Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Shape Alignment via Allen-Cahn Nonlinear-Convection

Daniel Solano, Laurent Younes, Jerome Darbon

TL;DR

Shape Alignment via Allen-Cahn Nonlinear-Convection develops a unified framework for registering shapes with possible topology changes by coupling a diffeomorphic flow $v$ with a topological control $u$ in a generalized convective Allen-Cahn model for the shape's characteristic function $f$. The authors establish global existence of mild solutions, formulate a time-discretized soft-endpoint optimal control problem, and derive a discrete Pontryagin maximum principle to characterize minimizers, complemented by numerical simulations enforcing a maximum-bounded principle. The method yields a meaningful discrepancy measure between shapes that accounts for both diffeomorphic deformation and topological alteration, providing a versatile tool for image analysis in contexts like computational anatomy. Numerical experiments on discs and gesture-like shapes demonstrate the ability to align shapes with differing topology by leveraging the combined effects of the two control channels and mean-curvature regularization, highlighting the distinct roles of $u$ and $v$ in topological versus geometric changes.

Abstract

This paper demonstrates the impact of a phase field method on shape registration to align shapes of possibly different topology. It yields new insights into the building of discrepancy measures between shapes regardless of topology, which would have applications in fields of image data analysis such as computational anatomy. A soft end-point optimal control problem is introduced whose minimum measures the minimal control norm required to align an initial shape to a final shape, up to a small error term. The initial data is spatially integrable, the paths in control spaces are integrable and the evolution equation is a generalized convective Allen-Cahn. Binary images are used to represent shapes for the initial data. Inspired by level-set methods and large diffeomorphic deformation metric mapping, the controls spaces are integrable scalar functions to serve as a normal velocity and smooth reproducing kernel Hilbert spaces to serve as velocity vector fields. The existence of mild solutions to the evolution equation is proved, the minimums of the time discretized optimal control problem are characterized, and numerical simulations of minimums to the fully discretized optimal control problem are displayed. The numerical implementation enforces the maximum-bounded principle, although it is not proved for these mild solutions. This research offers a novel discrepancy measure that provides valuable ways to analyze diverse image data sets. Future work involves proving the existence of minimums, existence and uniqueness of strong solutions and the maximum bounded principle.

Shape Alignment via Allen-Cahn Nonlinear-Convection

TL;DR

Shape Alignment via Allen-Cahn Nonlinear-Convection develops a unified framework for registering shapes with possible topology changes by coupling a diffeomorphic flow with a topological control in a generalized convective Allen-Cahn model for the shape's characteristic function . The authors establish global existence of mild solutions, formulate a time-discretized soft-endpoint optimal control problem, and derive a discrete Pontryagin maximum principle to characterize minimizers, complemented by numerical simulations enforcing a maximum-bounded principle. The method yields a meaningful discrepancy measure between shapes that accounts for both diffeomorphic deformation and topological alteration, providing a versatile tool for image analysis in contexts like computational anatomy. Numerical experiments on discs and gesture-like shapes demonstrate the ability to align shapes with differing topology by leveraging the combined effects of the two control channels and mean-curvature regularization, highlighting the distinct roles of and in topological versus geometric changes.

Abstract

This paper demonstrates the impact of a phase field method on shape registration to align shapes of possibly different topology. It yields new insights into the building of discrepancy measures between shapes regardless of topology, which would have applications in fields of image data analysis such as computational anatomy. A soft end-point optimal control problem is introduced whose minimum measures the minimal control norm required to align an initial shape to a final shape, up to a small error term. The initial data is spatially integrable, the paths in control spaces are integrable and the evolution equation is a generalized convective Allen-Cahn. Binary images are used to represent shapes for the initial data. Inspired by level-set methods and large diffeomorphic deformation metric mapping, the controls spaces are integrable scalar functions to serve as a normal velocity and smooth reproducing kernel Hilbert spaces to serve as velocity vector fields. The existence of mild solutions to the evolution equation is proved, the minimums of the time discretized optimal control problem are characterized, and numerical simulations of minimums to the fully discretized optimal control problem are displayed. The numerical implementation enforces the maximum-bounded principle, although it is not proved for these mild solutions. This research offers a novel discrepancy measure that provides valuable ways to analyze diverse image data sets. Future work involves proving the existence of minimums, existence and uniqueness of strong solutions and the maximum bounded principle.

Paper Structure

This paper contains 23 sections, 12 theorems, 106 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Summary of notation
  4. Useful properties of convolutions
  5. Main result
  6. Optimal control problem
  7. Existence and Uniqueness of Mild Solutions
  8. A discrete-time optimal control problem
  9. Discrete-time evolution
  10. Variational derivatives of the discrete evolution equation
  11. Characterization of Minimizers
  12. Numerical simulation
  13. Maximum bounded principle enforcing time step
  14. Spatial discretization in two dimensions
  15. Grid setup, initial and target data
  16. ...and 8 more sections

Key Result

Lemma 1

(Chapter 15, 1.15, Third Equation,taylor) Let $q\geq \tilde{q}\in [1,\infty)$, $k\geq l\in \mathbb{R}$, and $\tau >0$. Then $f \mapsto \eta_\tau \star f$ is a bounded linear mapping from $W^{l,\tilde{q}}(\mathbb{R}^d)$ to $W^{k,q}(\mathbb{R}^d)$ with, for some $C>0$, $f\in W^{l,\tilde{q}}(\mathbb{R}^d)$ and where

Figures (10)

  • Figure 1: Here our initial shape was the unit disk and the final shape was two unit disks under a Gaussian blur. Presented is the trajectory of the function f (superimposed with its main contour in magenta) at seven time points and the last image is the target image. (Using default parameters.)
  • Figure 2: Illustration of the impact of the penalty parameter. The rows correspond, from top to bottom, to $C_{\mathrm{top}} = 10^6$, $10^8$ and $10^{10}$, using default parameters otherwise. Each row provides, from left to right, the final image ($t=1$) resulting from the state equation, the diffeomorphism associated to the smooth vector field $v$ (solution to the flow equation at $t=1$), visualized by applying if to a regular grid, with different colors for connected components of the original image, and the result of only applying the normal motion (setting $v=0$) in the state equation.
  • Figure 3: Here our initial shape was three unit disks and the final shape was four unit disks under a Gaussian blur. (Using default parameters.)
  • Figure 4: Reverse comparison from that of \ref{['fig:3Cto4C']}. (Using default parameters.)
  • Figure 5: Here our initial shape was an hand giving the 'okay' gesture and the final shape was an open hand. (Using default parameters.)
  • ...and 5 more figures

Theorems & Definitions (22)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • proof
  • Lemma 3
  • ...and 12 more