Title Text
Curvature regularization with multigrid convergent estimator
Daniel Martins Antunes
Université Savoie Mont Blanc, LAMA
CoMeDic - ESIEE, Paris.
January 15, 2019
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Goal
Use curvature regularization in image processing tasks, i.e, inpainting, segmentation, stero.
segmentation
How?
Minimizing the energy functional
u⋆=argus∈Ωmin∫Ω∣∣u−us∣∣2dx+∫∂usα+βκ2ds
\displaystyle u^\star = arg \min_{u_s \in \Omega}\int_{\Omega}{ || u - u_s ||^2 dx } + \int_{\partial u_s}{ \alpha + \beta \kappa^2 ds}
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Data term
Data + Perimeter term
Data + Curvature term
[El-Zehiry, 2010]
Why curvature?
Completion property
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Why is it challenging?
u⋆=argus∈Ωmin∫Ω∣∣u−us∣∣2dx+∫∂usα+βκ2ds
\displaystyle u^\star = arg \min_{u_s \in \Omega}\int_{\Omega}{ || u - u_s ||^2 dx } + \int_{\partial u_s}{ \alpha + \beta \kappa^2 ds}
Non-convex term
Difficult to optimize
Second order term. Should be careful with discretization scheme
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
MDCA + Graph-Cut
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
MDCA + Graph-Cut
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
MDCA + Graph-Cut
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
MDCA + Graph-Cut
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
II + LP Relaxation
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Grid cell representation
Binary variables for pixels
xp,xe∈Ωminxp∈Ω∑(up−xp)2+xe∈Ω∑κ^II2(xe)⋅xe
\displaystyle \min_{ x_p,x_e \in \Omega}{ \sum_{x_p \in \Omega}{ (u_p - x_p)^2} + \sum_{x_e \in \Omega}{ \hat{\kappa}_{II}^2(x_e) \cdot x_e} }
(xp∈Ω)
( x_p \in \Omega )
and linels
(xe∈Ω)
( x_e \in \Omega )
subject to T(Ω),x∈{0,1}
\text{ subject to } T(\Omega), x \in \{0,1\}
Consistency constraints
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
II + LP Relaxation
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
xp,xe∈Ωminxp∈Ω∑(up−xp)2+xe∈Ω∑κ^2(xe)⋅xe,
\displaystyle \min_{ x_p,x_e \in \Omega}{ \sum_{x_p \in \Omega}{ (u_p - x_p)^2} + \sum_{x_e \in \Omega}{ \hat{\kappa}^2(x_e) \cdot x_e} },
xp,xe∈Ωminxe∈Ω∑C0⋅C1+C2⋅xp∈Br(xe)∑xp+2⋅xp,xq∈Br(xe)∑xpxq⋅xe
\displaystyle \min_{ x_p,x_e \in \Omega}{ \sum_{x_e \in \Omega}{C_0 \cdot \left( C_1 + C_2 \cdot \sum_{x_p \in B_r(x_e)}{x_p} + 2\cdot \sum_{x_p,x_q \in B_r(x_e)}{x_px_q} \right) \cdot x_e } }
subject to T(Ω),x∈{0,1}
\text{ subject to } T(\Omega), x \in \{0,1\}
subject to T(Ω),x∈{0,1}
\text{ subject to } T(\Omega), x \in \{0,1\}
L(Ω)
L(\Omega)
x∈[0,1]
x \in [0,1]
+ thresholding
linearization constraints
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Summary
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
MDCA + Graph cut: too local, generate many artifacts
II + LP: global optimization, but long running time
II + LP Relaxation: poor results
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Optimize digital boundary
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Optimization region
Symmetry issue
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Optimize digital boundary
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Optimization region
Estimator evaluation region
Y⋆=argY∈{0,1}∣O∣minp∈A∑(1/2+∣Fr(p)∣−c2)⋅yi∈Yr(p)∑yi+yi,yj∈Yr(p);i<j∑yiyj.
\displaystyle Y^{\star} = arg \min_{Y \in \{0,1\}^{|O|}} \sum_{p \in A}{ \left( (1/2+ |F_r(p)|-c_2) \cdot \sum_{y_i \in Y_r(p)}{y_i} + \sum_{ y_i,y_j \in Y_r(p); i < j }{y_iy_j} \right) }.
Solve using QPBOP
Y⋆=argY∈{0,1}∣O∣minp∈A∑κ^II(p)2
\displaystyle Y^{\star} = arg \min_{Y \in \{0,1\}^{|O|}} \sum_{p \in A}{ \hat{\kappa}_{II}(p)^2 }
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Optimize digital boundary
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Labeled one pixel
Labeled zero pixel
Invert
solution
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Optimize digital boundary
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Optimize digital boundary
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Boundary Regularization
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Daniel Martins Antunes
CoMeDiC - ESIEE, Paris. January 15, 2019
Boundary Regularization
Curvature regularization
with multigrid convergent estimator
Motivation
Modeling issues
Evolution model
Title Text Curvature regularization with multigrid convergent estimator Daniel Martins Antunes Université Savoie Mont Blanc, LAMA CoMeDic - ESIEE, Paris. January 15, 2019
Curvature Regularization with Multigrid Convergent Estimators
By Daniel Martins Antunes
Curvature Regularization with Multigrid Convergent Estimators
Some previous attempts on models using multigrid convergent estimators. Presentation for the CoMeDiC research group.
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