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Lecture 9: Extrinsic Deformations' - ina

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Review

Source

- Non-rigid deformation
- Intrinsic methods: deforming the boundary points
- An optimization problem
- Minimize shape distortion
- Maximize fit
- Example: Laplacian-based deformation

Target

Before

After

Extrinsic Deformation

- Computing deformation of each point in the plane or volume
- Not just points on the boundary curve or surface

Credits: Adams and Nistri, BMC Evolutionary Biology (2010)

Extrinsic Deformation

- Applications
- Registering contents between images and volumes
- Interactive spatial deformation

Techniques

- Thin-plate spline deformation
- Free form deformation
- Cage-based deformation

Thin-Plate Spline

- Given corresponding source and target points
- Computes a spatial deformation function for every point in the 2D plane or 3D volume

Credits: Sprengel et al, EMBS (1996)

Thin-Plate Spline

- A minimization problem
- Minimizing distances between source and target points
- Minimizing distortion of the space (as if bending a thin sheet of metal)
- There is a closed-form solution
- Solving a linear system of equations

Thin-Plate Spline

- Input
- Source points: p1,…,pn
- Target points: q1,…,qn
- Output
- A deformation function f[p] for any point p

pi

qi

p

f[p]

Thin-Plate Spline

- Minimization formulation
- Ef: fitting term
- Measures how close is the deformed source to the target
- Ed: distortion term
- Measures how much the space is warped
- : weight
- Controls how much non-rigid warping is allowed

Thin-Plate Spline

- Fitting term
- Minimizing sum of squared distances between deformed source points and target points

Thin-Plate Spline

- Distortion term
- Minimizing a physical bending energy on a metal sheet (2D):
- The energy is zero when the deformation is affine
- Translation, rotation, scaling, shearing

Thin-Plate Spline

- Finding the minimizer for
- Uniquely exists, and has a closed form:
- M: an affine transformation matrix
- vi: translation vectors (one per source point)
- Both M and vi are determined by pi,qi,

where

Thin-Plate Spline

- Result
- At higher , the deformation is closer to an affine transformation

Credits: Sprengel et al, EMBS (1996)

Thin-Plate Spline

- Application: image registration
- Manual or automatic feature pair detection

Source

Target

Deformed source

Credits: Rohr et al, TMI (2001)

Thin-Plate Spline

- Advantages
- Smooth deformations, with physical analogy
- Closed-form solution
- Few free parameters (no tuning is required)
- Disadvantages
- Solving the equations still takes time (hence cannot perform “Interactive” deformation)

Free Form Deformation

- Uses a control lattice that embeds the shape
- Deforming the lattice points warps the embedded shape

Credits: Sederberg and Parry, SIGGRAPH (1986)

Free Form Deformation

- Warping the space by “blending” the deformation at the control points
- Each deformed point is a weighted sum of deformed lattice points

Free Form Deformation

- Input
- Source lattice points: p1,…,pn
- Target lattice points: q1,…,qn
- Output
- A deformation function f[p] for any point p in the lattice grid:
- wi[p]: determined by relative position of p with respect to pi

p

f[p]

qi

pi

Free Form Deformation

- Desirable properties of the weights wi[p]
- Greater when p is closer to pi
- So that the influence of each control point is local
- Smoothly varies with location of p
- So that the deformation is smooth
- So that f[p] = wi[p] qi is an affine combination of qi
- So that f[p]=p if the lattice stays unchanged

p

f[p]

qi

pi

s

Free Form Deformation- Finding weights (2D)
- Let the lattice points be pi,j for i=0,…,k and j=0,…,l
- Compute p’s relative location in the grid (s,t)
- Let (xmin,xmax), (ymin,ymax) be the range of grid

p0,2

p1,2

p2,2

p3,2

p

p0,1

p1,1

p2,1

p3,1

p0,0

p1,0

p2,0

p3,0

Free Form Deformation

- Finding weights (2D)
- Let the lattice points be pi,j for i=0,…,k and j=0,…,l
- Compute p’s relative location in the grid (s,t)
- The weight wi,j for lattice point pi,j is:
- i,j: importance of pi,j
- B: Bernstein basis function:

p0,2

p1,2

p2,2

p3,2

p

t

p0,1

p1,1

p2,1

p3,1

p0,0

p1,0

p2,0

p3,0

s

Free Form Deformation

- Finding weights (2D)
- Weight distribution for one control point (max at that control point):

p3,2

p3,1

p1,1

p0,2

p3,0

p0,1

p2,0

p1,0

p0,0

Free Form Deformation

- A deformation example

Free Form Deformation

- Registration
- Embed the source in a lattice
- Compute new lattice positions over the target
- Fitting term
- Distance between deformed source shape and the target shape
- Agreement of the deformed image content and the target image
- Distortion term
- Thin-plate spline energy
- Non-folding constraint
- Local rigidity constraint (to prevent stretching)

Free Form Deformation

- Registration example

Deformed w/o rigidity

Source

Deformed with rigidity

Target

Credits: Loeckx et al, MICCAI (2004)

Free Form Deformation

- Advantages
- Smooth deformations
- Easy to implement (no equation solving)
- Efficient and localized controls for interactive editing
- Can be coupled with different fitting or energy objectives
- Disadvantages
- Too many lattice points in 3D

Cage-based Deformation

- Use a control mesh (“cage”) to embed the shape
- Deforming the cage vertices warps the embedded shape

Credits: Ju, Schaefer, and Warren, SIGGRAPH (2005)

Cage-based Deformation

- Warping the space by “blending” the deformation at the cage vertices
- wi[p]: determined by relative position of p with respect to pi

pi

p

qi

f[p]

Cage-based Deformation

- Finding weights (2D)
- Problem: given a closed polygon (cage) with vertices pi and an interior point p, find weights wi[p] such that:
- 1)
- 2)

pi

p

Cage-based Deformation

- Finding weights (2D)
- A simple case: the cage is a triangle
- The weights are unique, and are the barycentric coordinates of p

p1

p

p3

p2

Cage-based Deformation

- Finding weights (2D)
- The harder case: the cage is an arbitrary (possibly concave) polygon
- The weights are not unique
- A good choice: Mean Value Coordinates (MVC)
- Can be extended to 3D

pi-1

pi

αi

αi+1

p

pi+1

Cage-based Deformation

- Application: character animation
- Using improved weights: Harmonic Coordinates

Cage-based Deformation

- Registration
- Embed source in a cage
- Compute new locations of cage vertices over the target
- Minimizing some fitting and energy objectives
- Not seen in literature yet… future work!

Cage-based Deformation

- Advantages over free form deformations
- Much smaller number of control points
- Flexible structure suitable for organic shapes
- Disadvantages
- Setting up the cage can be time-consuming
- The cage needs to be a closed shape
- Not as flexible as point handles

Further Readings

- Thin-plate spline deformation
- “Principal warps: thin-plate splines and the decomposition of deformations”, by Bookstein (1989)
- “Landmark-Based Elastic Registration Using Approximating Thin-Plate Splines”, by Rohr et al. (2001)
- Free form deformation
- “Free-Form Deformation of Solid Geometric Models”, by Sederberg and Parry (1986)
- “Extended Free-Form Deformation: A sculpturing Tool for 3D Geometric Modeling”, by Coquillart (1990)
- Cage-based deformation
- “Mean value coordinates for closed triangular meshes”, by Ju et al. (2005)
- “Harmonic coordinates for character animation”, by Joshi et al. (2007)
- “Green coordinates”, by Lipman et al. (2008)

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