Jiansong Deng

Jiansong Deng, Falai Chen, Liangbing Jin

This paper discusses the dimensions of the spline spaces over T-meshes with lower degree. Two new concepts are proposed: extension of T-meshes and spline spaces with homogeneous boundary conditions. In the dimension analysis, the key strategy is linear space embedding with the operator of mixed partial derivative. The dimension of the original space equals the difference between the dimension of the image space and the rank of the constraints which ensuring any element in the image space has a preimage in the original space. Then the dimension formula and basis function construction of bilinear spline spaces of smoothness order zero over T-meshes are discussed in detail, and a dimension lower bound of biquadratic spline spaces over general T-meshes is provided. Furthermore, using level structure of hierarchical T-meshes, a dimension formula of biquadratic spline space over hierarchical T-meshes are proved. A topological explantation of the dimension formula is shown as well.

Mao Shi, Jiansong Deng

In this paper, we propose a method to obtain a constrained approximation of a rational Bézier curve by a polynomial Bézier curve. This problem is reformulated as an approximation problem between two polynomial Bézier curves based on weighted least-squares method, where weight functions $ρ(t)=ω(t)$ and $ρ(t)=ω(t)^{2}$ are studied respectively. The efficiency of the proposed method is tested using some examples.

Dong Liu, Yuanchao Wu, Bowen Tong et al.

Regularization methods using prior knowledge are essential in solving ill-posed inverse problems such as Electrical Impedance Tomography (EIT). However, designing effective regularization and integrating prior information into EIT remains challenging due to the complexity and variability of anatomical structures. In this work, we introduce SDEIT, a novel semantic-driven framework that integrates Stable Diffusion 3.5 into EIT, marking the first use of large-scale text-to-image generation models in EIT. SDEIT employs natural language prompts as semantic priors to guide the reconstruction process. By coupling an implicit neural representation (INR) network with a plug-and-play optimization scheme that leverages SD-generated images as generative priors, SDEIT improves structural consistency and recovers fine details. Importantly, this method does not rely on paired training datasets, increasing its adaptability to varied EIT scenarios. Extensive experiments on both simulated and experimental data demonstrate that SDEIT outperforms state-of-the-art techniques, offering superior accuracy and robustness. This work opens a new pathway for integrating multimodal priors into ill-posed inverse problems like EIT.

Qian Ni, Xuhui Wang, Jiansong Deng

The local refinement of PHT-splines (polynomial splines over hierarchical T-meshes) is achieved by a simple cross insertion, which may introduce superfluous control points or coefficients. By allowing split-in-half in mesh refinement, modified hierarchical T-meshes are defined. Using this approach, polynomial splines defined over the modified hierarchical T-meshes (modified PHT-splines) are introduced to increase the flexibility of PHT-splines. Numerical examples demonstrate the advantages of our new splines when applied to surface fitting and isogeometric analysis problems with anisotropic features.

Chao Zeng, Fang Deng, Xin Li et al.

This paper discusses the dimensions of biquadratic C1 spline spaces and bicubic C2 spline spaces over hierarchical T-meshes using the smoothing cofactor-conformality method. We obtain the dimension formula of biquadratic C1 spline spaces over hierarchical T-meshes in a concise way. In addition, we provide a dimension formula for bicubic C2 spline spaces over hierarchical T-mesh with fewer restrictions than that in the previous literature. A dimension formula for bicubic C2 spline spaces over a new type hierarchical T-mesh is also provided.