image deformation using moving least squares scott schaefer, travis mcphail, joe warren siggraph...
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Image Deformation Using Moving Least Squares
Scott Schaefer, Travis McPhail, Joe Warren
SIGGRAPH 2006
Presented by Nirup Reddy
Input Affine Similarity Rigid
Outline
Introduction Previous work MLS with points MLS with line segments Conclusions
Introduction
Image deformation Animation, morphing, medical imaging… Controlled by handles: points, lines… Function f
Interpolation: f(pi)=qi (handles)
Smoothness ~ smooth deformations Identity: qi=pif(x)=x
Similar to scattered data interpolation
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
Previous Work
As-Rigid-As-Possible Shape Manipulation
Thin-plate spline As-rigid-as-possible
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
i
ii ppF 2|ˆ)(|
i
iii ppFw 2|ˆ)(|
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
i
iivi qplw 2|)(| 2||
1
vpw
ii
Affine Transform Affine transform: lv(x)=xM+T Translation can be removed: T=q*-p*M
So, lv(x)=(x-p*)M+q*
The new cost function:
M could be different class of transformations
i i
i ii
w
pwp*
i i
i ii
w
qwq*
2|ˆˆ|
iiii qMpw
*ˆ ppp ii *ˆ qqq ii
Affine Deformations Solution
The deformation function
Aj can be precomputed
Contains non-uniform scaling and shear
j
jTjj
iii
Ti qpwpwpM ˆˆ)ˆˆ( 1
*ˆ)( qqAvfj
jja Ti
iii
Tij ppwppvA ˆ)ˆˆ)(( 1
*
Similarity Deformations
Only include translation, rotation, and uniform scaling
Remove shear from deformation
Similarity Deformations
Only include translation, rotation, and uniform scaling
Remove shear from deformation
Similarity Deformations (Cont.) A special subset of affine transforms
Translation Rotation Constraints: Uniform-scaling
Requirement: MTM=λ2I Define , where M2=M1
┴ Cost function (Least squares problem) still
quadratic in M1
where (x,y)┴=(-y,x)
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
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 MTM=I
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 MTM=I
Rigid Deformations (Cont.) Solution for matrix M
Solution for deformation function
where , and Ai is as in similarity deformations.
Limited precomputation
Example: Actual Image
Deformation’s
Affine Similarity Rigid
Comparison:
Affine Similarity RigidNon-uniform
scaling and shearLocal scaling can
often lead to undesirable
Deformations But Preserves
angles
Deformation is quiterealistic
Computation time very low abt 1.5ms
Highest was computed as 3.4ms
Highest was computed 3.8ms
Some Rigid Deformation Results
Deformation with Curves
Handles are control curves instead of control points
Cost function
T still can be removed
Limitations:
May be difficult to evaluate for arbitrary functions
We restrict these functions to be line segments and derive closed-form solutions for the deformations in terms of the end-points of these segments
Affine Lines
Represent line segments as matrix products
Cost function
Minimizer
Affine Lines (Cont.)
Deformation
Similarity Lines
Cost function
Minimizer
Similarity Lines (Cont.)
Deformation
Rigid Lines
Derive from similarity lines
Deformation
Implementation
Pixel by pixel --- expensive Deformation on a downsampled
grid Fill quads using bilinear
interpolation
Timing
Grid size: 100x100
Limitation: Foldbacks
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
Applications Of MLS: Very powerful tool in
computer graphics Animation Medical
Imaging Image Warping Image Morphing
Conclusions (Cont.)
Future Work Adding topological information Generalizing to 3D to deform surfaces Handles can be any curves