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CSE554 Extrinsic Deformations Slide 1
CSE 554
Lecture 9: Extrinsic Deformations
CSE 554
Lecture 9: Extrinsic Deformations
Fall 2012
CSE554 Extrinsic Deformations Slide 2
ReviewReview
• 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
CSE554 Extrinsic Deformations Slide 3
Extrinsic DeformationExtrinsic 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)
CSE554 Extrinsic Deformations Slide 4
Extrinsic DeformationExtrinsic Deformation
• Applications
– Registering contents between images and volumes
– Interactive spatial deformation
CSE554 Extrinsic Deformations Slide 5
TechniquesTechniques
• Thin-plate spline deformation
• Free form deformation
• Cage-based deformation
CSE554 Extrinsic Deformations Slide 6
Thin-Plate SplineThin-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)
CSE554 Extrinsic Deformations Slide 7
Thin-Plate SplineThin-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
CSE554 Extrinsic Deformations Slide 8
Thin-Plate SplineThin-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]
CSE554 Extrinsic Deformations Slide 9
Thin-Plate SplineThin-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
E Ef Ed
CSE554 Extrinsic Deformations Slide 10
Thin-Plate SplineThin-Plate Spline
• Fitting term
– Minimizing sum of squared distances between deformed source points and target points
Ef i1
n fpi qi2
CSE554 Extrinsic Deformations Slide 11
Thin-Plate SplineThin-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
Ed 2f
x2
2
22f
x y
2
2f
y2
2 2
x y
CSE554 Extrinsic Deformations Slide 12
Thin-Plate SplineThin-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,
E Ef Ed
fp M p i1
n
p pi vir r2 Logr
2 1 1 2
0 .5
1 .0
1 .5
2 .0
2 .5
where
r
CSE554 Extrinsic Deformations Slide 13
Thin-Plate SplineThin-Plate Spline
• Result
– At higher , the deformation is closer to an affine transformation
0
0.001 0.1Credits: Sprengel et al, EMBS (1996)
CSE554 Extrinsic Deformations Slide 14
Thin-Plate SplineThin-Plate Spline
• Application: image registration
– Manual or automatic feature pair detection
Source Target Deformed source
Credits: Rohr et al, TMI (2001)
CSE554 Extrinsic Deformations Slide 15
Thin-Plate SplineThin-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)
CSE554 Extrinsic Deformations Slide 16
Free Form DeformationFree Form Deformation
• Uses a control lattice that embeds the shape
• Deforming the lattice points warps the embedded shape
Credits: Sederberg and Parry, SIGGRAPH (1986)
CSE554 Extrinsic Deformations Slide 17
Free Form DeformationFree Form Deformation
• Warping the space by “blending” the deformation at the control points
– Each deformed point is a weighted sum of deformed lattice points
CSE554 Extrinsic Deformations Slide 18
Free Form DeformationFree 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
piqi
p f[p]
fp i1
n
wip qi
CSE554 Extrinsic Deformations Slide 19
Free Form DeformationFree Form Deformation
• Desirable properties of the weights w i[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 i1
nwip pi
1 i1
nwip
piqi
p f[p]
CSE554 Extrinsic Deformations Slide 20
Free Form DeformationFree 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
p0,0
p0,1
p0,2
p1,0
p1,1
p1,2
p2,0
p2,1
p2,2
p3,0
p3,1
p3,2
p
s
t
s px xmin
xmax xmin
t py ymin
ymax ymin
CSE554 Extrinsic Deformations Slide 21
Free Form DeformationFree 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:Biks k
i si 1 ski
p0,0
p0,1
p0,2
p1,0
p1,1
p1,2
p2,0
p2,1
p2,2
p3,0
p3,1
p3,2
p
s
t
ui,jp i,j Biks Bjlt
wi,jp ui,jp i0
k j0l ui,jp
CSE554 Extrinsic Deformations Slide 22
Free Form DeformationFree Form Deformation
• Finding weights (2D)
– Weight distribution for one control point (max at that control point):
p0,0
p0,1
p0,2
p1,0
p1,1
p2,0
p3,0
p3,1
p3,2
w1,1p
CSE554 Extrinsic Deformations Slide 23
Free Form DeformationFree Form Deformation
• A deformation example
CSE554 Extrinsic Deformations Slide 24
Free Form DeformationFree 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)
CSE554 Extrinsic Deformations Slide 25
Free Form DeformationFree Form Deformation
• Registration example
Credits: Loeckx et al, MICCAI (2004)
Source
Target
Deformed w/o rigidity
Deformed with rigidity
CSE554 Extrinsic Deformations Slide 26
Free Form DeformationFree 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
CSE554 Extrinsic Deformations Slide 27
Cage-based DeformationCage-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)
CSE554 Extrinsic Deformations Slide 28
Cage-based DeformationCage-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
fp i1
n
wip qip
f[p]
pi
qi
CSE554 Extrinsic Deformations Slide 29
Cage-based DeformationCage-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)
1 i1
nwip
p i1
nwip pi pi
p
CSE554 Extrinsic Deformations Slide 30
Cage-based DeformationCage-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
p2
p3
pw1p
Areap,p2,p3
Areap1,p2,p3
CSE554 Extrinsic Deformations Slide 31
• 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
Cage-based DeformationCage-based Deformation
pi
p
αi
αi+1
uip Tani 2 Tani1 2pi p
wip uip i1n uip
pi-1
pi+1
CSE554 Extrinsic Deformations Slide 32
Cage-based DeformationCage-based Deformation
• Finding weights (2D)
– Weight distribution of one cage vertex in MVC:
pi
wip
CSE554 Extrinsic Deformations Slide 33
Cage-based DeformationCage-based Deformation
• Application: character animation
– Using improved weights: Harmonic Coordinates
CSE554 Extrinsic Deformations Slide 34
Cage-based DeformationCage-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!
CSE554 Extrinsic Deformations Slide 35
Cage-based DeformationCage-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
CSE554 Extrinsic Deformations Slide 36
Further ReadingsFurther 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)