Image Deformation Using Moving Least Squares

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Image Deformation Using Moving Least Squares

• Scott Schaefer, Travis McPhail, Joe Warren
• SIGGRAPH 2006

Presented by Yun-Feng Chou
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Input
Affine
Similarity
Rigid
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Outline

• Introduction
• Previous work
• MLS with points
• MLS with line segments
• Conclusions

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Introduction

• Image deformation
• Animation, morphing, medical imaging
• Controlled by handles points, lines
• Function f
• Interpolation f(pi)qi (handles)
• Smoothness smooth deformations
• Identity qipi?f(x)x
• Similar to scattered data interpolation

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Previous Work

• Grid/Polygon handles
• Free-form deformations Sederberg and Parry 1986
  Lee et al. 1995
• Require aligning grid lines with image features
• Line handles
• Beier and Neely 1992
• Undesirable folding
• Point handles
• RBF
• Thin-plate splines Bookstein 1989
• Local non-uniform scaling and shearing
• As-rigid-as-possible deformations
• Igarashi et al. 2005
• Solve on the whole triangulation
• Discontinuities
• 2D shape deformation using nonlinear least
  squares optimization
• Weng et al. 2006

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As-Rigid-As-Possible Shape Manipulation
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Thin-plate spline
As-rigid-as-possible
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Previous Work

• Grid/Polygon handles
• Free-form deformations Sederberg and Parry 1986
  Lee et al. 1995
• Require aligning grid lines with image features
• Line handles
• Beier and Neely 1992
• Undesirable folding
• Point handles
• RBF
• Thin-plate splines Bookstein 1989
• Local non-uniform scaling and shearing
• As-rigid-as-possible deformations
• Igarashi et al. 2005
• Solve on the whole triangulation
• Discontinuities
• 2D shape deformation using nonlinear least
  squares optimization
• Weng et al. 2006

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Moving Least Squares Deformation

• Build image deformations based on handles (pi,qi)
• Interpolation
• Smoothness
• Identity
• Given a point v, solve for the best lv(x)
• Deformation function f(v)lv(v)
• Locally defined --- MOVING Least Squares
• Satisfy the three properties

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Affine Transform

• Affine transform lv(x)xMT
• Translation can be removed Tq-pM
• So, lv(x)(x-p)Mq
• The new cost function
• M could be different class of transformations

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Affine Deformations

• Solution
• The deformation function
• Aj can be precomputed
• Contains non-uniform scaling and shear

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Similarity Deformations

• Only include translation, rotation, and uniform
  scaling
• Remove shear from deformation

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Similarity Deformations (Cont.)

• A special subset of affine transforms
• Translation
• Rotation
• Constraints Uniform-scaling
• Requirement MTM?2I
• Define , where M2M1-
• Cost function (Least squares problem) still
  quadratic in M1
• where (x,y)-(-y,x)

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Similarity Deformations (Cont.)

• Solution for matrix M
• where
• Solution for deformation function
• where
• Property
• Preserves angles better than affine deformations
• Local scaling can hurt realism

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Rigid Deformations

• Deformations should not include scaling and shear
• Alexa 2000 Igarashi et al. 2005
• Best rigid transformation can be found from best
  similarity transformation
• Remove local uniform scaling MTMI

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Rigid Deformations (Cont.)

• Solution for matrix M
• Solution for deformation function
• where , and Ai is as in similarity
  deformations.
• Limited precomputation

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Some Rigid Deformation Results
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Deformation with Curves

• Handles are control curves instead of control
  points
• Cost function
• T still can be removed

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Affine Lines

• Represent line segments as matrix
  products
• Cost function
• Minimizer

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Affine Lines (Cont.)

• Deformation

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Similarity Lines

• Cost function
• Minimizer

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Similarity Lines (Cont.)

• Deformation

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Rigid Lines

• Derive from similarity lines
• Deformation

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Implementation

• Pixel by pixel --- expensive
• Deformation on a downsampled grid
• Fill quads using bilinear interpolation

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Timing
Grid size 100x100
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Limitation
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Conclusions

• Moving Least Squares
• 2x2 linear system at each grid point
• Real-time
• Similarity and rigid transformations
• More realistic results
• Extension line segments
• Closed-form expressions

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Conclusions (Cont.)

• Future Work
• Adding topological information
• Generalizing to 3D to deform surfaces
• Handles can be any curves