CS 468, Spring 2013 Differential Geometry for Computer Science Justin Solomon and Adrian Butscher

Transcription

1 CS 468, Spring 2013 Differential Geometry for Computer Science Justin Solomon and Adrian Butscher

2 Homework 4: June 5 Project: June 6 Scribe notes: One week after, June 6 at latest Course reviews

3 Á 1 = 1Á 1 Á 2 = 2Á 2 Á 3 = 3Á 3 Á 4 = 4Á 4 Á 5 = 5Á 5 Mostly static geometry

4 Articulation, skinning, etc. are difficult and repetitive

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6 Abstract surface Topology plus distances Embedding How the surface sits in R 3 Change one, the other, or both

7 Modify embedding directly but characterize effect on intrinsic structure.

8 dá t dt = H tn t = t ~x t ~x i+1 ~x i ±t = i ~x i+i Implicit Fairing of Irregular Meshes Desbrun, Meyer, Schroder, Barr; SIGGRAPH 99 Change embedding

9 w ;r (H) ( 1; for jhj 2 r( jhj) ; for jhj > ~H(p) = 1 2 X e=pq w(h e )H e ~ Ne Anisotropic Filtering of Non-Linear Surface Features Hildebrandt, Polthier Eurographics 2004

10 Weighted Laplacian Lumped area weights Explicit time step

11 By definition, mean curvature flow wants to decrease surface area.

12 dp dt = M 1 ( ~ H(P) f(p) ~ V (P)) MC normal from cotangent Laplacian Smoothed MC target function Gradient of volume Anisotropic Filtering of Non-Linear Surface Features Hildebrandt, Polthier; EG 2004

13 Surface editing

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15 J(Á) Z S W(x; E(x)) dx Energy measures deformation

16 E bend Z S ka orig A deformed k 2 dx E stretch Z S kg orig g deformed k 2 dx

17 Deformation energies are highly nonlinear.

18 Robust Fast Not rotationinvariant!

19 Laplacian Surface Editing Sorkine et al. SGP 2004

20 ± LV = (I D 1 A)V Poisson equation for V Approximation of normal minimize X i k± i L(v 0 i )k2 + X i2c kv 0 i u ik 2 Least squares Preserve coordinates Respect user constraints Translation-invariant coordinates

21 Normal preservation is not rotation or scale invariant

22 minimize X i kt i ± i L(v 0 i )k2 + X i2c kv 0 i u ik 2 New variable Trick: Restrict T to set of desirable transformations (or approximation thereof) M sim = s( I + H + ~ h > ~ h) Transform source δ with vertices

23 minimize X i kt i ± i L(v 0 i )k2 + X i2c kv 0 i u ik 2 New variable Trick: Restrict T to set of desirable transformations (or approximation thereof) M sim = s( I + H + ~ h > ~ h) Transform source δ with vertices

24 On Linear Variational Surface Deformation Methods Botsch and Sorkine TVCG

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26 Want: For each cell C i around vertex i p 0 i p0 j = R i(p i p j ) 8j 2 N(i) Rotation Center Neighbor Try to enforce local isometry

27 E(C i ; C 0 i ) = X j2n (i) Pointwise Laplacian w ij k(p 0 i p0 j ) R i(p i p j )k 2 E(S 0 ) = X i w i E(C i ; C 0 i) Area weight

28 E(C i ; C 0 i ) = X j2n (i) Pointwise Laplacian w ij k(p 0 i p0 j ) R i(p i p j )k 2 E(S 0 ) = X i w i E(C i ; C 0 i) Area weight

29 E(C i ; C 0 i ) = X j2n (i) w ij k(p 0 i p0 j ) R i(p i p j )k 2 E(S 0 ) = X i w i E(C i ; C 0 i) Vertex positions: Linear

30 arg min Ri X j2n (i) w ij ke 0 ij R ie ij k 2 = arg max Ri X j2n (i) 0 = arg max Ri i = arg max Ri Tr (R i S i ) w ij e 0> ij R ie ij X j2n (i) w ij e ij e 0> ij 1 A S i = U V > Tr(R i S i ) = Tr(R i U V > ) = Tr(V > R i U ) = Tr( R ) ~ = diag R ~ diag X ¾i for R ~ = I ~R = I ) R i = V U > Rotation matrices: SVD

31 Vertices fixed by user Initialize optimization with simpler method Alternating optimization

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34 W = Z Z Z = 1 4 S S Z H 2 da S K da k ~xk 2 da 2¼Â(S) S ( 1 2) 2 da Conformal-invariant bending energy

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36 E b = Z S H 2 da

37 K 0

38 Quadratic in vertex positions Invariant under isometry Isometry = preserved edge lengths Invariant under rigid motion Invariant under uniform scaling

39 E b = 1 2 X ij Q ij h~x i ; ~x j i Q must be positive semi-definite Convex energy Q must be constructed from intrinsic properties Edge lengths, interior angles, areas, etc. Rows and columns sum to 0 By translation invariance

40 E b = 1 2 ~x> all(l > M 1 L)~x > all Scale-independent Scaling term Looks like finite elements.

41 If the surface deforms isometrically, then Q is unchanged.

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43 Fixing edge lengths doesn t always approximate smooth isometry.

44 Smoothing and fairing Static deformation Dynamic deformation Different models and discretizations

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46 CS 468, Spring 2013 Differential Geometry for Computer Science Justin Solomon and Adrian Butscher