Euclidean Space Research Articles

BTSegDiff: Brain tumor segmentation based on multimodal MRI Dynamically guided diffusion probability model.

In the treatment of brain tumors, accurate diagnosis and treatment heavily rely on reliable brain tumor segmentation, where multimodal Magnetic Resonance Imaging (MRI) plays a pivotal role by providing valuable complementary information. This integration significantly enhances the performance of brain tumor segmentation. However, due to the uneven grayscale distribution, irregular shapes, and significant size variations in brain tumor images, this task remains highly challenging. In order to overcome these obstacles, we have introduced a novel framework for automated segmentation of brain tumors that leverages the diverse information from multi-modal MRI scans. Our proposed method is named BTSegDiff and it is based on a Diffusion Probability Model (DPM). First, we designed a dynamic conditional guidance module consisting of an encoder. This encoder is used to extract information from multimodal MRI images and guide the DPM in generating accurate and realistic segmentation masks. During the guidance process, we need to fuse the diffused generated features with the extracted multimodal features. However, diffusion process itself introduces a significant amount of Gaussian noise, which can affect the fusion results. Therefore, we designed a Fourier domain feature fusion module to transfer this fusion process to Euclidean space and reduce the impact of high-frequency noise on fusion. Lastly, we have taken into account that the DPM, as a generative model, produces non-unique results with each sampling. In the meticulous field of medicine, this is highly detrimental. Therefore, we have designed a Stepwise Uncertainty Sampling module based on Monte Carlo uncertainty calculation to generate unique outcomes and enhance segmentation accuracy simultaneously. To validate the effectiveness of our approach, we perform a validation on the popular BraTs2020 and BraTS2021 benchmarks. The experimental results show that our method outperforms many existing brain tumor segmentation methods. Our code is available at https://github.com/jaceqin/BTSegDiff.

Quantum pseudo-harmonic oscillator potential in non-Euclidean space: application to diatomic molecules

Abstract The present work analyzes a physical system with a quantum pseudo-harmonic oscillator in three-dimensional constant curvature spaces within the framework of non-relativistic theory. We present expressions for the energy equation and radial wavefunctions that depend on the curvature parameter k, using the functional analysis approach and the asymptotic iteration method. Additionally, we calculate the energy eigenvalues for diatomic molecules N2, H2, and ScH as a function of the constant curvature k. Using the Hellmann-Feynmann theorem, we derive expressions for the curvature-dependent expectation values of r-2 and p2, which we detail for the diatomic molecule system in this work. Furthermore, we perform a comparative analysis of the results for non-Euclidean space (spherical and hyperbolic spaces with constant curvature) and Euclidean space.

On the hidden layer-to-layer topology of the representations of reality realised within neural networks

PurposeConsider an information processing algorithm that is designed to process an input data object onto an output data object via a number of successive internal {\it layers} and mappings between them. The possible activation state within each layer can be represented as a cube within Euclidean space of a high dimension (e.g. equal to the number of artificial neurons at that level). Multiple instances of such input objects produce a point cloud within each layer’s cube: this is the “representation of the reality” at that layer, as sampled by the set of input objects.Design/methodology/approachMost neural networks reduce the dimension of each layer’s cube from layer to successive layer. This gives the false impression of refining the inner representations of reality, distilling it down to fewer dimensions from which to discriminate or to infer outcomes (whatever is the aim). However, the representation of reality realised within each layer’s cube is a manifold, a curved subset embedded within it and of much lower dimension. Investigations show that such manifolds may not always be reducing in their local dimension. Instead, the manifold may become folded over and over, filling up further dimensions and creating non-realistic (unforeseeable) proximities.FindingsWe discuss some of the likely consequences of these relatively unforeseen characteristics and, in particular, the possible vulnerability of such algorithms to non-realistic perturbations. We consider a possible response to this issue.Practical implicationsNew forms of calibration are necessary, using geometric/topological loss functions, as opposed to simple (variation-limiting) regularisation terms.Originality/valueWe apply persistent homology methods to understand how the images of the point cloud (representing the sampled reality) change as they pass from layer to layer.

Geometric and topological aspects of the passage from local to global invertibility

As a refinement of the global invertibility problem, we address the issue of estimating the cardinality of a prescribed fiber F-1(q) of a locally invertible map solely in terms of objects that are naturally associated to q itself. The following is a prototypical result. Let F:Rn→Rn be a local diffeomorphism, n≥3, and q∈F(Rn). We show that q is assumed exactly once by F if the pre-image of every 2-plane containing q, when viewed as a geometric surface in Euclidean n-space, is conformally diffeomorphic to R2. The proofs of this and other theorems involve geometric constructions, the Poincaré-Hopf theorem, the Bôcher theorem on positive harmonic functions, condensers on Riemann surfaces, and elliptic estimates. We conclude with a section that is devoted to invertibility problems related to various aspects of dynamics, algebraic and differential geometry, real and complex analysis. The paper is written in a semi-expository style, as an invitation to global injectivity.

On minimal hypersurfaces in Euclidean spaces and Riemannian manifolds

On minimal hypersurfaces in Euclidean spaces and Riemannian manifolds

Generalization of dual numbers and dual Euclidean 3-Space

Generalization of dual numbers and dual Euclidean 3-Space

Riemannian Shape Optimization of Thin Shells Using Isogeometric Analysis

ABSTRACTIn this contribution, we consider the optimal shape design of thin elastic shell structures based on a linearized shell model of Koiter's type, whose shape can be described by a surface immersed in three‐dimensional Euclidean space. We regard the set of unparametrized immersions of the surface as an infinite‐dimensional Riemannian shape space and perform optimization in this setting using the Riemannian shape gradient. Nonuniform rational basis splines (NURBS) are employed to discretize the shell and numerically solve the underlying equations that govern its mechanical behavior via isogeometric analysis. By representing NURBS patches as B‐spline patches in real projective space, NURBS weights can also be incorporated into the optimization routine. We discuss the practical implementation of the method and demonstrate our approach on the compliance minimization of a half‐cylindrical shell under static load and fixed area constraint.

Novel exploration of topological degree method for noninstantaneous impulsive fractional integro-differential equation through the application of filtering system

The primary focus of this paper is to explore the concept of an initial condition for a noninstantaneous impulsive fractional integro-differential equation of order 0<ϑ<1 in an n-dimensional Euclidean space. Using the Laplace transform method, we derive the solution representation of the given dynamical system and investigate the existence and uniqueness of the mild solution through the topological degree method and Gronwall’s inequality. Furthermore, we present a filter model featuring a finite impulsive response, which serves as a practical demonstration of the proposed system because it effectively captures the memory effects inherent in fractional-order systems and enhances system reliability with minimal input. Finally, the numerical computations with graphical illustrations provide concrete examples that validate the theoretical results, showcasing how the behavior of the system is impacted by changes in fractional order and emphasizing the versatility of our proposed method.

The metric projection over a polyhedral set through the relative interiors of its faces

Abstract We first characterize the region of the n-dimensional Euclidean space for which two optimization problems with the square distance function as common objective function, but different constraints, are equivalent. The affine hull of a certain face of a closed convex set $$C\subseteq {\mathbb {R}}^n$$ C ⊆ R n is the constraint associated to one problem and the whole closed convex set C is the constraint associated to the other problem. Such optimization problems are best approximation problems which can be reformulated in terms of the metric projection. Using the language of the metric projection, we characterize the region of $${\mathbb {R}}^n$$ R n which is metrically projected over a face of C in the same way that it is projected over the affine hull of the face itself. The metric projection over such a face is the one associated to the entire closed convex set, and the metric projection over the affine hull of such a face is the one associated to the affine hull. It turns out that this region is the closure of the inverse image, through the metric projection over the entire closed convex set, of the relative interior of the face. We also characterize analytically the closure of the regions of $${\mathbb {R}}^n$$ R n that are projected over the relative interiors of the faces of a polyhedral set, through the metric projection of the polyhedral set itself. We show that these regions are polyhedral convex sets by explicitly characterizing them through systems of linear inequalities.

The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics.

We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well.

Uncovering the Euclidean Geometry of Data

Suppose you have a collection of objects in some exotic metric space and you’d like to see what they would look like if they were instead in Euclidean space. Or suppose these objects aren’t even points in a metric space, they are just entities for which you have some rough, intuitive notion of distance between each pair of them—one that need not satisfy any official mathematical properties like the triangle inequality. There is a linear algebraic optimization procedure called multidimensional scaling that embeds these objects in Euclidean space in a manner that approximates their original distances. This uncovers the Euclidean geometry hidden in data. This article explores how it works and what it’s useful for, particularly in a data science context.

Naimark-spatial families of equichordal tight fusion frames

An equichordal tight fusion frame (▪) is a finite sequence of equi-dimensional subspaces of a Euclidean space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every ▪ is a type of optimal Grassmannian code, being a way to arrange a given number of members of a Grassmannian so that the minimal chordal distance between any pair of them is as large as possible. Any nontrivial ▪ has both a Naimark complement and spatial complement which themselves are ▪s. We show that taking iterated alternating Naimark and spatial complements of any ▪ of at least five subspaces yields an infinite family of ▪s with pairwise distinct parameters. Generalizing a method by King, we then construct ▪s from difference families for finite abelian groups, and use our Naimark-spatial theory to gauge their novelty.

Growth functions of periodic space tessellations.

This work analyzes the rules governing the growth of the numbers of vertices, edges and faces in all possible periodic tessellations of the 2D Euclidean space, and encodes those rules in several types of polynomial growth functions. These encodings map the geometric, combinatorial and topological properties of the tessellations into sets of integer coefficients. Several general statements about these encodings are given with rigorous mathematical proof. The variation of the growth functions is represented graphically and analyzed in orphic diagrams, so named because of their similarity to orphic art. Several examples of 3D space groups are included, to emphasize the complexity of the growth functions in higher dimensions. A freely available Python library is presented to facilitate the discovery of the growth functions and the generation of orphic diagrams.

Geometric properties of the Minkowski operator

This article is about Minkowski difference of sets, which is one of the Minkowski operators. The necessary and sufficient conditions for the existence of the Minkowski difference of given regular polygons in the plane were derived. The method of finding the Minkowski difference of given regular tetrahedrons in the Euclidean space R3 was explained. At the end of the article, the obtained results were summarized and a geometric method for finding the Minkowski difference of the convex set M and compact set N given in Rn was shown. The theory of foliations was applied to find the Minkowski difference of sets. New geometric concepts such as “dense embedding” and “completely dense embedding” were introduced. An important geometric property of the Minkowski operator was introduced and proved as a theorem.

ON THE GEODESIC CURVES OF CONSTANT BREADTH ACCORDING TO THE EXTENDED DARBOUX FRAME

In this study, we investigate the geodesic curves of constant breadth on an oriented hypersurface in Euclidean 4-space according to the extended Darboux frame and give the general differential equation characterizing these curves.

On some properties of Alexandroff space

The generalized definition of topology is based on the properties of standard Euclidean topology. The goal of this paper is to study spaces that have topologies, which satisfies the stronger condition namely arbitrary intersection of open sets are open. The topological space with this strong property is known as Alexandroff space. With this restriction we lose some important spaces such as Euclidean spaces, but the specialized spaces in turn display interesting properties that are not necessary for a standard Euclidean topology.

Anisotropic Diffusion in Riemannian Colour Geometry

Anisotropic diffusion has long been an important tool in image processing. More recently, it has also found its way to colour imaging. Until now, mainly Euclidean colour spaces have been considered in this context, but recent years have seen a renewed interest in and importance of non-Euclidean colour geometry. The main contribution of this paper is the derivation of the equations for anisotropic diffusion in Riemannian colour geometry. It is demonstrated that it contains several well-known solutions such as Perona–Malik diffusion and Tschumperlé–Deriche diffusion as special cases. Furthermore, it is shown how it is non-trivially connected to Sochen’s general framework for low-level vision. The main significance of the method is that it decouples the coordinates used for solving the diffusion equation from the ones that define the metric of the colour manifold, and thus directs the magnitude and direction of the diffusion through the diffusion tensor. It also enables the use of non-Euclidean colour manifolds and metrics for applications such as denoising, inpainting, and demosaicing, based on anisotropic diffusion.

A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold

Deep Learning (DL) has achieved remarkable success in tackling complex Artificial Intelligence tasks. The standard training of neural networks employs backpropagation to compute gradients and utilizes various optimization algorithms in the Euclidean space \(\mathbb {R}^n \) . However, this optimization process faces challenges, such as the local optimal issues and the problem of gradient vanishing and exploding. To address these problems, Riemannian optimization offers a powerful extension to solve optimization problems in deep learning. By incorporating the prior constraint structure and the metric information of the underlying geometric information, Riemannian optimization-based DL offers a more stable and reliable optimization process, as well as enhanced adaptability to complex data structures. This article presents a comprehensive survey of applying geometric optimization in DL, including the basic procedure of geometric optimization, various geometric optimizers, and some concepts of the Riemannian manifold. In addition, it investigates various applications of geometric optimization in different DL networks for diverse tasks and discusses typical public toolboxes that implement optimization on the manifold. This article also includes a performance comparison among different deep geometric optimization methods in image recognition scenarios. Finally, this article elaborates on future opportunities and challenges in this field.

HARDY INEQUALITIES AND IDENTITIES RELATED TO THE BAOUENDI-GRUSHIN VECTOR FIELDS AND LANDAU-HAMILTONIAN

In this paper, we present a weighted Hardy identity related to the Baouendi-Grushin vector fields and its applications in the context of differential inequalities. By selecting appropriate parameters, the Hardy identity related to the Baouendi-Grushin operator implies numerous sharp remainder formulae for Hardy type inequalities. In the commutative case, we obtain improved weighted Hardy inequalities in the setting of the Euclidean space. For example, in a special case, by dropping non-negative remainder terms, related to the Baouendi-Grushin operator, and choosing suitable parameters our identity allows us to derive an improved critical Hardy inequality for the radial derivative operator with a sharp constant that does not depend on the topological dimension. We employ the method of factorizing differential expressions, as used by Gesztesy and Littlejohn in [1]. In this paper, we demonstrate the application of the factorization method in the noncommutative Baouendi-Grushin setting. As an application of the Hardy identity related to the Baouendi-Grushin vector fields, we establish a Hardy inequality for the generalized Landau Hamiltonian (or the twisted Laplacian) with remainder terms.

Rotational Surfaces in Terms of Coordinate Finite Chen II-Type

In this study, we first establish several formulae according to the first and second Beltrami operators. We discuss the class of surfaces of revolution in the 3-dimensional Euclidean space E3 without parabolic points, in which the position vector X satisfies ∆IIX = DX, with ∆II is the Laplace operator of the metric II of the surface and D is a square matrix of order 3. We prove that surfaces satisfying the preceding relation are either part of a sphere or catenoid.