1. Morphing and Shape Processing
Tutorial 9
© Maks Ovsjanikov, Alex & Michael Bronstein
tosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapes
Stanford University, Winter 2009 1

2. Planar Intersection-Free Morphing:
Outline
Planar Intersection-Free Morphing:
How to morph tilings injectively, Floater and Gotsman, J. of Computational and Applied Mathematics, 1999
3D Morphing
Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007

3. Problem: How to morph tilings injectively, Floater and Gotsman, 1999
Given 2 planar tilings with the same connectivity and boundary, morph between them, preserving mesh structure.
Linear Morph:
Proposed Morph:

4. How to morph tilings injectively, Floater and Gotsman, 1999
General Approach:
Express each internal vertex as a convex combination of its neighbors:
Note that given the boundary vertices, each represents a system of linear equations.

5. How to morph tilings injectively, Floater and Gotsman, 1999
General Approach:
Express each internal vertex as a convex combination of its neighbors:
Given the boundary vertices, each represents a system of linear equations.

6. How to morph tilings injectively, Floater and Gotsman, 1999
General Approach:
Given the boundary vertices, each represents a system of linear equations.
Floater proved that given any convex :
only if , then the above system has a unique solution
Further, the solution obtained this way is guaranteed to be non-intersecting!
M. S. Floater, Parametrization and smooth approximation of surface triangulations,
Comp. Aided Geom. Design 14 (1997), 231{250.

7. Main Idea: How to morph tilings injectively, Floater and Gotsman, 1999
Interpolate coefficients linearly:
Let:
Note that are still convex
Find intermediate vertices by solving:
The intermediate meshes are guaranteed to exist and be valid triangulations with the same connectivity.

8. Results: How to morph tilings injectively, Floater and Gotsman, 1999
Linear Morph
Convex Morph

9. Results: How to morph tilings injectively, Floater and Gotsman, 1999
Linear Morph
Convex Morph

10. Need a common static boundary (heuristics to overcome).
How to morph tilings injectively, Floater and Gotsman, 1999
Conclusions:
Blending convex weights instead of vertex coordinates guarantees an intersection-free triangulation.
Fast and efficient way to generate different drawings of a planar graph.
Limitations:
Need a common static boundary (heuristics to overcome).
Only works in 2D (convex combinations are more intricate).

11. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Problem:
Given 2 shapes in different poses, find as isometric as possible morphing
Extrapolation: continue the deformation in a natural way.
Constraint shape deformation
Deformation transfer from one shape to another

12. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
General Approach:
Extrapolation
Interpolation
Explore the space of all shapes. Deformations are geodesics in the shape space.
Crucial question: What is the appropriate metric in this space?

13. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Approach Overview:
Define 2 metrics in shape-space:
Rigid
Isometric
Given these metrics, find geodesics in shape space by numerical optimization of the path, with respect to the metric.
Boundary Value Problem: Fix initial and final position, interpolate.
Initial Value Problem: Fix the initial position and the motion direction. Extrapolate.

14. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Defining Metrics:
On a Riemannian manifold, metric g is an inner product between
tangent vectors:
In the Shape Space, each point is a shape, and the tangent space
is the set of vector fields on S.
Each tangent vector assigns a deformation vector
To each point

15. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Defining Metrics:
Metric g is an inner product on tangent vectors (deformation fields):
Design Paradigm.
Given a property to be preserved, translate it to deformation fields.
Deformation fields, preserving this property should have small norm.
Define inner product so that is small if
is good. Define a score for each deformation field.

16. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Rigid Motion Metric:
Well known result:
A deformation of a shape is rigid if and only if:
where is the translation vector (uniform over the shape)
and is the rotation vector.
Conversely:
Given a deformation field , can find optimal and , by solving:
which is quadratic in and . Leads to a linear system.

17. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Rigid Motion Metric:
Given a deformation field , find optimal and , by solving:
Define the modified deformation field
Rigid Motion Inner Product:
So, if and only if
In general, want: , so add a regularization:
where is small.

18. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Isometric Motion Metric:
Could also try to subtract the part that preserves isometry, but difficult to
express Killing fields explicitly. Try to be “as isometric as possible.”
Recall that the deformation is isometric if an only if for every edge (p, q):
So define:
Implying is isometric. Penalize non-isometric fields.
Regularize:

19. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Defining Lengths:
By defining the metric, we have a way to measure the lengths of curves:
If two shapes are points in shape space, morph between them by
finding the geodesic that connects them. Minimize the length.
Boundary Value Problem:
Given shapes and find the geodesic connecting them given
by shapes and their deformations:
minimizing:

20. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Boundary Value Problem:
Given shapes and find the geodesic connecting them given
by shapes and their deformations:
minimizing:
Note: once shapes are fixed, the deformations and metrics
are fixed. Thus, minimization is done over vertices of
General strategy: Start with coarse versions of and Then:
- Refine in Time: add an intermediate shape
- Refine in Space: increase the resolution of all shapes.
- Re-optimize: deform the vertices of to minimize

21. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Boundary Value Problem:
Minimize:
General strategy: Start with coarse versions of and
Refinement in Time: add an intermediate shape :
- Linearly blend between and
- For each vertex = + 1 2

22. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Boundary Value Problem:
Minimize:
General strategy: Start with coarse versions of and
Refinement in Space: add detail (vertices) to the meshes.
- Add detail to the boundary shapes and
- Propagate to the intermediate meshes. Need a way to propagate detail while preserving the deformations.
- For each new vertex, that was added, project it onto and find the face that it projects to. Find its barycentric coordinates in that face.
- Also, store the coordinate in the normal direction:

23. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Boundary Value Problem:
Minimize:
General strategy: Start with coarse versions of and
Refinement in Space: add detail (vertices) to the meshes.
- For each new vertex, find its local coordinates on
- For each intermediate shape blend influences:
- where is the transfer of the new from to

24. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Boundary Value Problem:
Minimize:
Pipeline:

25. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Boundary Value Problem:
Minimize:
Optimization:
is a non-linear problem in Use a non-linear solver.
Need the gradient. Depends on the choice of metric. Non-trivial.
In practice, use LBFGS (limited memory), and can be expensive.

26. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Initial Value Problem:
Suppose we want to extrapolate: continue the geodesic path past the final pose or before the original. In that case, we get an IVP.
Formulation:
Suppose we want to minimize:
Beltrami Identity (calculus of variations):
The minimum u of satisfies:

27. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Initial Value Problem:
The minimzer X(t) of must satisfy:
A system of Ordinary Differential Equations. Allows us to
advance in time.
At first is given, so we can find that are preserved.
Then at each step, find by minimizing:

28. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Initial Value Problem:
At each step, find by minimizing:
Again, a non-linear problem in Need a multi-resolution approach.
Refinement in Space
Progressively increase the resolution of , and
Propagate details to the already computed path
Refine the solution, by resolving for with an initial guess.

29. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Initial Value Problem:
Pipeline:

30. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Applications:
Shape Interpolation and Extrapolation:
Blue shapes are input poses. Green – interpolated, pink – extrapolated.

31. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Applications:
Constraint Deformation:
Suppose, position of some points has been fixed. Find the shortest path joining with a shape such that
Add a term to the energy:

32. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Applications:
Shape Exploration:
Map a pre-defined set of shapes to a region in 2D. Edge length is length of geodesic path. Find a Delaunay triangulation. Refine edges, if time refinement was not necessary. Store all the refined meshes. Offline.

33. Geometric Modeling in Shape Space Kilian M. et al., SIGGRAPH, 2007
Conclusions:
Defining metric on the space of shapes, allows to measure geodesics.
Framework for analyzing Deformation Fields.
Multiresolution numerical method for finding shortest paths.
Plausible results with high degrees of freedom.
Many possible extensions
Limitations:
Very slow
Requires full knowledge of correspondences, across two shapes.