1. CSE 554 Lecture 10: Extrinsic Deformations
Fall 2016

2. Review Non-rigid deformation
Intrinsic methods: deforming the boundary points
An optimization problem
Minimize shape distortion
Maximize fit
Example: Laplacian-based deformation
Source
Target
Before
After

3. Extrinsic Deformation
Given source and target point pairs (handles)
Compute deformation of any point in the plane or volume
Not just points on the boundary curve or surface

4. Extrinsic Deformation
Applications
Registering contents between images and volumes
Interactive animation

5. Techniques Thin-plate spline deformation Free form deformation
Cage-based deformation

6. Thin-Plate Spline A minimization problem
Minimizing distances between source and target points
Minimizing distortion of the space
There is a closed-form solution
Solving a linear system of equations

7. Thin-Plate Spline Input Output Source points: p1,…,pn
Corresponding target points: q1,…,qn
Output
A deformation function f[p] (p is any point) pi qi p
f[p]

8. 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 distortion is allowed

9. Thin-Plate Spline Fitting term
Minimizing sum of squared distances between deformed source points and target points

10. Thin-Plate Spline Distortion term Penalizes non-linear deformation:
The energy is zero when the deformation is a linear transformation
Translation, rotation, scaling, shearing

11. Thin-Plate Spline Finding the minimizer for
Uniquely exists, and has a closed form:
M: a global transformation
vi: translation vectors (one per source point)
Both M and vi are determined by pi,qi,
By solving a linear equation system
where

12. Thin-Plate Spline Result
At higher , the deformation is closer to an affine transformation
Credits: Sprengel et al, EMBS (1996)

13. Thin-Plate Spline Application: landmark-based image registration
Manual or automatic detection of landmarks and correspondences
Source
Target
Deformed source
Credits: Rohr et al, TMI (2001)

14. Thin-Plate Spline Application: landmark-based image registration
Manual or automatic detection of landmarks and correspondences
Source
Target
Source
Credits: Rohr et al, TMI (2001)

15. Thin-Plate Spline Pros Cons
Only requires scattered point correspondences
Closed-form solution makes it efficient to compute the deformation
Once the global transformation (M) and local vectors (vi) are solved
Cons
Solving M, vi could be time-consuming for large number of handles
Not for interactive deformation (e.g., user moving the handles in real-time)

16. Free Form Deformation Uses a control lattice that embeds the shape
Deforming the lattice points (control points) warps the embedded shape
Credits: Sederberg and Parry, SIGGRAPH (1986)

17. Free Form Deformation
Blending the deformation at the control points (handles)
Each deformed point is a weighted sum of deformed lattice points

18. Free Form Deformation Input Output Source control points: p1,…,pn
Target control points: q1,…,qn
Output
A deformation function f[p] for any point p within the space of the grid.
wi[p]: “influence” of pi on p (regardless of qi, so it can be pre- computed) p
f[p] pi qi

19. Free Form Deformation Desirable properties of the weights wi[p]
Decreases with distance from p 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 pi qi p
f[p]

20. 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 s t
p0,1
p1,1
p2,1
p3,1
p0,0
p1,0
p2,0
p3,0

21. 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

22. Free Form Deformation Finding weights (2D)
Weight distribution for one control point (max at that control point):
p3,2
p3,1
p0,2
p1,1
p3,0
p0,1
p2,0
p1,0
p0,0

23. Free Form Deformation
A deformation example

24. Free Form Deformation Image registration Embed the source in a lattice
Compute new lattice positions over the target
Manually or automatically
Deform each source pixel using FFD

25. Free Form Deformation Pros Cons
Fast updates when control points are moved
Cons
Too many control points (a lot of them are interior to the shape)
Cartesian grid is too inflexible for complex shapes

26. 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)

27. Cage-based Deformation
“Blending” the deformation at the cage vertices
wi[p]: pre-computed “influence” of pi on p
More challenging to compute than FFD: not regular lattice structure pi p qi
f[p]

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

29. Cage-based Deformation
Finding weights (2D)
A simple case: the cage is a triangle
The weights are unique (3 eqs, 3 vars)
wi are known as the barycentric coordinates of p p1 p p3 p2

30. Cage-based Deformation
Finding weights (2D)
The harder case: the cage is an arbitrary (possibly concave) polygon
The weights are not unique (“generalized barycentric coordinates”)
A good choice: Mean Value Coordinates (MVC) [Floater, 2003]
pi-1 pi αi
αi+1 p
pi+1

31. Cage-based Deformation
Finding weights (2D)
Weight distribution of one cage vertex in MVC: pi

32. Cage-based Deformation
MVC be extended to 3D [Ju, 2005; Floater, 2005]
Other types of coordinates: Harmonic coordinates; Green coordinates; etc.
Application: character animation

33. Cage-based Deformation
Registration
Embed source in a cage
Compute new locations of cage vertices over the target
Deform source pixels using MVC or other generalized barycentric coordinates

34. Cage-based Deformation
Pros
Fast update when control points are moving
Fewer control points and better fitting to shape than FFD
Cons
Constructing the cage is mostly a manual process
Although there is some recent progress in automation
Sacht et al., Siggraph Asia 2015

35. Further Readings Thin-plate spline deformation Free form 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)