An Algebraic Model for Parameterized Shape Editing

1. An Algebraic Model for Parameterized Shape Editing Martin Bokeloh, Stanford Univ. Michael Wand, Saarland Univ. & MPI Hans-Peter Seidel, MPI VladlenKoltun, Stanford University

2. generating variations of individual shape • Structure-aware deformation Kraevoy et al. 2008 Gal et al 2009. Restricted to deformations with fixed topology

3. generating variations of individual shape • Structure-aware deformation • Inverse procedural modeling Stava et al. 2010 Bokeloh et al. 2010 Controllability: finding a production of a shape grammar that fits user constraints remains a difficult problem.

4. generating variations of individual shape • Structure-aware deformation • Inverse procedural modeling • Structure-preserved retargeting Lin et al. 2011 Rely on user-provided constraints, and limited to axis-aligned resizing.

5. generating variations of individual shape • Structure-aware deformation • Inverse procedural modeling • Structure-preserved retargeting • Pattern-aware shape deformation Bokeloh et al. 2011

6. Pattern-aware Deformation Model • Calculus of variations: User constraints Elastic energy Continuous patterns Discrete patterns Does not explicitly model the pattern structure of the object but rather uses elastic deformation to adjust patterns locally.

9. Goal • Parameterize an input 3D structure composed of regular patterns so that high-level shape editing that adapts the structure of the shape while maintaining its global characteristics can be supported.

10. Manipulating a single regular pattern A regular pattern P(o, l, t) o - origin of the pattern t - translational symmetry l - number of repetitions o n=4 t Manipulations Change l Change t

11. Parameterizing a structure consists of multiple regular patternsis not easy. (The key: relationships among intersecting patterns)

12. Algebraic Model = Regular patterns + link analysis Decompose the entire input shape into regular patterns

13. Algebraic Model = Regular patterns + link analysis Parameterize each regular pattern

14. Regular Patterns

15. Algebraic Model = Regular patterns + link analysis Detect link relationships among regular patterns

16. Link constraints – pattern constraints • (1-1)-interaction, line to line patch: • Collinear: the overlapping interval. • Intersect: the intersection point. • (1-2)-interaction, line to area patch: • Coplanar: the overlapping interval. • Intersect: the intersection point. • (2-2)-interaction, area to area patch: • Coplanar: the intersection points of the boundaries. • Intersect: (1-1)-interaction . • (0-1)- and (0-2)-interactions with rigid patches: • link the origin of the rigid pattern to the intersection line or surface.

17. Algebraic Model = Regular patterns + link analysis The complete shape is represented by a linear system.

18. Algebraic Model = Regular patterns + link analysis The null space of the linear system defines the space of valid variations of the shape.

19. Shape editing Interactive Constraints: the user selects a pattern element and drags it to a specific target point y. Difference constraints: The user selects two pattern elements , and specifies their difference vector. Regularization constraints: aim to keep the original values of the length variables. Objective function: pattern element closest to the selection point Two pattern elements The diff

21. Automated visualization of degrees of freedom for test shapes

29. Limitation • restricted to translational regular pattern • can only handle rigidly symmetric parts, ruling out organic shapes • not consider maintaining irregularity and global symmetries. • Can not handle highly detailed geometry with many interleaving patterns