In this paper, we study conformally flat (α, β)-metrics in the form of F = αφ(β/α), where α is a Riemannian metric and β is a 1-form on the manifold. We prove that conformally flat weak Landsberg (α, β)-metrics must be either Riemannian metrics or locally Minkowski metrics. Further, we prove that, if φ(s) is a polynomial in s, then conformally flat (α, β)-metrics with relatively isotropic mean Landsberg curvature must also be either Riemannian metrics or locally Minkowski metrics.Mathematics Subject Classification: 53B40, 53C60.
Convolutional network models have been widely used in image segmentation. However, there are many types of boundary contour features in medical images which seriously affect the stability and accuracy of image segmentation models, such as the ambiguity of tumors, the variability of lesions, and the weak boundaries of fine blood vessels. In this paper, in order to solve these problems we first introduce the dual-tree complex wavelet scattering transform module, and then innovatively propose a learning scattering wavelet network model. In addition, a new improved active contour loss function is further constructed to deal with complex segmentation. Finally, the equilibrium coefficient of our model is discussed. Experiments on the BraTS2020 dataset show that the LSW-Net model has improved the Dice coefficient, accuracy, and sensitivity of the classic FCN, SegNet, and At-Unet models by at least 3.51%, 2.11%, and 0.46%, respectively. In addition, the LSW-Net model still has an advantage in the average measure of Dice coefficients compared with some advanced segmentation models. Experiments on the DRIVE dataset prove that our model outperforms the other 14 algorithms in both Dice coefficient and specificity measures. In particular, the sensitivity of our model provides a 3.39% improvement when compared with the Unet model, and the model’s effect is obvious.
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