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