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CSE 554 Lecture 9: Extrinsic Deformations. Fall 2012. 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.

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review
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
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 deformation1
Extrinsic Deformation
  • Applications
    • Registering contents between images and volumes
    • Interactive spatial deformation
techniques
Techniques
  • Thin-plate spline deformation
  • Free form deformation
  • Cage-based deformation
thin plate spline
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 spline1
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 spline2
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 spline3
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 spline4
Thin-Plate Spline
  • Fitting term
    • Minimizing sum of squared distances between deformed source points and target points
thin plate spline5
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 spline6
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 spline7
Thin-Plate Spline
  • Result
    • At higher , the deformation is closer to an affine transformation

Credits: Sprengel et al, EMBS (1996)

thin plate spline8
Thin-Plate Spline
  • Application: image registration
    • Manual or automatic feature pair detection

Source

Target

Deformed source

Credits: Rohr et al, TMI (2001)

thin plate spline9
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
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 deformation1
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 deformation2
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 deformation3
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

free form deformation4

t

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 deformation5
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 deformation6
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 deformation7
Free Form Deformation
  • A deformation example
free form deformation8
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 deformation9
Free Form Deformation
  • Registration example

Deformed w/o rigidity

Source

Deformed with rigidity

Target

Credits: Loeckx et al, MICCAI (2004)

free form deformation10
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
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 deformation1
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 deformation2
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 deformation3
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 deformation4
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 deformation5
Cage-based Deformation
  • Finding weights (2D)
    • Weight distribution of one cage vertex in MVC:

pi

cage based deformation6
Cage-based Deformation
  • Application: character animation
    • Using improved weights: Harmonic Coordinates
cage based deformation7
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 deformation8
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
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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