Cse 554 lecture 9 extrinsic deformations
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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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CSE 554 Lecture 9: Extrinsic Deformations

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Cse 554 lecture 9 extrinsic deformations

CSE 554Lecture 9: Extrinsic Deformations

Fall 2012


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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