Shizuo Kaji

Sho Ozaki, Shizuo Kaji, Toshikazu Imae et al.

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.

Shizuo Kaji, Kenji Kajiwara, Shota Shigetomi

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.

Shizuo Kaji, Takeki Sudo, Kazushi Ahara

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.

Sho Ozaki, Shizuo Kaji, Kanabu Nawa et al.

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?"

Nozomi Hata, Shizuo Kaji, Akihiro Yoshida et al.

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.

Shizuo Kaji, Toshiaki Maeno, Koji Nuida et al.

It is known that any $n$-variable function on a finite prime field of characteristic $p$ can be expressed as a polynomial over the same field with at most $p^n$ monomials. However, it is not obvious to determine the polynomial for a given concrete function. In this paper, we study the concrete polynomial expressions of the carries in addition and multiplication of $p$-ary integers. For the case of addition, our result gives a new family of symmetric polynomials, which generalizes the known result for the binary case $p = 2$ where the carries are given by elementary symmetric polynomials. On the other hand, for the case of multiplication of $n$ single-digit integers, we give a simple formula of the polynomial expression for the carry to the next digit using the Bernoulli numbers, and show that it has only $(n+1)(p-1)/2 + 1$ monomials, which is significantly fewer than the worst-case number $p^n$ of monomials for general functions. We also discuss applications of our results to cryptographic computation on encrypted data.

Koji Nuida, Takuro Abe, Shizuo Kaji et al.

In this paper, we specify a class of mathematical problems, which we refer to as "Function Density Problems" (FDPs, in short), and point out novel connections of FDPs to the following two cryptographic topics; theoretical security evaluations of keyless hash functions (such as SHA-1), and constructions of provably secure pseudorandom generators (PRGs) with some enhanced security property introduced by Dubrov and Ishai [STOC 2006]. Our argument aims at proposing new theoretical frameworks for these topics (especially for the former) based on FDPs, rather than providing some concrete and practical results on the topics. We also give some examples of mathematical discussions on FDPs, which would be of independent interest from mathematical viewpoints. Finally, we discuss possible directions of future research on other cryptographic applications of FDPs and on mathematical studies on FDPs themselves.