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Image segmentation is an important task in the domain of computer vision and medical imaging. In natural and medical images, intensity inhomogeneity, i.e. the varying image intensity, occurs often and it poses considerable challenges for image segmentation. In this paper, we propose an efficient variational method for segmenting images with intensity inhomogeneity. The method is inspired by previous works on two-stage segmentation and variational Retinex. Our method consists of two stages. In the first stage, we decouple the image into reflection and illumination parts by solving a convex energy minimization model with either total variation or tight-frame regularisation. In the second stage, we segment the original image by thresholding on the reflection part, and the inhomogeneous intensity is estimated by the smoothly varying illumination part. We adopt a primal dual algorithm to solve the convex model in the first stage, and the convergence is guaranteed. Numerical experiments clearly show that our method is robust and efficient to segment both natural and medical images.

Recently, Huang, Lutwak, Yang & Zhang discovered the duals of Federer’s curvature measures within the dual Brunn-Minkowski theory and stated the “Minkowski problem” associated with these new measures. As they showed, this dual Minkowski problem has as special cases the Aleksandrov problem (when the index is 0) and the logarithmic Minkowski problem (when the index is the dimension of the ambient space)—two problems that were never imagined to be connected in any way. Huang, Lutwak, Yang & Zhang established sufficient conditions to guarantee existence of solution to the dual Minkowski problem in the even setting. In this work, existence of solution to the even dual Minkowski problem is established under new sufficiency conditions. It was recently shown by B ̈or ̈oczky, Henk & Pollehn that these new sufficiency conditions are also necessary.

In this paper, we consider a family of neural networks for solving nonlinear complementarity problems (NCP). The neural networks are constructed from the merit functions based on three classes of NCP-functions: the generalized natural residual function and its two symmetrizations. In this paper, we first characterize the stationary points of the induced merit functions. Growth behavior of the complementarity functions is also described, as this will play an important role in describing the level sets of the merit functions. In addition, the stability of the steepest descent-based neural network model for NCP is analyzed. We provide numerical simulations to illustrate the theoretical results, and also compare the proposed neural networks with existing neural networks based on other well-known NCP-functions. Numerical results indicate that the performance of the neural network is better when the parameter <i>p</i> associated

A surgery on a knot in 3-sphere is called SU(2)-cyclic if it gives a manifold whose fundamental group has no non-cyclic SU(2) representations. Using holonomy perturbations on the Chern-Simons functional, we prove that the distance of two SU(2)-cyclic surgery coefficients is bounded by the sum of the absolute values of their numerators. This is an analog of Culler-Gordon-Luecke-Shalen's cyclic surgery theorem.