As-Rigid-As-Possible Shape Manipulation

Transcription

1 As-Rigid-As-Possible Shape Manipulation T. Igarashi 1, T. Mascovich 2 J. F. Hughes 3 1 The University of Tokyo 2 Brown University 3 PRESTO, JST SIGGRAPH 2005 Presented by: Prabin Bariya

2 Interactive shape manipulation technique for 2D shapes, without using a skeleton or FFD. Introduces internal model rigidity into shape manipulation. Uses a quadractic error metric so the minimization problem is formulated as a set of simultaneous linear equations. The problem is split into two least-squares minimization problems (rotation and scale) that are solved sequentially.

3 The user chooses handle points within the shape and moves each handle to a desired location. The system the moves, rotates and deforms the shape to match the new handle locations while minimizing distortions.

4 Triangulation The boundary of the shape should be represented as a simple closed polygon. The system generates a triangulated mesh inside the boundary of the shape. Near equilateral triangles are used for better results. Mesh should not be too large. Resulting triangulation is the rest shape.

5 Registration and Compilation The system performs a pre-computation, called registration, to accelerate computation during the interaction. When handles are added or removed, additional pre-computations are performed, called compilation. Compilation allows computation of resulting shape, given handle configurations.

6 Input: xy-coordinates of the handles. Output: xy-coordinates of the free vertices that minimize distortion. Uses two independent quadratic error functions for the rotation and scaling. Final result is obtained by sequentially solving two least-squares problems.

7 Scale-Free Construction Generates an intermediate result by minimizing an error function that allows rotation and uniform scaling. For a triangle in rest shape {v 0,v 1,v 2 }, v 2 = v 0 + x 01 v 0 v 1 + y 01 R 90 v 0 v 1. Given a deformed triangle {v 0,v 1,v desired 2 }, v2 can be computed similarly.

8 Scale-Free Construction Then, E v2 = v desired 2 v 2 2. And, E {v0,v 1,v 2 } = i=1,2,3 v desired i v i 2 Error for the entire mesh is sum of error for all triangles, expressed in matrix form as: E 1{v } = v T Gv G u + Bq = 0 Only q changes during manipulation. G can be pre-computed during registration and G 1 B during compilation.

9 Scale Adjustment doesn capture changes in scale and is handled here. First each triangle in the rest shape is fitted in the intermediate result, allowing rotation and translation. For a triangle {v 0,v 1,v 2 } in the intermediate result, we need to find a new triangle {v0 fitted,v1 fitted,v2 fitted } that is congruent to the corresponding triangle {v 0,v 1,v 2 } in the rest shape.

10 Fitting triangles The fitted triangle should minize E = f{v fitted,v 0 1 fitted,v 2 fitted } i=1,2,3 vi fitted v i 2. Approximate by first minimizing the error allowing uniform scaling. Like before, we have: v2 fitted = v0 fitted + x 01 v0 fitted v1 fitted + y 01 R 90 v0 fitted v1 fitted So the fitting functional is a quadratic in the four free variables w = (v 0 x fitted,v 0 y fitted,v 1 x fitted,v 1 y fitted )

11 Fitting triangles Minimize E f by setting partial derivaties over w to zero. de f dw = Fw + C = 0 F is fixed for a given mesh, so can be pre-computed and inverted during registration. C is defined by the result of and is computed during manipulation. Scale the fitted triangle obtained by solving this equation by a factor of v0 fitted v1 fitted / v 0 v 1 This makes the fitted triangle congruent to the corresponding triangle in the rest shape.

12 Generating final result Need to reconcile vertices of several triangles that correspond to the same mesh vertex. For a triangle {v 0,v 1,v 2 } and corresponding fitted triangle {v0 fitted,v1 fitted,v2 fitted }, a quadratic edge error function is defined: E = 2{v 0,v 1,v 2 } (i,j) {(0,1),(1,2),(2,0)} v i v j vi fitted vj fitted 2

13 Generating final result The optimal position for a vertex is some average of the positions desired by each triangle in which it appears. The error for the entire mesh is given by: E 2v = v T Hv + fv +c H is defined by the connectivity of the original mesh and, f and c are determined by the fitted triangles.

14 Generating final result To minimize E 2, setting partial derivatives over u to zero, we get: H u + Dq + f 0 = 0 H can be precomputed during registration and, H and D during compilation. f 0 is computed during manipulation using fitted triangles. Final result obtained by solving the last equation using pre-computed LU factorization of H.

15 Pentium III 1GHz processor, 756MB memory and Java implementation. Real-time performance. Interaction deterioriates starting at around 300 vertices. Fairly robust against uneven triangulation and irregularly spaced mesh.

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