arXiv subjects

Xiaomeng Li

Publications and source records attributed to Xiaomeng Li.

165 records · Page 10Linked to original sources

arXiv2018

H-DenseUNet: Hybrid Densely Connected UNet for Liver and Tumor Segmentation from CT Volumes

Liver cancer is one of the leading causes of cancer death. To assist doctors in hepatocellular carcinoma diagnosis and treatment planning, an accurate and automatic liver and tumor segmentation method is highly demanded in the clinical practice. Recently, fully convolutional neural networks (FCNs), including 2D and 3D FCNs, serve as the back-bone in many volumetric image segmentation. However, 2D convolutions can not fully leverage the spatial information along the third dimension while 3D convolutions suffer from high computational cost and GPU memory consumption. To address these issues, we propose a novel hybrid densely connected UNet (H-DenseUNet), which consists of a 2D DenseUNet for efficiently extracting intra-slice features and a 3D counterpart for hierarchically aggregating volumetric contexts under the spirit of the auto-context algorithm for liver and tumor segmentation. We formulate the learning process of H-DenseUNet in an end-to-end manner, where the intra-slice representations and inter-slice features can be jointly optimized through a hybrid feature fusion (HFF) layer. We extensively evaluated our method on the dataset of MICCAI 2017 Liver Tumor Segmentation (LiTS) Challenge and 3DIRCADb Dataset. Our method outperformed other state-of-the-arts on the segmentation results of tumors and achieved very competitive performance for liver segmentation even with a single model.

cs.CV↗

arXiv2018

Extremal functions for Adams' inequalities in dimension four

Let $Ω\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(Ω)$ be the usual Sobolev space. For any positive integer $\ell$, $λ_{\ell}(Ω)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{λ_1(Ω)}\oplus E_{λ_2(Ω)}\oplus\cdots\oplus E_{λ_{\ell}(Ω)}$, where $E_{λ_i(Ω)}$ is eigenfunction space associated with $λ_i(Ω)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(Ω)$. For $0\leqα<λ_{\ell+1}(Ω)$, we define a norm by $\|u\|_{2,α}^{2}=\|Δu\|^2_2-α\|u\|^2_2$ for $u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities $$\sup_{u\in E_{\ell}^{\bot},\,\| u\|_{2,α}\leq 1}\int_Ωe^{32π^2u^2}dx<+\infty;$$ moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overlineΩ)$ with $\|u_0\|_{2,α}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).

math.AP↗

arXiv2016

Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space

In a previous work (Int. Math. Res. Notices 13 (2010) 2394-2426), Adimurthi-Yang proved a singular Trudinger-Moser inequality in the entire Euclidean space $\mathbb{R}^N$ $(N\geq 2)$. Precisely, if $0\leq β<1$ and $0<γ\leq1-β$, then there holds for any $τ>0$, $$\sup_{u\in W^{1,N}(\mathbb{R}^N),\,\int_{\mathbb{R}^N}(|\nabla u|^N+τ|u|^N)dx\leq 1}\int_{\mathbb{R}^N}\frac{1}{|x|^{Nβ}}\left(e^{α_Nγ|u|^{\frac{N}{N-1}}}-\sum_{k=0}^{N-2}\frac{α_N^kγ^k|u|^{\frac{kN}{N-1}}} {k!}\right)dx<\infty,$$ where $α_N=Nω_{N-1}^{1/(N-1)}$ and $ω_{N-1}$ is the area of the unit sphere in $\mathbb{R}^N$. The above inequality is sharp in the sense that if $γ>1-β$, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case $β=0$ has been considered by Li-Ruf (Indiana Univ. Math. J. 57 (2008) 451-480) and Ishiwata (Math. Ann. 351 (2011) 781-804). We shall investigate the singular case $0<β<1$ and prove that for all $τ>0$, $0<β<1$ and $0<γ\leq 1-β$, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.

math.AP↗