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An Algebraic Model for Parameterized Shape Editing
Martin Bokeloh, Stanford Univ.Michael Wand, Saarland Univ. & MPIHans-Peter Seidel, MPIVladlen Koltun, Stanford University
generating variations of individual shape
• Structure-aware deformation
Gal et al 2009.
Restricted to deformations with fixed topology
Kraevoy et al. 2008
generating variations of individual shape
• Structure-aware deformation• Inverse procedural modeling
Controllability: finding a production of a shape grammar that fits user constraints remains a difficult problem.
Bokeloh et al. 2010 Stava et al. 2010
generating variations of individual shape
• Structure-aware deformation• Inverse procedural modeling• Structure-preserved retargeting
Rely on user-provided constraints, and limited to axis-aligned resizing.
Lin et al. 2011
generating variations of individual shape
• Structure-aware deformation• Inverse procedural modeling• Structure-preserved retargeting• Pattern-aware shape deformation
Bokeloh et al. 2011
Pattern-aware Deformation Model• Calculus of variations:
dcru EEEEfE )(
User constraints
Elastic energy
Continuous patterns
Discrete patterns
)( min fE
Does not explicitly model the pattern structure of the object but rather uses elastic deformation to adjust patterns locally.
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.
Manipulating a single regular patternA regular pattern P(o, l, t)o - origin of the patternt - translational symmetryl - number of repetitions
o
t n=4
Manipulations
Change l
Change t
Parameterizing a structure consists of multiple regular patterns is not easy. (The key: relationships among intersecting patterns)
Algebraic Model = Regular patterns + link analysis
Decompose the entire input shape into regular patterns
Algebraic Model = Regular patterns + link analysis
Parameterize each regular pattern
Regular Patterns
Algebraic Model = Regular patterns + link analysis
Detect link relationships among regular patterns
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.
Algebraic Model = Regular patterns + link analysis
The complete shape is represented by a linear system.
Algebraic Model = Regular patterns + link analysis
The null space of the linear system defines the space of valid variations of the shape.
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:
Shape editing
pattern element closest to the selection point
Two pattern elements The diff
Automated visualization of degrees of freedom for test shapes
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