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

\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?

\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

\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} }

( x_p \in \Omega )

and linels

( x_e \in \Omega )

\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

\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} },

\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 } }

\text{ subject to } T(\Omega), x \in \{0,1\}

L(\Omega)

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

\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

\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