In this paper, we propose a novel iterative convolution-thresholding method (ICTM) that is applicable to a range of variational models for image segmentation. A variational model usually minimizes an energy functional consisting of a fidelity term and a regularization term. In the ICTM, the interface between two different segment domains is implicitly represented by their characteristic functions. The fidelity term is usually written as a linear functional of the characteristic functions and the regularized term is approximated by a functional of characteristic functions in terms of heat kernel convolution. This allows us to design an iterative convolution-thresholding method to minimize the approximate energy. The method is simple, efficient and enjoys the energy-decaying property. Numerical experiments show that the method is easy to implement, robust and applicable to various image segmentation models.

Fixing a weakly unobstructed Lagrangian torus in a symplectic manifold X, we define a holomorphic function W known as the Floer potential. We construct a canonical A∞-functor from the Fukaya category of X to the category of matrix factorizations of W. It provides a unified way to construct matrix factorizations from Lagrangian Floer theory. The technique is applied to toric Fano manifolds to transform Lagrangian branes to matrix factorizations. Using the method, we also obtain an explicit expression of the matrix factorization mirror to the real locus of the complex projective space.

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