polygons to 3D trangular meshes dentcal to secton 3..) Next, we derve an effcent, stable method for evaluatng the resultng mean value nterpolant n terms of the postons and assocated values of vertces of the mesh. Fnally, we consder several practcal applcatons of such coordnates ncludng a smple method for generatng classes of deformatons useful n character anmaton. Mean value nterpolaton (c) Fgure : Interpolatng hue values at polygon vertces usng Wachspress coordnates (a, b) versus mean value coordnates (c, d) on a convex and a concave polygon. the constructon to such a polygon yelds an nterpolant that has poles (dvsons by zero) on the nteror of the polygon. The top porton of Fgure shows Wachspress s nterpolant appled to two closed polygons. Note the poles on the outsde of the convex polygon on the left as well as along the extensons of the two top edges of the nonconvex polygon on the rght. More recently, several papers, [Floater 997; Floater 998; Floater 003], [Malsch and Dasgupta 003] and [Hormann 004], have focused on buldng nterpolants for nonconvex D polygons. In partcular, Floater proposed a new type of nterpolant based on the mean value theorem [Floater 003] that generates smooth coordnates for starshaped polygons. Gven a polygon wth vertces p j and assocated values f j, Floater s nterpolant defnes a set of weght functons w j of the form tan w j = [ α j ] + tan p j v [ ] α j (d). () where α j s the angle formed by the vector p j v and p j+ v. Normalzng each weght functon w j by the sum of all weght functons yelds the mean value coordnates of v wth respect to p j. In hs orgnal paper, Floater prmarly ntended ths nterpolant to be used for mesh parameterzaton and only explored the behavor of the nterpolant on ponts n the kernel of a starshaped polygon. In ths regon, mean value coordnates are always nonnegatve and reproduce lnear functons. Subsequently, Hormann [Hormann 004] showed that, for any smple polygon (or nested set of smple polygons), the nterpolant ˆf[v] generated by mean value coordnates s welldefned everywhere n the plane. By mantanng a consstent orentaton for the polygon and treatng the α j as sgned angles, Hormann also shows that mean value coordnates reproduce lnear functons everywhere. The bottom porton of Fgure shows mean value coordnates appled to two closed polygons. Note that the nterpolant generated by these coordnates possesses no poles anywhere even on nonconvex polygons. Contrbutons Horman s observaton suggests that Floater s mean value constructon could be used to generate a smlar nterpolant for a wder class of shapes. In ths paper, we provde Gven a closed surface P n R 3, let p[x] be a parameterzaton of P. (Here, the parameter x s twodmensonal.) Gven an auxlary functon f[x] defned over P, our problem s to construct a functon ˆf[v] where v R 3 that nterpolates f[x] on P,.e.; ˆf[p[x]] = f[x] for all x. Our basc constructon extends an dea of Floater developed durng the constructon of D mean value coordnates. To construct ˆf[v], we project a pont p[x] of P onto the unt sphere S v centered at v. Next, we weght the pont s assocated value f[x] by p[x] v and ntegrate ths weghted functon over S v. To ensure affne nvarance of the resultng nterpolant, we dvde the result by the ntegral of the weght functon p[x] v taken over S v. Puttng the peces together, the mean value nterpolant has the form x ˆf[v] = w[x,v] f[x]ds v x w[x,v]ds (3) v where the weght functon w[x, v] s exactly p[x] v. Observe that ths formula s essentally an ntegral verson of the dscrete formula of Equaton. Lkewse, the contnuous weght functon w[x, v] and the dscrete weghts w j of Equaton dffer only n ther numerators. As we shall see, the tan [ ] α terms n the numerators of the w j are the result of takng the ntegrals n Equaton 3 wth respect to ds v. The resultng mean value nterpolant satsfes three mportant propertes. Interpolaton: As v converges to the pont p[x] on P, ˆf[v] converges to f[x]. Smoothness: The functon ˆf[v] s welldefned and smooth for all v not on P. Lnear precson: If f[x] = p[x] for all x, the nterpolant ˆf[v] s dentcally v for all v. Interpolaton follows from the fact that the weght functon w[x, v] approaches nfnty as p[x] v. Smoothness follows because the projecton of f[x] onto S v s contnuous n the poston of v and takng the ntegral of ths contnuous process yelds a smooth functon. The proof of lnear precson reles on the fact that the ntegral of the unt normal over a sphere s exactly zero (due to symmetry). Specfcally, p[x] v x p[x] v ds v = 0 snce p[x] v p[x] v s the unt normal to S v at parameter value x. Rewrtng ths equaton yelds the theorem. v = x p[x] / p[x] v ds v x p[x] v ds v
3 Notce that f the projecton of P onto S v s onetoone (.e.; v s n the kernel of P), then the orentaton of ds v s nonnegatve, whch guarantees that the resultng coordnate functons are postve. Therefore, f P s a convex shape, then the coordnate functons are postve for all v nsde P. However, f v s not n the kernel of P, then the orentaton of ds v s negatve and the coordnates functons may be negatve as well. 3 Coordnates for pecewse lnear shapes In practce, the ntegral form of Equaton 3 can be complcated to evaluate symbolcally. However, n ths secton, we derve a smple, closed form soluton for pecewse lnear shapes n terms of the vertex postons and ther assocated functon values. As a smple example to llustrate our approach, we frst rederve mean value coordnates for closed polygons va mean value nterpolaton. Next, we apply the same dervaton to construct mean value coordnates for closed trangular meshes. 3. Mean value coordnates for closed polygons Consder an edge E of a closed polygon P wth vertces {p, p } and assocated values { f, f }. Our frst task s to convert ths dscrete data nto a contnuous form sutable for use n Equaton 3. We can lnearly parameterze the edge E va p[x] = φ [x]p where φ [x] = ( x) and φ [x] = x. We then use ths same parameterzaton to extend the data values f and f lnearly along E. Specfcally, we let f[x] have the form f[x] = φ [x] f. Now, our task s to evaluate the ntegrals n Equaton 3 for 0 x. Let E be the crcular arc formed by projectng the edge E onto the unt crcle S v, we can rewrte the ntegrals of Equaton 3 restrcted to E as xw[x,v] f[x]de x w[x,v]de = w f (4) w where weghts w = φ [x] x p[x] v de. Our next goal s to compute the correspondng weghts w for edge E n Equaton 4 wthout resortng to symbolc ntegraton (snce ths wll be dffcult to generalze to 3D). Observe that the followng dentty relates w to a vector, w (p v) = m. (5) where m = p[x] v x de s smply the ntegral of the outward unt p[x] v normal over the crcular arc E. We call m the mean vector of E, as scalng m by the length of the arc yelds the centrod of the crcular arc E. Based on D trgonometry, m has a smple expresson n terms of p and p. Specfcally, To evaluate the ntegral of Equaton 3, we can relate the dfferental ds v to dx va ds v = p [x].(p[x] v) p[x] v dx where p [x] s the cross product of the n tangent vectors p[x] to P at x p[x]. Note that the sgn of ths expresson correctly captures whether P has folded back durng ts projecton onto S v. m = tan[α/]( (p v) p v + (p v) p v ) where α denotes the angle between p v and p v. Hence we obtan w = tan[α/]/ p v whch agrees wth the Floater s weghtng functon defned n Equaton for D mean value coordnates when restrcted to a sngle edge of a polygon. Equaton 4 allows us to formulate a closed form expresson for the nterpolant ˆf[v] n Equaton 3 by summng the ntegrals for all edges E k n P (note that we add the ndex k for enumeraton of edges): ˆf[v] = k w k f k k w k (6) where w k and f k are weghts and values assocated wth edge E k. 3. Mean value coordnates for closed meshes We now consder our prmary applcaton of mean value nterpolaton for ths paper; the dervaton of mean value coordnates for trangular meshes. These coordnates are the natural generalzaton of D mean value coordnates. Gven trangle T wth vertces {p, p, p 3 } and assocated values { f, f, f 3 }, our frst task s to defne the functons p[x] and f[x] used n Equaton 3 over T. To ths end, we smply use the lnear nterpolaton formula of Equaton. The resultng functon f[x] s a lnear combnaton of the values f tmes bass functons φ [x]. As n D, the ntegral of Equaton 3 reduces to the sum n Equaton 6. In ths case, the weghts w have the form φ [x] w = x p[x] v dt where T s the projecton of trangle T onto S v. To avod computng ths ntegral drectly, we nstead relate the weghts w to the mean vector m for the sphercal trangle T by nvertng Equaton 5. In matrx form, {w,w,w 3 } = m {p v, p v, p 3 v} (7) All that remans s to derve an explct expresson for the mean vector m for a sphercal trangle T. The followng theorem solves ths problem. Theorem 3. Gven a sphercal trangle T, let θ be the length of ts th edge (a crcular arc) and n be the nward unt normal to ts th edge (see Fgure 3 (b)). Then, m = θ n (8) where m, the mean vector, s the ntegral of the outward unt normals over T. Proof: Consder the sold trangular wedge of the unt sphere wth cap T. The ntegral of outward unt normals over a closed surface s always exactly zero [Flemng 977, p.34]. Thus, we can partton the ntegral nto three trangular faces whose outward normals are n wth assocated areas θ. The theorem follows snce m θ n s then zero. Note that a smlar result holds n D, where the mean vector m defned by Equaton 3. for a crcular arc E on the unt crcle can be nterpreted as the sum of the two nward unt normals of the vectors p v (see Fgure 3 (a)). In 3D, the lengths θ of the edges of the sphercal trangle T are the angles between the vectors p v and p + v whle the unt normals n are formed by takng the cross
4 m E (a) n n v Fgure 3: Mean vector m on a crcular arc E wth edge normals n (a) and on a sphercal trangle T wth arc lengths θ and face normals n. product of p v and p + v. Gven the mean vector m, we now compute the weghts w usng Equaton 7 (but wthout dong the matrx nverson) va w = ψ 3 θ θ m n m n (p v) At ths pont, we should note that projectng a trangle T onto S v may reverse ts orentaton. To guarantee lnear precson, these foldedback trangles should produce negatve weghts w. If we mantan a postve orentaton for the vertces of every trangle T, the mean vector computed usng Equaton 8 ponts towards the projected sphercal trangle T when T has a postve orentaton and away from T when T has a negatve orentaton. Thus, the resultng weghts have the approprate sgn. 3.3 Robust mean value nterpolaton The dscusson n the prevous secton yelds a smple evaluaton method for mean value nterpolaton on trangular meshes. Gven pont v and a closed mesh, for each trangle T n the mesh wth vertces {p, p, p 3 } and assocated values { f, f, f 3 },. Compute the mean vector m va Equaton 8. Compute the weghts w usng Equaton 9 3. Update the denomnator and numerator of ˆf[v] defned n Equaton 6 respectvely by addng w and w f To correctly compute ˆf[v] usng the above procedure, however, we must overcome two obstacles. Frst, the weghts w computed by Equaton 9 may have a zero denomnator when the pont v les on plane contanng the face T. Our method must handle ths degenerate case gracefully. Second, we must be careful to avod numercal nstablty when computng w for trangle T wth a small projected area. Such trangles are the domnant type when evaluatng mean value coordnates on meshes wth large number of trangles. Next we dscuss our solutons to these two problems and present the complete evaluaton algorthm as pseudocode n Fgure 4. Stablty: When the trangle T has small projected area on the unt sphere centered at v, computng weghts usng Equaton 8 and 9 becomes numercally unstable due to cancellng of unt normals n that are almost coplanar. To ths end, we next derve a stable formula for computng weghts w. Frst, we substtute Equaton 8 nto Equaton 9, usng trgonometry we obtan T ψ ψ θ 3 (b) n n n 3 v (9) w = θ cos[ψ + ]θ cos[ψ ]θ + sn[ψ + ]sn[θ ] p k v, (0) // Robust evaluaton on a trangular mesh for each vertex p j wth values f j d j p j x f d j < ε return f j u j (p j x)/d j totalf 0 totalw 0 for each trangle wth vertces p, p, p 3 and values f, f, f 3 l u + u // for =,,3 θ arcsn[l /] h ( θ )/ f π h < ε // x les on t, use D barycentrc coordnates w sn[θ ]d d + return ( w f )/( w ) c (sn[h]sn[h θ ])/(sn[θ + ]sn[θ ]) s sgn[det[u,u,u 3 ]] c f, s ε // x les outsde t on the same plane, gnore t contnue w (θ c + θ c θ + )/(d sn[θ + ]s ) totalf+ = w f totalw+ = w f x totalf/totalw Fgure 4: Mean value coordnates on a trangular mesh where ψ ( =,,3) denotes the angles n the sphercal trangle T. Note that the ψ are the dhedral angles between the faces wth normals n and n +. We llustrate the angles ψ and θ n Fgure 3 (b). To calculate the cos of the ψ wthout computng unt normals, we apply the halfangle formula for sphercal trangles [Beyer 987], cos[ψ ] = sn[h]sn[h θ ], () sn[θ + ]sn[θ ] where h = (θ +θ +θ 3 )/. Substtutng Equaton nto 0, we obtan a formula for computng w that only nvolves lengths p v and angles θ. In the pseudocode from Fgure 4, angles θ are computed usng arcsn, whch s stable for small angles. Coplanar cases: Observe that Equaton 9 nvolves dvson by n (p v), whch becomes zero when the pont v les on plane contanng the face T. Here we need to consder two dfferent cases. If v les on the plane nsde T, the contnuty of mean value nterpolaton mples that ˆf[v] converges to the value f[x] defned by lnear nterpolaton of the f on T. On the other hand, f v les on the plane outsde T, the weghts w become zero as ther ntegral defnton φ [x] p[x] v dt becomes zero. We can easly test for the frst case because the sum Σ θ = π for ponts nsde of T. To test for the second case, we use Equaton to generate a stable computaton for sn[ψ ]. Usng ths defnton, v les on the plane outsde T f any of the dhedral angles ψ (or sn[ψ ]) are zero. 4 Applcatons and results Whle mean value coordnates fnd ther man use n boundary value nterpolaton, these coordnates can be appled to a varety of applcatons. In ths secton, we brefly dscuss several of these applcatons ncludng constructng volumetrc textures and surface deformaton. We conclude wth a secton on our mplementaton of these coordnates and provde evaluaton tmes for varous shapes.
5 Fgure 5: Orgnal model of a cow (topleft) wth hue values specfed at the vertces. The planar cuts llustrate the nteror of the functon generated by 3D mean value coordnates. 4. Boundary value nterpolaton As mentoned n Secton, these coordnate functons may be used to perform boundary value nterpolaton for trangular meshes. In ths case, functon values are assocated wth the vertces of the mesh. The functon constructed by our method s smooth, nterpolates those vertex values and s a lnear functon on the faces of the trangles. Fgure 5 shows an example of nterpolatng hue specfed on the surface of a cow. In the topleft s the orgnal model that serves as nput nto our algorthm. The rest of the fgure shows several slces of the cow model, whch reveal the volumetrc functon produced by our coordnates. Notce that the functon s smooth on the nteror and nterpolates the colors on the surface of the cow. 4. Volumetrc textures These coordnate functons also have applcatons to volumetrc texturng as well. Fgure 6 (topleft) llustrates a model of a bunny wth a D texture appled to the surface. Usng the texture coordnates (u,v ) as the f for each vertex, we apply our coordnates and buld a functon that nterpolates the texture coordnates specfed at the vertces and along the polygons of the mesh. Our functon extrapolates these surface values to the nteror of the shape to construct a volumetrc texture. Fgure 6 shows several slces revealng the volumetrc texture wthn. 4.3 Surface Deformaton Surface deformaton s one applcaton of mean value coordnates that depends on the lnear precson property outlned n Secton. In ths applcaton, we are gven two shapes: a model and a control mesh. For each vertex v n the model, we frst compute ts mean value weght functons w j wth respect to each vertex p j n the undeformed control mesh. To perform the deformaton, we move the vertces of the control mesh to nduce the deformaton on the orgnal surface. Let ˆp j be the postons of the vertces from the deformed control mesh, then the new vertex poston ˆv n the deformed model s computed as ˆv = ˆp j. Notce that, due to lnear precson, f ˆp j = p j, then ˆv = v. Fgures and 7 show several examples of deformatons generated wth ths Fgure 6: Textured bunny (topleft). Cuts of the bunny to expose the volumetrc texture constructed from the surface texture. process. Fgure (a) depcts a horse before deformaton and the surroundng control mesh shown n black. Movng the vertces of the control mesh generates the smooth deformatons of the horse shown n (b,c,d). Prevous deformaton technques such as freeform deformatons [Sederberg and Parry 986; MacCracken and Joy 996] requre volumetrc cells to be specfed on the nteror of the control mesh. The deformatons produced by these methods are dependent on how the control mesh s decomposed nto volumetrc cells. Furthermore, many of these technques restrct the user to creatng control meshes wth quadrlateral faces. In contrast, our deformaton technque allows the artst to specfy an arbtrary closed trangular surface as the control mesh and does not requre volumetrc cells to span the nteror. Our technque also generates smooth, realstc lookng deformatons even wth a small number of control ponts and s qute fast. Generatng the mean value coordnates for fgure took 3.3s and.9s for fgure 7. However, each of the deformatons only took 0.09s and 0.03s respectvely, whch s fast enough to apply these deformatons n realtme. 4.4 Implementaton Our mplementaton follows the pseudocode from Fgure 4 very closely. However, to speed up computatons, t s helpful to precompute as much nformaton as possble. Fgure 8 contans the number of evaluatons per second for varous models sampled on a 3GHz Intel Pentum 4 computer. Prevously, practcal applcatons nvolvng barycentrc coordnates have been restrcted to D polygons contanng a very small number of lne segments. In ths paper, for the frst tme, barycentrc coordnates have been appled to truly large shapes (on the order of 00, 000 polygons). The coordnate computaton s a global computaton and all vertces of the surface must be used to evaluate the functon at a sngle pont. However, much of the tme spent s determnng whether or not a pont les on the plane of one of the trangles n the mesh and, f so, whether or not that pont s nsde that trangle. Though we have not done so, usng varous spatal parttonng data structures to reduce the number of trangles that
6 mportant generalzaton would be to derve mean value coordnates for pecewse lnear mesh wth arbtrary closed polygons as faces. On these faces, the coordnates would degenerate to standard D mean value coordnates. We plan to address ths topc n a future paper. Acknowledgements We d lke to thank John Morrs for hs help wth desgnng the control meshes for the deformatons. Ths work was supported by NSF grant ITR References BEYER, W. H CRC Standard Mathematcal Tables (8th Edton). CRC Press. Fgure 7: Orgnal model and surroundng control mesh shown n black (topleft). Deformng the control mesh generates smooth deformatons of the underlyng model. Model Trs Verts Eval/s Horse control mesh (fg ) Armadllo control mesh (fg 7) Cow (fg 5) Bunny (fg 6) Fgure 8: Number of evaluatons per second for varous models. must be checked for coplanarty could greatly enhance the speed of the evaluaton. 5 Conclusons and Future Work Mean value coordnates are a smple, but powerful method for creatng functons that nterpolate values assgned to the vertces of a closed mesh. Perhaps the most ntrgung feature of mean value coordnates s that fact that they are welldefned on both the nteror and the exteror of the mesh. In partcular, mean value coordnates do a reasonable job of extrapolatng value outsde of the mesh. We ntend to explore applcatons of ths feature n future work. Another nterestng pont s the relatonshp between mean value coordnates and Wachspress coordnates. In D, both coordnate functons are dentcal for convex polygons nscrbed n the unt crcle. As a result, one method for computng mean value coordnates s to project the vertces of the closed polygon onto a crcle and compute Wachspress coordnates for the nscrbed polygon. However, n 3D, ths approach fals. In partcular, nscrbng the vertces of a trangular mesh onto a sphere does not necessarly yeld a convex polyhedron. Even f the nscrbed polyhedron happens to be convex, the resultng Wachspress coordnates are ratonal functons of the vertex poston v whle the mean value coordnates are transcendental functons of v. Fnally, we only consder meshes that have trangular faces. One COQUILLART, S Extended freeform deformaton: a sculpturng tool for 3d geometrc modelng. In SIGGRAPH 90: Proceedngs of the 7th annual conference on Computer graphcs and nteractve technques, ACM Press, DESBRUN, M., MEYER, M., AND ALLIEZ, P. 00. Intrnsc Parameterzatons of Surface Meshes. Computer Graphcs Forum, 3, FLEMING, W., Ed Functons of Several Varables. Second edton. Sprnger Verlag. FLOATER, M. S., AND HORMANN, K Surface parameterzaton: a tutoral and survey. In Advances n Multresoluton for Geometrc Modellng, N. A. Dodgson, M. S. Floater, and M. A. Sabn, Eds., Mathematcs and Vsualzaton. Sprnger, Berln, Hedelberg, FLOATER, M. S., KOS, G., AND REIMERS, M Mean value coordnates n 3d. To appear n CAGD. FLOATER, M Parametrzaton and smooth approxmaton of surface trangulatons. CAGD 4, 3, FLOATER, M Parametrc Tlngs and Scattered Data Approxmaton. Internatonal Journal of Shape Modelng 4, FLOATER, M. S Mean value coordnates. Comput. Aded Geom. Des. 0,, 9 7. HORMANN, K., AND GREINER, G MIPS  An Effcent Global Parametrzaton Method. In Curves and Surfaces Proceedngs (Sant Malo, France), HORMANN, K Barycentrc coordnates for arbtrary polygons n the plane. Tech. rep., Clausthal Unversty of Technology, September. hormann/papers/barycentrc.pdf. KHODAKOVSKY, A., LITKE, N., AND SCHROEDER, P Globally smooth parameterzatons wth low dstorton. ACM Trans. Graph., 3, KOBAYASHI, K. G., AND OOTSUBO, K tffd: freeform deformaton by usng trangular mesh. In SM 03: Proceedngs of the eghth ACM symposum on Sold modelng and applcatons, ACM Press, LOOP, C., AND DEROSE, T A multsded generalzaton of Bézer surfaces. ACM Transactons on Graphcs 8, MACCRACKEN, R., AND JOY, K. I Freeform deformatons wth lattces of arbtrary topology. In SIGGRAPH 96: Proceedngs of the 3rd annual conference on Computer graphcs and nteractve technques, ACM Press, MALSCH, E., AND DASGUPTA, G Algebrac constructon of smooth nterpolants on polygonal domans. In Proceedngs of the 5th Internatonal Mathematca Symposum. MEYER, M., LEE, H., BARR, A., AND DESBRUN, M. 00. Generalzed Barycentrc Coordnates for Irregular Polygons. Journal of Graphcs Tools 7,, 3. SCHREINER, J., ASIRVATHAM, A., PRAUN, E., AND HOPPE, H Intersurface mappng. ACM Trans. Graph. 3, 3, SEDERBERG, T. W., AND PARRY, S. R Freeform deformaton of sold geometrc models. In SIGGRAPH 86: Proceedngs of the 3th annual conference on Computer graphcs and nteractve technques, ACM Press, WACHSPRESS, E A Ratonal Fnte Element Bass. Academc Press, New York. WARREN, J., SCHAEFER, S., HIRANI, A., AND DESBRUN, M Barycentrc coordnates for convex sets. Tech. rep., Rce Unversty. WARREN, J Barycentrc Coordnates for Convex Polytopes. Advances n Computatonal Mathematcs 6,
BERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAMReport 201448 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationGraph Theory and Cayley s Formula
Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll
More informationPLANAR GRAPHS. Plane graph (or embedded graph) A graph that is drawn on the plane without edge crossing, is called a Plane graph
PLANAR GRAPHS Basc defntons Isomorphc graphs Two graphs G(V,E) and G2(V2,E2) are somorphc f there s a onetoone correspondence F of ther vertces such that the followng holds:  u,v V, uv E, => F(u)F(v)
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationABC. Parametric Curves & Surfaces. Overview. Curves. Many applications in graphics. Parametric curves. Goals. Part 1: Curves Part 2: Surfaces
arametrc Curves & Surfaces Adam Fnkelsten rnceton Unversty COS 46, Sprng Overvew art : Curves art : Surfaces rzemyslaw rusnkewcz Curves Splnes: mathematcal way to express curves Motvated by loftsman s
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationEE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN
EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson  3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson  6 Hrs.) Voltage
More informationState function: eigenfunctions of hermitian operators> normalization, orthogonality completeness
Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators> normalzaton, orthogonalty completeness egenvalues and
More informationChapter 7. RandomVariate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 RandomVariate Generation
Chapter 7 RandomVarate Generaton 7. Contents Inversetransform Technque AcceptanceRejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationLoop Parallelization
  Loop Parallelzaton C52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I,J]+B[I,J] ED FOR ED FOR analyze
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationThe Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets
. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely
More informationConversion between the vector and raster data structures using Fuzzy Geographical Entities
Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationz(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1
(4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationINTRODUCTION. governed by a differential equation Need systematic approaches to generate FE equations
WEIGHTED RESIDUA METHOD INTRODUCTION Drect stffness method s lmted for smple D problems PMPE s lmted to potental problems FEM can be appled to many engneerng problems that are governed by a dfferental
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationA Computer Technique for Solving LP Problems with Bounded Variables
Dhaka Unv. J. Sc. 60(2): 163168, 2012 (July) A Computer Technque for Solvng LP Problems wth Bounded Varables S. M. Atqur Rahman Chowdhury * and Sanwar Uddn Ahmad Department of Mathematcs; Unversty of
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationA frequency decomposition time domain model of broadband frequencydependent absorption: Model II
A frequenc decomposton tme doman model of broadband frequencdependent absorpton: Model II W. Chen Smula Research Laborator, P. O. Box. 134, 135 Lsaker, Norwa (1 Aprl ) (Proect collaborators: A. Bounam,
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationGRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 NORM
GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 NORM BARRIOT JeanPerre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jeanperre.barrot@cnes.fr 1/Introducton The
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCullochPtts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses
More informationDiVA Digitala Vetenskapliga Arkivet
DVA Dgtala Vetenskaplga Arkvet http://umudvaportalorg Ths s a book chapter publshed n Hghperformance scentfc computng: algorthms and applcatons (ed Berry, MW; Gallvan, KA; Gallopoulos, E; Grama, A; Phlppe,
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More informationIMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1
Nov Sad J. Math. Vol. 36, No. 2, 2006, 009 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationGeneralizing the degree sequence problem
Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts
More informationIntroduction: Analysis of Electronic Circuits
/30/008 ntroducton / ntroducton: Analyss of Electronc Crcuts Readng Assgnment: KVL and KCL text from EECS Just lke EECS, the majorty of problems (hw and exam) n EECS 3 wll be crcut analyss problems. Thus,
More informationHYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION
HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR Emal: ghapor@umedumy Abstract Ths paper
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationSCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar
SCALAR A phscal quantt that s completel charactered b a real number (or b ts numercal value) s called a scalar. In other words, a scalar possesses onl a magntude. Mass, denst, volume, temperature, tme,
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationInterIng 2007. INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 1516 November 2007.
InterIng 2007 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 1516 November 2007. UNCERTAINTY REGION SIMULATION FOR A SERIAL ROBOT STRUCTURE MARIUS SEBASTIAN
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationTime Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University
Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationAlgebraic Point Set Surfaces
Algebrac Pont Set Surfaces Gae l Guennebaud Markus Gross ETH Zurch Fgure : Illustraton of the central features of our algebrac MLS framework From left to rght: effcent handlng of very complex pont sets,
More informationA machine vision approach for detecting and inspecting circular parts
A machne vson approach for detectng and nspectng crcular parts DuMng Tsa Machne Vson Lab. Department of Industral Engneerng and Management YuanZe Unversty, ChungL, Tawan, R.O.C. Emal: edmtsa@saturn.yzu.edu.tw
More informationComplex Number Representation in RCBNS Form for Arithmetic Operations and Conversion of the Result into Standard Binary Form
Complex Number epresentaton n CBNS Form for Arthmetc Operatons and Converson of the esult nto Standard Bnary Form Hatm Zan and. G. Deshmukh Florda Insttute of Technology rgd@ee.ft.edu ABSTACT Ths paper
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationVision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION
Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble
More informationA Note on the Decomposition of a Random Sample Size
A Note on the Decomposton of a Random Sample Sze Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract Ths note addresses some results of Hess 2000) on the decomposton
More informationYves Genin, Yurii Nesterov, Paul Van Dooren. CESAME, Universite Catholique de Louvain. B^atiment Euler, Avenue G. Lema^tre 46
Submtted to ECC 99 as a regular paper n Lnear Systems Postve transfer functons and convex optmzaton 1 Yves Genn, Yur Nesterov, Paul Van Dooren CESAME, Unverste Catholque de Louvan B^atment Euler, Avenue
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract  Stock market s one of the most complcated systems
More informationComputerAided Design. Computer aided clothing pattern design with 3D editing and pattern alteration
ComputerAded Desgn 44 (2012) 721 734 Contents lsts avalable at ScVerse ScenceDrect ComputerAded Desgn journal homepage: www.elsever.com/locate/cad Computer aded clothng pattern desgn wth 3D edtng and
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and
More informationPOLYSA: A Polynomial Algorithm for Nonbinary Constraint Satisfaction Problems with and
POLYSA: A Polynomal Algorthm for Nonbnary Constrant Satsfacton Problems wth and Mguel A. Saldo, Federco Barber Dpto. Sstemas Informátcos y Computacón Unversdad Poltécnca de Valenca, Camno de Vera s/n
More informationSection 5.3 Annuities, Future Value, and Sinking Funds
Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme
More informationLecture 7 March 20, 2002
MIT 8.996: Topc n TCS: Internet Research Problems Sprng 2002 Lecture 7 March 20, 2002 Lecturer: Bran Dean Global Load Balancng Scrbe: John Kogel, Ben Leong In today s lecture, we dscuss global load balancng
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationA MultiCamera System on PCCluster for Realtime 3D Tracking
The 23 rd Conference of the Mechancal Engneerng Network of Thaland November 4 7, 2009, Chang Ma A MultCamera System on PCCluster for Realtme 3D Trackng Vboon Sangveraphunsr*, Krtsana Uttamang, and
More informationAryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006
Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,
More information) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance
Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationOn Mean Squared Error of Hierarchical Estimator
S C H E D A E I N F O R M A T I C A E VOLUME 0 0 On Mean Squared Error of Herarchcal Estmator Stans law Brodowsk Faculty of Physcs, Astronomy, and Appled Computer Scence, Jagellonan Unversty, Reymonta
More informationFace Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)
Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationStochastic epidemic models revisited: Analysis of some continuous performance measures
Stochastc epdemc models revsted: Analyss of some contnuous performance measures J.R. Artalejo Faculty of Mathematcs, Complutense Unversty of Madrd, 28040 Madrd, Span A. Economou Department of Mathematcs,
More informationTraffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationNONCONSTANT SUM REDANDBLACK GAMES WITH BETDEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 OCOSTAT SUM REDADBLACK GAMES WITH BETDEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationAn Integrated Semantically Correct 2.5D Object Oriented TIN. Andreas Koch
An Integrated Semantcally Correct 2.5D Object Orented TIN Andreas Koch Unverstät Hannover Insttut für Photogrammetre und GeoInformaton Contents Introducton Integraton of a DTM and 2D GIS data Semantcs
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More informationAn interactive system for structurebased ASCII art creation
An nteractve system for structurebased ASCII art creaton Katsunor Myake Henry Johan Tomoyuk Nshta The Unversty of Tokyo Nanyang Technologcal Unversty Abstract NonPhotorealstc Renderng (NPR), whose am
More informationMultiple discount and forward curves
Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of
More informationIMPACT ANALYSIS OF A CELLULAR PHONE
4 th ASA & μeta Internatonal Conference IMPACT AALYSIS OF A CELLULAR PHOE We Lu, 2 Hongy L Bejng FEAonlne Engneerng Co.,Ltd. Bejng, Chna ABSTRACT Drop test smulaton plays an mportant role n nvestgatng
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationSection B9: Zener Diodes
Secton B9: Zener Dodes When we frst talked about practcal dodes, t was mentoned that a parameter assocated wth the dode n the reverse bas regon was the breakdown voltage, BR, also known as the peaknverse
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationDamage detection in composite laminates using cointap method
Damage detecton n composte lamnates usng contap method S.J. Km Korea Aerospace Research Insttute, 45 EoeunDong, YouseongGu, 35333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The contap test has the
More informationMultivariate EWMA Control Chart
Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant
More informationYIELD CURVE FITTING 2.0 Constructing Bond and Money Market Yield Curves using Cubic BSpline and Natural Cubic Spline Methodology.
YIELD CURVE FITTING 2.0 Constructng Bond and Money Market Yeld Curves usng Cubc BSplne and Natural Cubc Splne Methodology Users Manual YIELD CURVE FITTING 2.0 Users Manual Authors: Zhuosh Lu, Moorad Choudhry
More informationChapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT
Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the
More informationMultiple View Image Reconstruction: A Harmonic Approach
Multple Vew Image Reconstructon: A Harmonc Approach Justn Domke and Yanns Alomonos Center for Automaton Research, Department of Computer Scence Unversty of Maryland, College Park, MD, 2742, USA www.cs.umd.edu/users/domke
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More information