This paper studies the effectiveness of accomplishing high-level tasks with a minimum of manual annotation and good feature representations for medical images. In medical image analysis, objects like cells are characterized by significant clinical features. Previously developed features like SIFT and HARR are unable to comprehensively represent such objects. Therefore, feature representation is especially important. In this paper, we study automatic extraction of feature representation through deep learning (DNN). Furthermore, detailed annotation of objects is often an ambiguous and challenging task. We use multiple instance learning (MIL) framework in classification training with deep learning features. Several interesting conclusions can be drawn from our work: (1) automatic feature learning outperforms manual feature; (2) the unsupervised approach can achieve performance that's close to fully supervised approach (93.56%) vs. (94.52%); and (3) the MIL performance of coarse label (96.30%) outweighs the supervised performance of fine label (95.40%) in supervised deep learning features.

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The elliptic interface problems with discontinuous and high-contrast coefficients appear in many applications and often lead to huge condition numbers of the corresponding linear systems. Thus, it is highly desired to construct high order schemes to solve the elliptic interface problems with discontinuous and high-contrast coefficients. Let Γ be a smooth curve inside a rectangular region Ω. In this paper, we consider the elliptic interface problem −∇ • (a∇u) = f in Ω \ Γ with Dirichlet boundary conditions, where the coefficient a and the source term f are smooth in Ω \ Γ and the two nonzero jump condition functions [u] and [a∇u • n] across Γ are smooth along Γ. To solve such elliptic interface problems, we propose a high order compact finite difference scheme for numerically computing both the solution u and the gradient ∇u on uniform Cartesian grids without changing coordinates into local coordinates. Our numerical experiments confirm the fourth order accuracy for computing the solution u, the gradient ∇u and the velocity a∇u of the proposed compact finite difference scheme on uniform meshes for the elliptic interface problems with discontinuous and high-contrast coefficients.

Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a sixth order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several stateof-the-art finite difference schemes in the literature, particularly in the critical pre-asymptotic region where kh is near 1 with k being the wavenumber and h the mesh size.

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