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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.

Most biochemical processes involve macromolecules in solution. The corresponding electrostatics is of central importance for understanding their structures and functions. An accurate and efficient numerical scheme is introduced to evaluate the corresponding electrostatic potential and force by solving the governing Poisson-Boltzmann equation. This paper focuses on the following issues: (i) the point charge singularity problem, (ii) the dielectric discontinuity problem across a molecular surface, and (iii) the infinite domain problem. Green's function associated with the point charges plus a harmonic function is introduced as the zeroth order approximation to the solution to solve the point charge singularity problem. A jump condition capturing finite difference scheme is adopted to solve the discontinuity problem across molecule surfaces, where a body-fitting grid is used. The infinite domain problem is solved by mapping the outer infinite domain into a finite domain. The corresponding stiffness matrix is symmetric and positive definite, therefore, fast algorithm such as preconditioned conjugate gradient method can be applied for inner iteration. Finally, the resulting scheme is second order accurate for both the potential and its gradient.

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In this paper, we consider a class of optimization problems of minimizing a (at least) twice continuously dierentiable function (probably nonconvex) f(x) : Rn ! R over a product of multiple balls/spheres constraints. Upon rescaling the balls/spheres, we cast without loss of generality such class of minimization problems in the following form: (BCOP) 8>< >: min x2Rn f(x) s:t: ci(x) := kx[i]k2 􀀀 1 = 0; i 2 E; ci(x) := kx[i]k2 􀀀 1  0; i 2 I; where E = f1; 2; : : : ;m1g, I = fm1 + 1;m1 + 2; : : : ;mg, x[i] 2 Rpi , x = (xT [1]; xT [2]; : : : ; xT [m])T , n = Pm i=1 pi, and k
k stands for the `2 vector norm. Here, we introduce the notation x[i] 2 Rpi to represent the ith subvector of x 2 Rn and formulate the product of multiple ball/sphere constraints as a set of equality and inequality constraints. To simplify subsequent presentation, we call the above programming the ball/sphere constrained optimization problem (BCOP). We emphasize that this problem does not allow overlap among the variables x[i] and therefore the constraints are separable. However, these variables x[i] may be linked together through the objective function f(x).