3D As-Rigid-As-Possible Deformations Using MLS - LCG - UFRJ

3D As-Rigid-As-Possible 

Deformations Using MLS 

Alvaro Cuno, Claudio Esperança, Antonio Oliveira, Paulo Cavalcanti 

Federal University of Rio de Janeiro - COPPE 

May - 2007 

COPPE-UFRJ, May - 2007 – p. 1

Introduction 

• Image deformation using Moving Least Squares (Schaefer et al. SIGGRAPH 2006) 


Introduction 

• Image deformation using Moving Least Squares (Schaefer et al. SIGGRAPH 2006) 

• Attractive because: 

+ Reasonable results 

+ No skeletons 

+ Meshless 

+ Easy to implement (no LA package) 

+ Closed-form solutions 

+ Interactive 

+ Affine, similarity and rigid transformations 


Deformation paradigm 

Blue points: initial control points positions {p i } 

Black points: moved control points positions {q i } 


Deformation paradigm 

Blue points: initial control points positions {p i } 

Black points: moved control points positions {q i } 

Requeriments of f 

• Interpolation: f(p i) = q i 

• Identity: q i = p i ⇒ f(v) = v 

• Smoothness: f should produce smooth deformations 

• f must be an affine transformation 


Schaefer’s method 

• Problem: Let {p i} ∈ ℜ 2 and {q i} ∈ ℜ 2 . Then, for pixel v, 

find the affine transformation 




find the affine transformation fv(x) = xM + T that 

minimizes 

 

wi|fv(pi) − qi| 2 , 

where wi = |p i − v| −u ,u ∈ Z + 

i 




find the affine transformation fv(x) = xM + T that 

minimizes 

 

wi|fv(pi) − qi| 2 , 

where wi = |p i − v| −u ,u ∈ Z + 

• MLS: wi = g(v) ⇒ one solution for each v 

i 


MLS “rigid ” deformation in 3D 

• Optimization problem 

min 

wi|ˆpiR − ˆqi| 2 = max 

(Translation solution = 2D) 

i 

i 

wiˆqiRˆp T i 


MLS “rigid ” deformation in 3D 

• Optimization problem 

min 

wi|ˆpiR − ˆqi| 2 = max 

(Translation solution = 2D) 

i 

i 

wiˆqiRˆp T i 

• Similar to “The Absolute Orientation Problem” 

• Find the best transformation for mapping a point cloud 

into another 

• Exact and approximate solutions 

• MLS-deformation technique 

• Solve the problem adaptively for each pixel v 

• Error is weighted by distance from v to the restrictions 


Rotation representation 

• Direction Cosine Matrix 




• Rotation Matrix ⇒ 9 coefficients 





• Euler angles 






• Quaternions 







• Dual Quaternions 







• Dual Quaternions 

• Axis-Angle : {(ex ey ez),α} 


Optimization problem 

max F(e,α) = e M e T + cos(α)(E − e M e T ) + sin(α)V e T 

s.t. e = 1, 

cos 2 (α) + sin 2 (α) = 1 




s.t. e = 1, 

cos 2 (α) + sin 2 (α) = 1 




where 

M= 

i wiˆq T i ˆpi = 

s.t. e = 1, 

⎛ 

⎜ 

⎝ 

 

i wiˆqixˆpix 

i wiˆqiyˆpix 

i wiˆqizˆpix 

cos 2 (α) + sin 2 (α) = 1 

 

 

 

i wiˆqixˆpiy 

i wiˆqiyˆpiy 

i wiˆqizˆpiy 

 

 

 

i wiˆqixˆpiz 

i wiˆqiyˆpiz 

i wiˆqizˆpiz 

E = 

i wiˆqi · ˆpi= Trace(M), 

V = 

i wiˆpi×ˆqi = (M23−M32 M31−M13 M12−M21) 

⎞ 

⎟ 

⎠ , 


Eigenvalue problem 

• Kuhn-Tucker optimality conditions gives us 

 

M + M T + VT 

V 

e 

λ − 2E 

T = λ e T , 

• Characteristic polynomial P(λ) (3th degree) 




 

M + M T + VT 

V 

e 

λ − 2E 

T = λ e T , 


• Depressed quartic equation (using y = λ − E) 

y 4 −2M 2 y 2 −8 det(M)y+M 4 −4 

where Mk is the k th column of M. 

i 

 

j



 

M + M T + VT 

V 

e 

λ − 2E 

T = λ e T , 



y 4 −2M 2 y 2 −8 det(M)y+M 4 −4 


• It can be shown F(e,α) = λ − E = y 

i 

 

j



 

M + M T + VT 

V 

e 

λ − 2E 

T = λ e T , 



y 4 −2M 2 y 2 −8 det(M)y+M 4 −4 


• It can be shown F(e,α) = λ − E = y 

i 

 

• ∴ max F(e,α) → y must be the largest real root 

j

Solving the depressed equation 

• Any depressed quartic equation y 4 + ay 2 + by + c may be 

factored as (y 2 + py + q)(y 2 − py + s), where p must satisfy 

p 6 + 2ap 4 + (a 2 − 4ac)p 2 − b 2 = 0. 





p 6 + 2ap 4 + (a 2 − 4ac)p 2 − b 2 = 0. 

• If p 2 = z then we must solve cubic equation 

z 3 − 4M 2 z 2 + 

16( 

i j




p 6 + 2ap 4 + (a 2 − 4ac)p 2 − b 2 = 0. 

• If p 2 = z then we must solve cubic equation 

z 3 − 4M 2 z 2 + 

16( 

i j

Rotation parameters 

• The original system „ 

M + M T + VT V 

λ−2E 

« 

− λI eT =0, 





λ−2E 

as 

(M + M T − λI) “ 

eT (λ−2E) 

−V· eT ” 

=V T 

« 

− λI eT =0, can be written 





λ−2E 

as 

(M + M T − λI) “ 

eT (λ−2E) 

−V· eT ” 

=V T 

• Thus, the rotation axis e can be found solving 

where e = u 

||u|| 

(M + M T − λI)u T = V T , 

« 






λ−2E 

as 

(M + M T − λI) “ 

eT (λ−2E) 

−V· eT ” 

=V T 

• Thus, the rotation axis e can be found solving 

where e = u 

||u|| 

(M + M T − λI)u T = V T , 

• Finally, the angular parameters 

cos(α) = 

« 


1 − u2 

−2u 

1 + u2, sin(α) = 

1 + u2 COPPE-UFRJ, May - 2007 – p. 10

Algorithm 

Algorithm 1: Compute deformed position 

Input: v, {pi}, {qi} 

Output: Deformed vertex position v ′ 

1 

2 

3 

4 

5 

6 

7 

8 

[p∗,q∗] ← WeightedCentroid(v, {pi}, {qi}); 

M ← CorrelationMatrix(p∗, q∗); 

λ ← LargestRootOfP(M) + Trace(M) ; // eigenvalue 

e ← RotationAxis(M, λ) ; 

[sin(α), cos(α)] ← SinCos(M, e) ; 

ˆv ← v − p∗; 

d ← (ˆve T )e + cos(α)(ˆv T − (ˆve T )e) + sin(α)(ˆv × e); 

v ′ ← q∗ + d; 

return v ′ 

9 COPPE-UFRJ, May - 2007 – p. 11

Results 


Results (cont.) 

Table 1: FPS* for deformation of several models 

Model Vertices Control points FPS 

Dolphin 2811 4 40 

Homer 5103 6 22 

Plane 1089 132 19.5 

Dino 14050 5 6.5 

Bar 6146 256 2 

Cylinder 1058 322 9.5 

*PC Pentium IV 3.2GHz, Linux, C++, OpenGL (naive implementation) 


Extensions 

Similarity deformation: min 

µs = 

 

y 

i wi|µsˆpiR − ˆqi| 2 , 

i wiˆpi 2, ∴ v′ ← q∗ + µsd 


Extensions 

Similarity deformation: min 

µs = 

 

y 

i wi|µsˆpiR − ˆqi| 2 , 

i wiˆpi 2, ∴ v′ ← q∗ + µsd 

Initial Affine Simil. Rigid COPPE-UFRJ, May - 2007 – p. 14

Extensions (cont.) 

Deformation using line segments 

min 

i 

1 

0 wi|pi(t)R + T − qi(t)| 2 


Discussion 

• Non-smooth results 

(a) w(d) = d −2 w(d) = d −3 w(d) = d −4 


Discussion (cont.) 

• Space deformation gives non-natural results for meshes 


Discussion (cont.) 

• Resistance to large translations 

Initial MLS-rigid RBF 


Future work 

• Mixing MLS with RBFs 


Future work 

• Mixing MLS with RBFs 

• Change the distance metric 


Future work 

• Skeleton-driven deformation 


Future work 

• Skeleton-driven deformation 

• Scalability, more controlable, character animation, etc. 


. . . Thanks

3D As-Rigid-As-Possible Deformations Using MLS - LCG - UFRJ

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?