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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.

Surface registration between cortical surfaces is crucial in medical imaging for performing systematic comparisons between brains. Landmark-matching registration that matches anatomical features, called the sulcal landmarks, is often required to obtain a meaningful 1-1 correspondence between brain surfaces. This is commonly done by parameterizing the surface onto a simple parameter domain, such as the unit sphere, in which the sulcal landmarks are consistently aligned. Landmark matching surface registration can then be obtained from the landmark aligned parameterizations. For genus-0 closed brain surfaces, the optimized spherical harmonic parameterization, which aligns landmarks to consistent locations on the sphere, has been widely used. This approach is limited by the loss of bijectivity under large deformations and the slow computation. In this paper, we propose FLASH, a fast algorithm to compute the optimized spherical harmonic parameterization with consistent landmark alignment. This is achieved by formulating the optimization problem to ℂ and thereby linearizing the problem. Errors introduced near the pole are corrected using quasiconformal theories. Also, by adjusting the Beltrami differential of the mapping, a diffeomorphic (1-1, onto) spherical parameterization can be effectively obtained. The proposed algorithm has been tested on 38 human brain surfaces. Experimental results demonstrate that the computation of the landmark aligned spherical harmonic parameterization is significantly accelerated using the proposed algorithm.

In this paper, we investigate the space of certain weak stability conditions on the triangulated category of D0-D2-D6 bound states on a smooth projective Calabi-Yau 3-fold. In the case of a quintic 3-fold, the resulting space is interpreted as a universal covering space of an infinitesimal neighborhood of the conifold point in the stringy K¨ahler moduli space. We then associate the DT type invariants counting semistable objects, which give new curve counting invariants on Calabi-Yau 3-folds. We also investigate the wall-crossing formula of our invariants and their interplay with the Seidel-Thomas twist.

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