Accurate pancreas segmentation from 3D CT volumes is important for pancreas diseases therapy. It is challenging to accurately delineate the pancreas due to the poor intensity contrast and intrinsic large variations in volume, shape, and location. In this paper, we propose a semiautomated deformable U-Net, i.e., DUNet for the pancreas segmentation. The key innovation of our proposed method is a deformable convolution module, which adaptively adds learned offsets to each sampling position of 2D convolutional kernel to enhance feature representation. Combining deformable convolution module with U-Net enables our DUNet to flexibly capture pancreatic features and improve the geometric modeling capability of U-Net. Moreover, a nonlinear Dice-based loss function is designed to tackle the class-imbalanced problem in the pancreas segmentation. Experimental results show that our proposed method outperforms all comparison methods on the same NIH dataset.
We solve constructively the piston problem for the one-dimensional isentropic Euler equations for a modified Chaplygin gas. We give a rigorous proof of the global existence and uniqueness of a shock wave separating constant states ahead of the piston when the piston advances into the gas. The results are quite different from those for a pure Chaplygin gas or a generalized Chaplygin gas, in which a Radon measure solution is constructed to deal with the concentration of mass on the piston. When the piston recedes from the gas, we show strictly that only a first-family rarefaction wave exists in front of the piston and that concentration will never occur. In addition, by studying the limiting behavior, we show that the piston solutions of the modified Chaplygin gas equations tend to the piston solutions of the generalized or pure Chaplygin gas equations as a single parameter of the pressure state function vanishes.
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