In this paper, we study critical points of second Neumann eigenfunctions on convex domains in two-dimensional space forms from three complementary perspectives: spectral and geometric criteria for the absence of interior critical points, quantitative localization of possible critical points, and explicit bounds for the hot spots constant. Precisely, we first establish monotonicity-radius localization principles in 𝕊^2 and ℍ^2. As a consequence, we obtain a unified diameter criterion for convex domains in these two space forms: if μ_2(Ω)D^2≤ j_1,1^2, then every second Neumann eigenfunction on Ω has no interior critical points. Moreover, when interior critical points may exist, we derive explicit quantitative location restrictions in terms of the domain diameters in 𝕊^2 and ℍ^2. Finally, we develop an analytic approach to study the hot spots constant ℭ(Ω) on convex domains. For planar convex domains, we improve the Euclidean upper bound to ℭ(Ω)<1.48. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, ℭ(Ω) < 4 for Ω⊂𝕊^2 contained in a hemisphere; ℭ(Ω) < 11.2 for Ω⊂ℍ^2. Our proofs combine the properties of Bessel and Legendre functions, eigenvalue estimates, and Green's identity. Our results quantitatively measure “how wrong” the hot spots conjecture can be.

In this paper, we determine the symmetry properties and non-vertex critical points of the second Neumann eigenfunction u, as well as the multiplicity of the corresponding eigenvalue, on isosceles trapezoids and kites. By exploiting reflection symmetry, we reduce the problem to a comparison between the second Neumann eigenvalue and the first mixed Dirichlet–Neumann eigenvalue on the half-domain. More precisely, for isosceles trapezoids with base angle α≤3, the second eigenfunction is antisymmetric. If 3<α<2, there exists a critical height ĥ(α) at which the two symmetry branches cross: u is antisymmetric when height h<ĥ(α) and symmetric when h>ĥ(α), while at h=ĥ(α) the second Neumann eigenvalue has multiplicity two. For a convex kite P_1P_2P_3P_4, where P_1=(0,0), P_2=(a,-h), P_3=(1,0), and P_4=(a,h), an analogous result holds: there exists a critical height h̃(a) such that u is symmetric with respect to the x-axis when hh̃(a), while at h=h̃(a) the second Neumann eigenvalue has multiplicity two. In all cases where the second Neumann eigenvalue is simple, we determine all non-vertex critical points of the corresponding eigenfunction, thereby verifying the hot spots conjecture.

Whole‑body organ segmentation is critical for radiotherapy planning and systemic disease assessment, but existing PET/CT‑tailored segmentation methods suffer from insufficient anatomical coverage, severe annotation bottlenecks, and an unfavorable trade‑off between accuracy and speed, limiting their clinical translation. To address these issues, we developed a weakly supervised framework for whole‑body segmentation of 147 anatomical structures on PET/CT and established normative SUV reference ranges in healthy individuals. This retrospective study included 2308 18F‑FDG PET/CT scans from two hospitals and 1746 public CT scans. Model development used 1846 scans for pseudo‑label training (1746 public CT + 100 PET/CT) and 2000 unlabeled PET/CT scans for weakly supervised learning; independent testing used 110 PET/CT scans. A healthy cohort of 98 individuals was included to establish normal SUV reference ranges. An uncertainty‑guided pseudo‑label learning algorithm was trained to segment 147 anatomical structures. Dice scores were calculated between model predictions and expert annotations. Paired t‑tests or Wilcoxon tests with Bonferroni correction were used for statistical analysis. The model achieved a mean Dice score of 0.874 across 147 structures on the test set, with segmentation completed within 120 s per scan. For 113 structures matched to TotalSegmentator, it outperformed TotalSegmentator (0.895 vs. 0.884; P < 0.001). The uncertainty-weighting strategy improved segmentation of small vascular structures (0.702 vs. 0.651; P < 0.001). The model also showed a lower absolute deviation in the 95th-percentile SUV (SUV95; 0.018 vs. 0.021) than TotalSegmentator in PET quantification. Normal SUV reference ranges for 147 structures were established from the healthy cohort. The proposed weakly supervised framework enables precise and efficient whole-body segmentation of 147 structures on PET/CT. The established normal SUV reference ranges provide a standardized benchmark for quantitative PET/CT interpretation. The source code and pretrained weights are publicly available at https://github.com/Wpx01/WiseSeg.

In this paper, one of our aims is to investigate the instability of the distribution of the critical point set $\mathcal{C}(u)$ of a solution $u$ to a semilinear equation with Dirichlet boundary condition in the planar annular domains. Precisely, we prove that $\mathcal{C}(u)$ in an eccentric circle annular domain, or a petal-like domain, or an annular domain where the interior and exterior boundaries are equally scaled ellipses contains only finitely many points rather than a Jordan curve. This result indicates that the critical point set $\mathcal{C}(u)$ is unstable when any boundary of planar concentric circle annular domain $\Omega$ has some small deformation or minor perturbation. Based on studying the distribution of the nodal sets $u^{-1}_\theta(0)(u_\theta=\nabla u\cdot \theta)$ and $u^{-1}(0)$, we prove that the solution $u$ on each symmetric axis has exactly two critical points under some conditions. Meanwhile, we further obtain that $\mathcal{C}(u)$ only has two critical points in an eccentric circle annular domain, has four critical points in an exterior petal-like domain with the exterior boundary $\gamma_E$ is an ellipse, and the maximum points are distributed on the long symmetric semi-axis and the saddle points on the short symmetric semi-axis. Moreover, we describe the geometric location of critical points of the solution $u$ by the moving plane method.

In this paper, we investigate the quantitative unique continuation, propagation of smallness and measure bounds of nodal sets of solutions to the Buckling-type equation Δ2u + λΔu − k2u = 0 in a bounded analytic domain Ω ⊆ ℝn with the homogeneous boundary conditions u = 0 and ∂ u∂ν=0 on ∂Ω, where λ, k are nonnegative real constants, and ν is the outer unit normal vector on ∂Ω. We obtain that, the upper bounds for the maximal vanishing order of u and the n − 1-dimensional Hausdorff measure of the nodal set of u are both C(√(λ) + √(k) + 1) , where C is a positive constant only depending on n and Ω. Moreover, we also give a quantitative result of the propagation of smallness of u.

In this paper, we focus on the quantitative unique continuation property of solutions to \begin{equation*} \Delta^2u=Vu, \end{equation*} where $V\in W^{1,\infty}$. We show that the maximal vanishing order of the solutions is not large than \begin{equation} C\left(\|V\|^{\frac{1}{4}}_{L^{\infty}}+\|\nabla V\|_{L^{\infty}}+1\right). \end{equation} Our key argument is to lift the original equation to that with a positive potential, then decompose the resulted fourth-order equation into a special system of two second-order equations. Based on the special system, we define a variant frequency function with weights and derive its almost monotonicity to establishing some doubling inequalities with explicit dependence on the Sobolev norm of the potential function.

In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equationΔX2u=Vu, where ΔX=Δx+|x|2αΔy (0<α≤1), with x∈Rm,y∈Rn, denotes the Baouendi-Grushin type subelliptic operators, and the potential V satisfies the strongly singular growth assumption |V|≤c0ρ4, whereρ=(|x|2(α+1)+(α+1)2|y|2)12(α+1) is the gauge norm. The main argument is to introduce an Almgren's type frequency function for the solutions, and show its monotonicity to obtain a doubling estimate based on setting up some refined Hardy-Rellich type inequalities on the gauge balls with boundary terms.

In this paper, we focus on estimating measure upper bounds of nodal sets of solutions to the following boundary value problem \begin{equation*} \left\{ \begin{array}{lll} \Delta u+Vu=0\quad \mbox{in}\ \Omega,\\[2mm] u=0\quad \mbox{on}\ \partial\Omega, \end{array}\right. \end{equation*} where $V\in W^{1,\infty}(\Omega)$ is a potential and $\Omega\subset \mathbb{R}^n (n\geq2)$ is a bounded domain. We show that upper bounds on the $(n-1)$-dimensional Hausdorff measure of the nodal sets of $u$ in $\Omega$ is less than or equal to $$C\Big(1+\log\left(\|\nabla V\|_{L^{\infty}(\Omega)}+1\right)\Big)\cdot\left(\|V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+|\nabla V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+1\right),$$ provided $\partial\Omega$ is $C^{2}$-smooth and $V$ is analytic. Here $C$ is a positive constant depending only on $n$ and $\Omega$. In particular, if $\|\nabla V\|_{L^{\infty}(\Omega)}$ is small, the measure upper bound of the of nodal set of $u$ is less than or equal to $C\left(\|V\|^{\frac{1}{2}}_{L^{\infty}(\Omega)}+1\right)$.

There are many unsolved problems in vascular image segmentation, including vascular structural connectivity, scarce branches and missing small vessels. Obtaining vessels that preserve their correct topological structures is currently a crucial research issue, as it provides an overall view of one vascular system. In order to preserve the topology and accuracy of vessel segmentation, we proposed a novel Morphology Edge Attention Network (MEA-Net) for the segmentation of vessel-like structures, and an Optimal Geometric Matching Connection (OGMC) model to connect the broken vessel segments. The MEA-Net has an edge attention module that improves the segmentation of edges and small objects by morphology operation extracting boundary voxels on multi-scale. The OGMC model uses the concept of curve touching from differential geometry to filter out fragmented vessel endpoints, and then employs minimal surfaces to determine the optimal connection order between blood vessels. Finally, we calculate the geodesic to repair missing vessels under a given Riemannian metric. Our method achieves superior or competitive results compared to state-of-the-art methods on four datasets of 3D vascular segmentation tasks, both effectively reducing vessel broken and increasing vessel branch richness, yielding blood vessels with a more precise topological structure.

Pancreatic duct dilation indicates a high risk of various pancreatic diseases. Segmentation for dilated pancreatic duct (DPD) on computed tomography (CT) image shows the potential to assist the early diagnosis, surgical planning and prognosis. Because of the DPD's tiny size, slender tubular structure and the surrounding distractions, most current researches on DPD segmentation achieve low accuracy and always have segmentation errors on the terminal DPD regions. To address these problems, we propose a cascaded terminal guidance network to efficiently improve the DPD segmentation performance. Firstly, a basic cascaded segmentation architecture is established to get the pancreas and coarse DPD segmentation, a DPD graph structure is build on the coarse DPD segmentation to locate the terminal DPD regions. Then, a terminal anatomy attention module is introduced for jointly learning the local intensity from the CT images, feature cues from the coarse DPD segmentation and global anatomy information from the designed pancreas anatomy-aware maps. Finally, a terminal distraction attention module which explicitly learns the distribution of the terminal distraction regions is proposed to reduce the false positive and false negative predictions. We also propose a new metric called tDice to measure the terminal segmentation accuracy for targets with tubular structures and two other metrics for segmentation error evaluation. We collect our dilated pancreatic duct segmentation dataset with 150 CT scans from patients with five types of pancreatic tumors. Experimental results on our dataset show that our proposed approach boosts DPD segmentation accuracy by nearly 20% compared with the existing results, and achieves more than 9% improvement for the terminal segmentation accuracy compared with the state-of-the-art methods.

Assessment of myocardial viability is essential in diagnosis and treatment management of patients suffering from myocardial infarction, and classification of pathology on myocardium is the key to this assessment. This work defines a new task of medical image analysis, i.e., to perform myocardial pathology segmentation (MyoPS) combining three-sequence cardiac magnetic resonance (CMR) images, which was first proposed in the MyoPS challenge, in conjunction with MICCAI 2020. The challenge provided 45 paired and pre-aligned CMR images, allowing algorithms to combine the complementary information from the three CMR sequences for pathology segmentation. In this article, we provide details of the challenge, survey the works from fifteen participants and interpret their methods according to five aspects, i.e., preprocessing, data augmentation, learning strategy, model architecture and post-processing. In addition, we analyze the results with respect to different factors, in order to examine the key obstacles and explore potential of solutions, as well as to provide a benchmark for future research. We conclude that while promising results have been reported, the research is still in the early stage, and more in-depth exploration is needed before a successful application to the clinics. Note that MyoPS data and evaluation tool continue to be publicly available upon registration via its homepage (www.sdspeople.fudan.edu.cn/zhuangxiahai/0/myops20/).

In this paper, we mainly study the critical points and critical zero points of solutions u to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain Ω in ℝ2. Based on the delicate analysis about the distributions of connected components of the super-level sets {x ∈ Ω: u(x) > t} and sub-level sets {x ∈ Ω: u(x) < t} for some t, we obtain the geometric structure of interior critical point sets of u. Precisely, let Ω be a multiply connected domain with the interior boundary γI and the external boundary γE, where u∣γI = ψ1(x), u∣γE = ψ2(x). When ψ1(x) and ψ2(x) have N1 and N2 local maximal points on γI and γE respectively, we deduce that $$\sum\nolimits_{i = 1}^k {{m_i} \le {N_1} + {N_2}} $$ , where m1,…,mk are the respective multiplicities of interior critical points x1,…,xk of u. In addition, when $${\min _{{\gamma _E}}}{\psi _2}\left( x \right) \ge {\max _{{\gamma _I}}}{\psi _1}\left( x \right)$$ and u has only N1 and N2 equal local maxima relative to $$\overline \Omega $$ on γI and γe respectively, we develop a new method to show that one of the following three results holds $$\sum\nolimits_{i = 1}^k {{m_i} = {N_1} + {N_2}} $$ or $$\sum\nolimits_{i = 1}^k {{m_i} + 1 = {N_1} + {N_2}} $$ or $$\sum\nolimits_{i = 1}^k {{m_i} + 2 = {N_1} + {N_2}} $$ . Moreover, we investigate the geometric structure of interior critical zero points of u. We obtain that the sum of multiplicities of the interior critical zero points of u is less than or equal to the half of the number of its isolated zero points on the boundaries.

Image segmentation means to partition an image into separate meaningful regions. Segmentation in medical images can extract different organs, lesions, and other regions of interest, which helps in subsequent disease diagnosis, surgery planning, and efficacy assessment. However, medical images have many unavoidable interference factors, such as imaging noise, artificial artifacts, and mutual occlusion of organs, which make accurate segmentation highly difficult. Incorporating prior knowledge and image information into segmentation model based on variational methods has proven efficient for more accurate segmentation. In recent years, segmentation based on deep learning has been significantly developed, and the combination of classical variational method-based models with deep learning is a hot topic. In this survey, we briefly review the segmentation methods based on a variational method making use of image information and regularity information. Subsequently, we clarify how the integration of variational methods into the deep learning framework leads to more precise segmentation results.

This paper relates the post-analysis of the first edition of the HEad and neCK TumOR (HECKTOR) challenge. This challenge was held as a satellite event of the 23rd International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI) 2020, and was the first of its kind focusing on lesion segmentation in combined FDG-PET and CT image modalities. The challenge's task is the automatic segmentation of the Gross Tumor Volume (GTV) of Head and Neck (H&N) oropharyngeal primary tumors in FDG-PET/CT images. To this end, the participants were given a training set of 201 cases from four different centers and their methods were tested on a held-out set of 53 cases from a fifth center. The methods were ranked according to the Dice Score Coefficient (DSC) averaged across all test cases. An additional inter-observer agreement study was organized to assess the difficulty of the task from a human perspective. 64 teams registered to the challenge, among which 10 provided a paper detailing their approach. The best method obtained an average DSC of 0.7591, showing a large improvement over our proposed baseline method and the inter-observer agreement, associated with DSCs of 0.6610 and 0.61, respectively. The automatic methods proved to successfully leverage the wealth of metabolic and structural properties of combined PET and CT modalities, significantly outperforming human inter-observer agreement level, semi-automatic thresholding based on PET images as well as other single modality-based methods. This promising performance is one step forward towards large-scale radiomics studies in H&N cancer, obviating the need for error-prone and time-consuming manual delineation of GTVs. (C) 2022 The Authors. Published by Elsevier B.V.

We prove the existence of solutions to the Monge problem with an absolutely continuous initial measure by solving a secondary variational problem with any strictly convex function, the so-called secondary variational method. The cost function is given by an arbitrary norm on R-n. In addition, if a norm satisfies the uniform smoothness and convexity estimates, and two measures are absolutely continuous, then for the Monge problem with such a norm cost function, we can find a same optimal transport map via the secondary variational method even with different strictly convex functions (the classical Monge problem is a special case). This optimal transport map is just the one which uniquely satisfies a monotone condition. Finally, we construct an example with the L-1 norm cost function, which is not a strictly convex norm, to show that one can get different optimal transport maps by solving secondary variational problems with different strictly convex functions. In view of this example, for the Monge problem between two absolutely continuous measures, if a norm cost function does not satisfy the uniform smoothness and convexity estimates, there can be no uniqueness of optimal transport maps obtained via the secondary variational method.

Active contour models have been widely used in image segmentation, and the level set method (LSM) is the most popular approach for solving the models, via implicitly representing the contour by a level set function. However, the LSM suffers from high computational burden and numerical instability, requiring additional regularization terms or re-initialization techniques. In this paper, we use characteristic functions to implicitly represent the contours, propose a new representation to the geodesic active contours and derive an efficient algorithm termed as the iterative convolution-thresholding method (ICTM). Compared to the LSM, the ICTM is simpler and much more efficient. In addition, the ICTM enjoys most desired features of the level set-based methods. Extensive experiments, on 2D synthetic, 2D ultrasound, 3D CT, and 3D MR images for nodule, organ and lesion segmentation, demonstrate that the proposed method not only obtains comparable or even better segmentation results (compared to the LSM) but also achieves significant acceleration.

We study a weighted eigenvalue problem of the β-biased infinity Laplacian operator arising from the β-biased tug-of-war. We characterize the principal eigenvalue by the comparison principle and show that β-biased infinity Laplacian operator possesses two principal eigenvalues, corresponding to a positive and a negative principal eigenfunction. When a parameter is less than the principal eigenvalue, certain existence and uniqueness results of the inhomogeneous equations related to this problem are established. As an application, we obtain the decay estimates for viscosity solutions of the parabolic problem associated to the β-biased infinity Laplacian. In the process, we also establish the Lipschitz regularity and Harnack inequality by barrier method.

Automatic segmentation of head and neck tumor plays an important role for radiomics analysis. In this short paper, we propose an automatic segmentation method for head and neck tumors from PET and CT images based on the combination of convolutional neural networks (CNNs) and hybrid active contours. Specifically, we first introduce a multi-channel 3D U-Net to segment the tumor with the concatenated PET and CT images. Then, we estimate the segmentation uncertainty by model ensembles, and define a segmentation quality score to select the cases with high uncertainties. Finally, we develop a hybrid active contour model to refine the high uncertainty cases. We evaluate the proposed method on the MICCAI 2020 HECKTOR challenge and achieve promising performance with average Dice Similarity Coefficient, precision and recall of 0.7525, 0.8384, 0.7471 respectively.