arXiv subjects

Jiansong Deng

Publications and source records attributed to Jiansong Deng.

12 recordsLinked to original sources

arXiv2026

Universal Dimension Formula for Planar $C^r$ Splines at Degree $3r+1$

We prove that Schumaker's lower bound gives the dimension of $S_{3r+1}^r(Δ)$ for every nondegenerate triangulation $Δ$ of a polygonal disk in $\mathbb{R}^2$ and every integer $r\ge3$. After accounting for the local vertex contributions, we reduce the remaining dimension defect to compatible vertex functionals. Apolarity associates a unique polynomial with each face, and compatibility forces differences of adjacent face polynomials to contain powers of both endpoint forms. We eliminate the leading residual coefficient using Hermite relations when $r$ is odd and an additional coefficient comparison when $r$ is even. A uniform induction then raises the endpoint powers until all face differences vanish; the zero boundary face potentials imply that every vertex functional vanishes and the global defect is zero. Together with the known cases $r=1,2$ and Hong's higher-degree theorem, this yields the dimension formula for all $r\ge1$ and $d\ge3r+1$.

math.NA↗

arXiv2026

The Dimension Landscape of Bivariate $C^2$ Splines

Schumaker's lower bound incorporates local geometric corrections and provides insight into the dimensions of bivariate spline spaces. Building on this bound, we study bivariate $C^2$ spline dimensions from three perspectives: universal attainment, generic attainment, and geometric jumps. We prove universal attainment in degree seven for every legal realization of every triangulation. We prove generic attainment in degrees five and six for every fixed combinatorial triangulation and give explicit polynomial conditions certifying generic failure on fixed triangulations in degrees three and four. For geometric jumps with matching local direction types at corresponding interior vertices, we prove nonexistence in degree seven and construct examples in degrees three and five. In degree six, universal attainment and the existence of such geometric jumps remain open questions.

math.NA↗

arXiv2026

A geometric criterion for the dimension of Morgan-Scott spline spaces $S_{2r}^r(Δ_{\mathrm{MS}})$

We establish a geometric dimension criterion for spline spaces on nondegenerate standard Morgan-Scott triangulations with four distinct edge directions at each interior vertex. For every integer $r\ge1$, concurrence of the three auxiliary lines joining corresponding interior and boundary vertices implies that $\dim S_{2r}^r(Δ_{\mathrm{MS}})$ exceeds the Schumaker lower bound by one. The proof constructs a quadratic form through the six dual spoke points and uses its $r$th power to define a nonzero functional annihilating the image of the smoothing-cofactor map. Diener's upper bound then gives the dimension formula. We also give a matrix representation of this functional and an explicit example in the quadratic case. Combined with the known necessary conditions and the negative-branch exclusion of Deng, Feng, and Kozak, the result yields a necessary and sufficient concurrence criterion for $r>2$.

math.NA↗

arXiv2026

Geometry-dependent rank defect in $C^1$ cubic spline space

Determining the dimension of the $C^1$ cubic spline space $S_3^1(\mathcal{T})$ on an arbitrary nondegenerate planar triangulation has remained unresolved since the 1970s. Schumaker's lower bound includes a local correction $σ$ for singular interior four-stars, and it was conjectured that this bound is always attained. We disprove this conjecture by constructing a one-parameter family of nondegenerate realizations of a fixed 18-triangle complex, with only the central vertex moving as $v_6(t)=(t,0)$ on the admissible interval $I=(-3/4,24/55)$. The family exhibits three distinct cases. For $t\in I\setminus\{1/5,3/83\}$, the lower bound is attained and $\dim S_3^1(\mathcal{T}(t))=33$. At $t=3/83$, the central four-star is singular, $σ=1$, and the resulting dimension 34 is exactly accounted for by the classical local correction. At $t=1/5$, however, all interior vertices are nonsingular and $σ=0$, yet $\dim S_3^1(\mathcal{T}(1/5))=34>P_{\mathcal{T}(1/5)}(1,3)=33$. The smoothing-cofactor calculation shows that the dependence at $t=3/83$ is confined to the central vertex block, whereas the dependence at $t=1/5$ couples all seven interior vertex cycles even though every individual block has full row rank. A complementary Bernstein--Bézier calculation gives the same dimension profile. Thus the singular-four-star correction does not capture every geometry-dependent contribution to $\dim S_3^1(\mathcal{T})$; genuinely global compatibility must also be taken into account.

math.NA↗

arXiv2025

SDEIT: Semantic-Driven Electrical Impedance Tomography

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.

cs.CV↗

arXiv2021

Space Mapping of Spline Spaces over Hierarchical T-meshes

In this paper, we construct a bijective mapping between a biquadratic spline space over the hierarchical T-mesh and the piecewise constant space over the corresponding crossing-vertex-relationship graph (CVR graph). We propose a novel structure, by which we offer an effective and easy operative method for constructing the basis functions of the biquadratic spline space. The mapping we construct is an isomorphism. The basis functions of the biquadratic spline space hold the properties such as linearly independent, completeness and the property of partition of unity, which are the same with the properties for the basis functions of piecewise constant space over the CVR graph. To demonstrate that the new basis functions are efficient, we apply the basis functions to fit some open surfaces.

cs.CG↗

arXiv2019

Modified PHT-splines

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.

math.NA↗

arXiv2019

Numerical integration over implicitly defined domains with topological guarantee

Numerical integration over the implicitly defined domains is challenging due to topological variances of implicit functions. In this paper, we use interval arithmetic to identify the boundary of the integration domain exactly, thus getting the correct topology of the domain. Furthermore, a geometry-based local error estimate is explored to guide the hierarchical subdivision and save the computation cost. Numerical experiments are presented to demonstrate the accuracy and the potential of the proposed method.

math.NA↗

arXiv2015

Dimensions of Biquadratic and Bicubic Spline Spaces over Hierarchical T-meshes

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.

math.NA↗

arXiv2014

Approximating rational Bezier curves by constrained Bezier curves of arbitrary degree

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.

math.NA↗

arXiv2011

The Dimension of Spline Spaces with Highest Order Smoothness over Hierarchical T-meshes

This paper discusses the dimension of spline spaces with highest order smoothness over hierarchical T-meshes over certain type of hierarchical T-meshes. The major step is to set up a bijection between the spline space with highest order smoothness over a hierarchical T-mesh and a univariate spline space whose definition depends on the l-edges of the extended T-mesh. We decompose the univariate spline space into direct sums in the sense of isomorphism using the theory of the short exact sequence in homological algebra. According to the decomposition of the univariate spline space, the dimension formula of the spline space with highest order smoothness over certain type of hierarchical T-mesh is presented. A set of basis functions of the spline space is also constructed.

cs.CG↗

arXiv2008

Dimensions of Biquadratic Spline Spaces over T-meshes

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.

math.NA↗