Differential Geometry: Numerical Integration and Surface Flow
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1 Differenial Geomery: Numerical Inegraion and Surface Flow [Implici Fairing of Irregular Meshes using Diffusion and Curaure Flow. Desbrun e al., 1999]
2 Energy Minimizaion Recall: We hae been considering he siuaion in which we are gien an energy E ha we would like o minimize. Mean Curaure Flow E:R 3V R, aking ere posiions o areas. Circle Packings E:R V R, aking radii a he erices o ere posiions o absolue angle-sum deficis. Circle Paerns E:R T R, aking log-radii a riangles o he inegral of kie-angle-sum deficis.
3 Energy Minimizaion Recall: We hae been considering he siuaion in which we are gien an energy E ha we would like o minimize. In each case, he negaie gradien of he energy old us how o modify he alues o reduce he energy a each ime sep: d = E and he Hessian gae us he change in he gradien direcion.
4 Numerical Inegraion: Suppose ha we hae an eoling sysem and a funcion ha ells us how he sysem changes a any paricular ime as a funcion of is curren sae: = Φ d and suppose ha we know he iniial sae, how should we eole he sysem?
5 Eplici Inegraion Forward Euler: If we assume ha: d = Φ +
6 Eplici Inegraion Forward Euler: If we assume ha: We ge: d = Φ + + d + = + Φ which is precisely how we eoled he surface when performing mean curaure flow.
7 Eplici Inegraion Forward Euler: If we assume ha: We ge: d + d Noe: = Φ + = + Φ which is precisely how we eoled he surface when performing In his inerpreaion, mean curaure we rea flow. he deriaie a ime as a predicor of he sae a he ne ime-sep +
8 Implici Inegraion Backward Euler: If we assume ha: We ge: d Φ + = + = Φ d + Φ + = d
9 Implici Inegraion Backward Euler: If we assume ha: We ge: d = Φ + d + = + Φ + The challenge is ha now our updaed sae depends on change ha is defined by he sae + ha we don know. +
10 Implici Inegraion Backward Euler: + + = + Φ + d + We could sole his using saring wih an iniial guess for he sae y= a + and use his o predic he new sae: + = + Φ y
11 Implici Inegraion Backward Euler: + + = + Φ + d + We could sole his using saring wih an iniial guess for he sae y= a + and use his o predic he new sae: + = + Φ y Then, depending on he difference beween y and he prediced sae +Φy, we modify our guess.
12 Implici Inegraion Backward Euler: + + = + Φ + d + Q: Gien he difference beween he guess and he prediced sae, how do we modify he guess?
13 Implici Inegraion Backward Euler: + + = + Φ + d + A: Assuming our guess is y hen he error in our predicion is: ε = + Φ y y Predicion Guess
14 Implici Inegraion Backward Euler: + + = + Φ + d + A: Assuming our guess is y hen he error in our predicion is: ε = + Φ y y Seing our new guess o be ŷ, we wan he difference beween he original predicion and he new predicion o be -ε: ε = Φ yˆ yˆ Φ y y New Predicion Old Predicion
15 Implici Inegraion Backward Euler: + + = + Φ + d + A: To ge he predicion difference: ε = Φ yˆ yˆ Φ y y New Predicion Old Predicion we linearize he funcion Φ. Seing ŷ=y+ : Φ y + Φ y + dφ y
16 Implici Inegraion Backward Euler: + + = + Φ + d + A: To ge he predicion difference: ε = Φ yˆ yˆ Φ y y New Predicion Old Predicion we linearize he funcion Φ. Seing ŷ=y+ : Φ y + Φ y + dφ Noe: If Φ is he negaie gradien of an energy, dφ is he energy s Hessian. y
17 Implici Inegraion Backward Euler: + + = + Φ + d + A: To ge he predicion difference: ε = Φ yˆ yˆ Φ y y Φ y + Φ y + dφ yˆ This gies: ε = Φ yˆ yˆ Φ y y = Φ y + y Φ y y Φ y + dφ y Φ y y = dφ yˆ yˆ
18 Implici Inegraion Backward Euler: + + = + Φ + d + A: To ge he predicion difference: ε = Φ yˆ yˆ Φ y y Φ y + Φ y + dφ yˆ This gies: ε = Φ yˆ yˆ Φ y y = Φ y + y Φ y y Φ y + dφ y Φ y y = dφ yˆ yˆ
19 Implici Inegraion Backward Euler: + + = + Φ + d + A: To ge he predicion difference: ε = Φ yˆ yˆ Φ y y Φ y + Φ y + dφ yˆ This gies: ε = Φ yˆ yˆ Φ y y = Φ y + y Φ y y Φ y + dφ y Φ y y = dφ yˆ yˆ
20 Implici Inegraion Backward Euler: + + = + Φ + d + A: To ge he predicion difference: ε = Φ yˆ yˆ Φ y y Φ y + Φ y + dφ yˆ This gies: ε = Φ yˆ yˆ Φ y y = Φ y + y Φ y y Φ y + dφ y Φ y y = dφ yˆ yˆ
21 Implici Inegraion Backward Euler: + + = + Φ + d + A: Puing all his ogeher, we obain he modified guess ŷ=y+, by soling he sysem: ε = dφ 1 o ge he offse ha akes us from he iniial guess y= o he improed guess ŷ= +. ŷ
22 Implici Inegraion Backward Euler: Problem: ε = dφ 1 Compuing he offse required ealuaing he deriaie dφ a ŷ, bu we don know ŷ! ŷ
23 Implici Inegraion Backward Euler: Soluion Ieraie: ε = dφ 1 Use an ieraie approach, defining a sequence of guesses {y,y 1,...} where we compue he improed guess y i+1 by using he deriaie mari compued a y i. ŷ
24 Implici Inegraion Backward Euler: Soluion Ieraie: ε = dφ 1 Use an ieraie approach, defining a sequence of guesses {y,y 1,...} where we compue he improed guess y i+1 by using he deriaie mari compued a y i. This means ha we hae o define and sole he linear sysem dφ y i a each inernal ieraion. ŷ
25 Implici Inegraion Backward Euler: Soluion Semi-Implici: ε = dφ 1 Jus use he deriaie mari from he iniial guess. ŷ
26 Implici Inegraion Backward Euler: Soluion Semi-Implici: Jus use he deriaie mari from he iniial guess. This corresponds o inerpreing posiional deriaies as backward-predicing while elociy deriaies as forward-predicing. + + Φ y + Φ y + dφ ε = dφ 1 d + ŷ y
27 Implici Inegraion Backward Euler: Soluion Semi-Implici: ε = dφ 1 Since our iniial guess is y=, our error is: ε = + Φ y y = Φ ŷ
28 Implici Inegraion Backward Euler: Soluion Semi-Implici: Since our iniial guess is y=, our error is: Since our modified guess will be he sae for he ne ime-sep, his gies: Φ = 1 ŷ d ε 1 1 d I d I Φ Φ + = Φ Φ + = + y y Φ = Φ + = ε
29 Deriaies: Gien a funcion F:R n R n, he deriaie of F is an nn mari df ha describes he iny change of each oupu coefficien as a funcion of a iny change in each of he inpu coefficiens.
30 Deriaies: Gien a funcion F:R n R n, he deriaie of F is an nn mari df ha describes he iny change of each oupu coefficien as a funcion of a iny change in each of he inpu coefficiens. In he case ha F is linear, hen F is is own deriaie.
31 Mean Curaure Flow: Gien a surface, we defined an energy ha was he area of surface and we showed ha he gradien of he energy was proporional o he mean curaure ecor: E { } L 1,..., n = where L is he coangen-weigh Laplacian defined by he erices.
32 Mean Curaure Flow: Thus, o minimize he area, we offse poins on he surface in he direcion of he negaie mean curaure: = L + 1
33 Mean Curaure Flow: Thus, o minimize he area, we offse poins on he surface in he direcion of he negaie mean curaure: = L + 1 Which amouns o eplici inegraion.
34 Mean Curaure Flow: Thus, o minimize he area, we offse poins on he surface in he direcion of he negaie mean curaure: = L + 1 Which amouns o eplici inegraion. Noe: Since he geomery changes a each ime-sep, we hae o compue he new coangen-weigh Laplacian, L a each ime-sep.
35 Mean Curaure Flow: Thus, o minimize he area, we offse poins on he surface in he direcion of he negaie mean curaure: = L + 1 In pracice, he sep-size has o be small proporional o he smalles edge-lengh. Can we do beer?
36 Mean Curaure Flow Semi-Implici: Using a semi-implici scheme, we can generae more sable inegraion by soling a linear sysem a each sep: 1 1 L dl I + + =
37 Mean Curaure Flow Semi-Implici: Using a semi-implici scheme, we can generae more sable inegraion by soling a linear sysem a each sep: I = + dl L To do his righ, we would differeniae he coangen-weigh enries in he Laplacian mari L since hey depend on he ere posiions.
38 Mean Curaure Flow Semi-Implici: Using a semi-implici scheme, we can generae more sable inegraion by soling a linear sysem a each sep: Howeer, if we preend ha hey are fied, hen he deriaie of he Laplacian mari is jus he Laplacian mari, and we ge: 1 1 L L I + + = 1 1 L dl I + + =
39 Mean Curaure Flow Semi-Implici: Using a semi-implici scheme, we can generae more sable inegraion by soling a linear sysem a each sep: I = + dl L Howeer, if we preend ha hey are fied, hen he Noe: deriaie of he Laplacian mari is jus he Laplacian An addiional mari, adanage and we ge: of his simplificaion 1 is ha he deriaie + 1 I = of he L Laplacian L is now of size nn insead of 3n3n.
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