1 / 32

Non-rigid Deformation Techniques: Thin-Plate Spline, Free-Form Deformation, and Cage-Based Deformation

This review explores three non-rigid deformation techniques: Thin-Plate Spline, Free-Form Deformation, and Cage-Based Deformation, and their applications in image registration and interactive spatial deformation.

rachelbrown
Download Presentation

Non-rigid Deformation Techniques: Thin-Plate Spline, Free-Form Deformation, and Cage-Based Deformation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CSE 554Lecture 10: Extrinsic Deformations Fall 2013

  2. 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

  3. 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)

  4. Extrinsic Deformation • Applications • Registering contents between images and volumes • Interactive spatial deformation

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

  6. 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)

  7. 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

  8. 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]

  9. 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

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

  11. 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

  12. 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

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

  14. 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)

  15. 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)

  16. 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

  17. 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]: pre-computed “influence” of pi on p p f[p] qi pi

  18. 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

  19. 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

  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) • 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

  21. 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

  22. Free Form Deformation • A deformation example

  23. Free Form Deformation • Image registration • Embed the source in a lattice • Compute new lattice positions over the target • Manually, or solve it as an optimization problem (maximally matching images contents while minimizing distortion) • Deform each source pixel using FFD www.slicer.org

  24. 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)

  25. Cage-based Deformation • Warping the space by “blending” the deformation at the cage vertices • wi[p]: pre-computed “influence” of pi on p pi p qi f[p]

  26. 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

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

  28. 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

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

  30. Cage-based Deformation • Application: character animation

  31. Cage-based Deformation • Registration • Embed source in a cage • Compute new locations of cage vertices over the target • Minimizing some fitting and energy objectives • Deform source pixels using MVC • Not seen in literature yet… future work!

  32. 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)

More Related