arXiv subjects

Shizuo Kaji

Publications and source records attributed to Shizuo Kaji.

At least 19 recordsLinked to original sources

arXiv2026

Topology-based Filtering of Graph Signals via Persistent Homology

We study topology-based filtering of vertex-defined signals on graphs and their two-dimensional analogues. Unlike graph-spectral filters, the proposed approach distinguishes features by topological persistence rather than by spatial wavelength or periodicity. We consider graphs with faces embedded in surfaces, a class that includes discrete models of images and meshes. We prove that, in general, exact simultaneous removal of low-persistence features in dimensions $0$ and $1$ is impossible. This motivates a relaxed formulation, for which we introduce the Low Persistence Filter (LPF). The LPF removes finite-persistence features below a prescribed threshold while controlling the resulting $\ell_\infty$ perturbation of the signal. We illustrate the method on one-dimensional signals, two-dimensional images, and signals on triangular meshes. A Python implementation is publicly available.

eess.SP↗

arXiv2026

Non-negative Matrix Factorisation with Topological Regularisation

We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

cs.LG↗

arXiv2026

An explicit construction of Kaleidocycles by elliptic theta functions

We consider the configuration space of ordered points on the two-dimensional sphere that satisfy a specific system of quadratic equations. We construct periodic orbits in this configuration space using elliptic theta functions and show that they simultaneously satisfy semi-discrete analogues of mKdV and sine-Gordon equations. The configuration space we investigate corresponds to the state space of a linkage mechanism known as the Kaleidocycle, and the constructed orbits describe the characteristic motion of the Kaleidocycle. A key consequence of our construction is the proof that Kaleidocycles exist for any number of tetrahedra greater than five. Our approach is founded on the relationship between the deformation of spatial curves and integrable systems, offering an intriguing example where an integrable system is explicitly solved to generate an orbit in the space of real solutions to polynomial equations defined by geometric constraints.

nlin.SI↗

arXiv2026

Topological flow data analysis for transient flow patterns: a graph-based approach

We introduce a method of time series analysis for two-dimensional transient flow patterns based on Topological Flow Data Analysis (TFDA), a new approach to topological data analysis. TFDA identifies local topological flow structures from an instantaneous streamline pattern and describes their global connections as a unique planar tree and its string representation. With TFDA, the evolution of two-dimensional flow patterns is reduced to a discrete dynamical system represented as a transition graph between topologically equivalent streamline patterns. We apply this method to study the lid-driven cavity flow for Reynolds numbers from $Re=14000$ to $16000$, a benchmark problem in the analysis of fluid dynamics. Our approach can extract some physical information from the lid-driven cavity flow: transition of the flow from periodic to quasi-periodic and chaotic; estimation of the period of periodic dynamics; relation between variations in energy and enstrophy and topological changes in flow patterns; statistical properties of intricate flow evolution at higher Reynolds number. In addition, we perform an observational causal inference to analyse changes in local flow patterns in the cavity corner. This work demonstrates the potential of TFDA-based time series analysis to uncover complex dynamical behaviours in fluid flow data from a topological perspective.

physics.flu-dyn↗

arXiv2026

Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain

We present a computational method for reconstructing a vector field on a convex polytope $\mathcal{P} \subset \mathbb{R}^d$ of arbitrary dimension from discrete samples. We specifically address the scenario where the vector field is subject to a no-penetration (slip) boundary condition, requiring it to be tangent to the boundary $\partial \mathcal{P}$. Given a degree bound $k$, our algorithm computes a polynomial vector field of degree at most $k$ that fits the observed data in the least-squares sense while exactly satisfying the tangency constraints. Central to our approach is an explicit characterization of the module of polynomial vector fields tangent to $\partial \mathcal{P}$, derived using algebraic concepts from the theory of hyperplane arrangements.

math.DS↗

arXiv2026

Shape-Aware Topological Representation for Pipeline Hyperbola Detection in GPR Data

Ground Penetrating Radar (GPR) is a widely used Non-Destructive Testing (NDT) technique for subsurface exploration, particularly in infrastructure inspection and maintenance. However, conventional interpretation methods are often limited by noise sensitivity and a lack of structural awareness. This study presents a novel framework that enhances the detection of underground utilities, especially pipelines, by integrating shape-aware topological features derived from B-scan GPR images using Topological Data Analysis (TDA), with the spatial detection capabilities of the YOLOv5 deep neural network (DNN). We propose a novel shape-aware topological representation that amplifies structural features in the input data, thereby improving the model's responsiveness to the geometrical features of buried objects. To address the scarcity of annotated real-world data, we employ a Sim2Real strategy that generates diverse and realistic synthetic datasets, effectively bridging the gap between simulated and real-world domains. Experimental results demonstrate significant improvements in mean Average Precision (mAP), validating the robustness and efficacy of our approach. This approach underscores the potential of TDA-enhanced learning in achieving reliable, real-time subsurface object detection, with broad applications in urban planning, safety inspection, and infrastructure management.

eess.SP↗

arXiv2025

Polyhedra of Constant Gaussian Curvature

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface embedded in $\mathbb{R}^3$ with constant Gaussian curvature must be a sphere, reflecting the strong rigidity of differential geometry. In contrast, the discrete setting, where curvature is represented as an angular defect concentrated at vertices, admits far greater flexibility. For instance, a flat torus can be realised as a polyhedral surface in $\mathbb{R}^3$ with zero curvature at every vertex. We establish a general result: any closed surface, whether orientable or non-orientable and of arbitrary genus, can be realised in $\mathbb{R}^3$ as a (possibly self-intersecting) polyhedral surface in which every vertex has the same angular defect. This highlights a fundamental distinction between discrete and smooth settings, showing that curvature constraints in the discrete realm impose fewer restrictions. Our proof is constructive and, once recognised, entirely elementary. Yet this fundamental fact appears to have gone unnoticed in the existing literature.

math.DG↗

arXiv2024

Iterative CT Reconstruction via Latent Variable Optimization of Shallow Diffusion Models

Image-generative artificial intelligence (AI) has garnered significant attention in recent years. In particular, the diffusion model, a core component of generative AI, produces high-quality images with rich diversity. In this study, we proposed a novel computed tomography (CT) reconstruction method by combining the denoising diffusion probabilistic model with iterative CT reconstruction. In sharp contrast to previous studies, we optimized the fidelity loss of CT reconstruction with respect to the latent variable of the diffusion model, instead of the image and model parameters. To suppress the changes in anatomical structures produced by the diffusion model, we shallowed the diffusion and reverse processes and fixed a set of added noises in the reverse process to make it deterministic during the inference. We demonstrated the effectiveness of the proposed method through the sparse-projection CT reconstruction of 1/10 projection data. Despite the simplicity of the implementation, the proposed method has the potential to reconstruct high-quality images while preserving the patient's anatomical structures and was found to outperform existing methods, including iterative reconstruction, iterative reconstruction with total variation, and the diffusion model alone in terms of quantitative indices such as the structural similarity index and peak signal-to-noise ratio. We also explored further sparse-projection CT reconstruction using 1/20 projection data with the same trained diffusion model. As the number of iterations increased, the image quality improved comparable to that of 1/10 sparse-projection CT reconstruction. In principle, this method can be widely applied not only to CT but also to other imaging modalities.

cs.CV↗

arXiv2022

Free-form Design of Discrete Architectural Surfaces by use of Circle Packing

This paper presents an efficient approach for the conceptual design of architectural surfaces which are composed of triangular panels. In the free-form design of discrete architectural surfaces, the Gaussian curvature plays an important role not only aesthetically but also in terms of stiffness and constructability. However, designing a surface manually with specific Gaussian curvatures can be a time-consuming task. We propose a method to find a triangulated surface with user-specified Gaussian curvatures (not limited to constant Gaussian curvatures) and boundary vertex positions. In addition, the conformal class of the final design can be specified; that is, the user has control over the shape (the corner angles) of each triangular panel. The panels could be encouraged to form a regular tessellation or kept close to those of the initial design. The controllability of the conformal class suppresses possible distortion of the panels, resulting in higher structural performance and aesthetics. Our method relies on the idea in computational conformal geometry called circle packing. In this line of research, the discrete Ricci flow has been widely used for surface modelling. However, it is not trivial to incorporate constraints such as boundary locations and convexity of the spanned surface, which are essential to architectural applications. We propose a perturbation of the discrete Ricci energy and develop a least-squares-based optimisation scheme to address these problems with an open-source implementation available online.

cs.CG↗

arXiv2022

Training of deep cross-modality conversion models with a small dataset, and their application in megavoltage CT to kilovoltage CT conversion

In recent years, deep-learning-based image processing has emerged as a valuable tool for medical imaging owing to its high performance. However, the quality of deep-learning-based methods heavily relies on the amount of training data; the high cost of acquiring a large dataset is a limitation to their utilization in medical fields. Herein, based on deep learning, we developed a computed tomography (CT) modality conversion method requiring only a few unsupervised images. The proposed method is based on CycleGAN with several extensions tailored for CT images, which aims at preserving the structure in the processed images and reducing the amount of training data. This method was applied to realize the conversion of megavoltage computed tomography (MVCT) to kilovoltage computed tomography (kVCT) images. Training was conducted using several datasets acquired from patients with head and neck cancer. The size of the datasets ranged from 16 slices (two patients) to 2745 slices (137 patients) for MVCT and 2824 slices (98 patients) for kVCT. The required size of the training data was found to be as small as a few hundred slices. By statistical and visual evaluations, the quality improvement and structure preservation of the MVCT images converted by the proposed model were investigated. As a clinical benefit, it was observed by medical doctors that the converted images enhanced the precision of contouring. We developed an MVCT to kVCT conversion model based on deep learning, which can be trained using only a few hundred unpaired images. The stability of the model against changes in data size was demonstrated. This study promotes the reliable use of deep learning in clinical medicine by partially answering commonly asked questions, such as "Is our data sufficient?" and "How much data should we acquire?"

cs.CV↗

arXiv2021

Mod 2 cohomology of 2-local finite groups of low rank

We determine the mod $2$ cohomology over the Steenrod algebra of the classifying spaces of the free loop groups $LG$ for compact groups $G=Spin(7)$, $Spin(8)$, $Spin(9)$, and $F_4$. Then, we show that they are isomorphic as algebras over the Steenrod algebra to the mod $2$ cohomology of the corresponding Chevalley groups of type $G(q)$, where $q$ is an odd prime power. In a similar manner, we compute the cohomology of the free loop space over $BDI(4)$ and show that it is isomorphic to that of $BSol(q)$ as algebras over the Steenrod algebra.

math.AT↗

arXiv2020

Nested Subspace Arrangement for Representation of Relational Data

Studies on acquiring appropriate continuous representations of discrete objects, such as graphs and knowledge base data, have been conducted by many researchers in the field of machine learning. In this study, we introduce Nested SubSpace (NSS) arrangement, a comprehensive framework for representation learning. We show that existing embedding techniques can be regarded as special cases of the NSS arrangement. Based on the concept of the NSS arrangement, we implement a Disk-ANChor ARrangement (DANCAR), a representation learning method specialized to reproducing general graphs. Numerical experiments have shown that DANCAR has successfully embedded WordNet in ${\mathbb R}^{20}$ with an F1 score of 0.993 in the reconstruction task. DANCAR is also suitable for visualization in understanding the characteristics of graphs.

cs.LG↗

arXiv2020

Cubical Ripser: Software for computing persistent homology of image and volume data

We introduce Cubical Ripser for computing persistent homology of image and volume data (more precisely, weighted cubical complexes). To our best knowledge, Cubical Ripser is currently the fastest and the most memory-efficient program for computing persistent homology of weighted cubical complexes. We demonstrate our software with an example of image analysis in which persistent homology and convolutional neural networks are successfully combined. Our open-source implementation is available online.

cs.CV↗

arXiv2019

Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves

A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage mechanisms which consist of $n$-copies of a rigid body joined together by hinges to form a ring. Each hinge joint has its own axis of revolution and rigid bodies joined to it can be freely rotated around the axis. The family includes the famous threefold symmetric Bricard6R linkage also known as the Kaleidocycle, which exhibits a characteristic "turning over" motion. We can model such a linkage as a discrete closed curve in $\mathbb{R}^3$ with a constant torsion up to sign. Then, its motion is described as the deformation of the curve preserving torsion and arc length. We describe certain motions of this object that are governed by the semi-discrete mKdV equations, where infinitesimally the motion of each vertex is confined in the osculating plane.

nlin.SI↗

arXiv2019

A closed linkage mechanism having the shape of a discrete Möbius strip

A closed linkage mechanism in three-dimensional space is an object comprising rigid bodies connected with hinges in a circular form like a rosary. Such linkages include Bricard6R and Bennett4R. To design such a closed linkage, it is necessary to solve a high-degree algebraic equation, which is generally difficult. In this lecture, the author proposes a new family of closed linkage mechanisms with an arbitrary number of hinges as an extension of a certain Bricard6R. They have singular properties, such as one-dimensional degree of freedom (1-DOF), and certain energies taking a constant value regardless of the state. These linkage mechanisms can be regarded as discrete Möbius strips and may be of interest in the context of pure mathematics as well. However, many of the properties described here have been confirmed only numerically, with no rigorous mathematical proof, and should be interpreted with caution.

math.HO↗

arXiv2019

Flag Bott manifolds of general Lie type and their equivariant cohomology rings

In this article we introduce flag Bott manifolds of general Lie type as the total spaces of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds, and admit nice torus actions. We calculate the torus equivariant cohomology rings of flag Bott manifolds of general Lie type.

math.AT↗

arXiv2019

Overview of image-to-image translation by use of deep neural networks: denoising, super-resolution, modality conversion, and reconstruction in medical imaging

Since the advent of deep convolutional neural networks (DNNs), computer vision has seen an extremely rapid progress that has led to huge advances in medical imaging. This article does not aim to cover all aspects of the field but focuses on a particular topic, image-to-image translation. Although the topic may not sound familiar, it turns out that many seemingly irrelevant applications can be understood as instances of image-to-image translation. Such applications include (1) noise reduction, (2) super-resolution, (3) image synthesis, and (4) reconstruction. The same underlying principles and algorithms work for various tasks. Our aim is to introduce some of the key ideas on this topic from a uniform point of view. We introduce core ideas and jargon that are specific to image processing by use of DNNs. Having an intuitive grasp of the core ideas of and a knowledge of technical terms would be of great help to the reader for understanding the existing and future applications. Most of the recent applications which build on image-to-image translation are based on one of two fundamental architectures, called pix2pix and CycleGAN, depending on whether the available training data are paired or unpaired. We provide computer codes which implement these two architectures with various enhancements. Our codes are available online with use of the very permissive MIT license. We provide a hands-on tutorial for training a model for denoising based on our codes. We hope that this article, together with the codes, will provide both an overview and the details of the key algorithms, and that it will serve as a basis for the development of new applications.

physics.med-ph↗

arXiv2019

Geometric representations of finite groups on real toric spaces

We develop a framework to construct geometric representations of finite groups $G$ through the correspondence between real toric spaces $X^{\mathbb R}$ and simplicial complexes with characteristic matrices. We give a combinatorial description of the $G$-module structure of the homology of $X^{\mathbb R}$. As applications, we make explicit computations of the Weyl group representations on the homology of real toric varieties associated to the Weyl chambers of type $A$ and $B$, which show an interesting connection to the topology of posets. We also realize a certain kind of Foulkes representation geometrically as the homology of real toric varieties.

math.AT↗