arXiv subjects

Gyeongha Hwang

Publications and source records attributed to Gyeongha Hwang.

At least 19 recordsLinked to original sources

arXiv2026

Design Choices in Splitting-Based Self-Supervised Sparse-View CT Reconstruction

Self-supervised data splitting has emerged as a promising paradigm for sparse-view CT reconstruction, enabling training from incomplete measurements without fully sampled ground truth. However, the influence of key design choices, including partitioning strategy, preprocessing, and inference, remains insufficiently understood. In this work, we introduce a unified framework that decomposes splitting-based reconstruction into these three components, enabling controlled comparison of existing methods and two incremental extensions: multi-partition splitting and an alternative inference strategy. Experiments on simulated LoDoPaB-CT data under independent and correlated noise, together with validation on the real-world 2DeteCT dataset, show that the optimal partitioning strategy strongly depends on the measurement noise structure. Lattice-based splitting performs favorably under independent noise, whereas angular masking is more robust under correlated noise and real measured data. Multi-partition splitting consistently improves over pure projection-wise splitting in several settings. Complementary perceptual and structural metrics, including LPIPS and HaarPSI, reveal differences between masking strategies that are less apparent from PSNR and SSIM alone. These results provide practical guidelines for designing self-supervised sparse-view CT reconstruction methods and highlight the limitations of common independence assumptions in realistic imaging environments.

cs.CV↗

arXiv2026

Denoising ICF Images with Multiplicative Uniform Noise: A Self-Supervised Study Based on the Log-Domain Noisier2Inverse Framework

This paper documents the implementation and evaluation of a self-supervised denoising framework on Inertial Confinement Fusion (ICF) images corrupted by Multiplicative Uniform noise: the \emph{Log-Domain Noisier2Inverse} framework. This framework is developed and analysed in this work; the key theoretical result -- that minimising the log-domain self-supervised loss is equivalent to supervised learning in the transformed domain -- is presented with full proof. We document significant implementation challenges arising from the unique characteristics of ICF imagery, describe the fixes applied at each stage, and report final quantitative results. The log-domain approach with per-image JSON Uniform noise loading (Variant~B) achieves the best result: a mean PSNR of $21.41\db$ and SSIM of $0.8358$, a $+19.46\db$ improvement over the noisy input baseline of $1.95\db$, substantially outperforming BM3D log-domain ($4.47\db$, SSIM $0.5181$) and Noise2Self ($4.75\db$, SSIM $0.0177$). Variant~A, using fixed Gaussian noise loading, achieves $21.39\db$ PSNR and SSIM $0.8436$. Of the three evaluated methods, Log-Domain Noisier2Inverse and Noise2Self are entirely self-supervised during training, requiring no clean ground truth data; BM3D is a classical filter-based method requiring no training at all. The clean reference images are used solely for quantitative evaluation of all three methods.

cs.CV↗

arXiv2026

SPLIT: Self-supervised Partitioning for Learned Inversion in Nonlinear Tomography

Machine learning has achieved impressive performance in tomographic reconstruction, but supervised training requires paired measurements and ground-truth images that are often unavailable. This has motivated self-supervised approaches, which have primarily addressed denoising and, more recently, linear inverse problems. We address nonlinear inverse problems and introduce SPLIT (Self-supervised Partitioning for Learned Inversion in Nonlinear Tomography), a self-supervised machine-learning framework for reconstructing images from nonlinear, incomplete, and noisy projection data without any samples of ground-truth images. SPLIT enforces cross-partition consistency and measurement-domain fidelity while exploiting complementary information across multiple partitions. Our main theoretical result shows that, under mild conditions, the proposed self-supervised objective is equivalent to its supervised counterpart in expectation. We regularize training with an automatic stopping rule that halts optimization when a no-reference image-quality surrogate saturates. As a concrete application, we derive SPLIT variants for multispectral computed tomography. Experiments on sparse-view acquisitions demonstrate high reconstruction quality and robustness to noise, surpassing classical iterative reconstruction and recent self-supervised baselines.

cs.CV↗

arXiv2026

Self-Supervised Angular Deblurring in Photoacoustic Reconstruction via Noisier2Inverse

Photoacoustic tomography (PAT) is an emerging imaging modality that combines the complementary strengths of optical contrast and ultrasonic resolution. A central task is image reconstruction, where measured acoustic signals are used to recover the initial pressure distribution. For ideal point-like or line-like detectors, several efficient and fast reconstruction algorithms exist, including Fourier methods, filtered backprojection, and time reversal. However, when applied to data acquired with finite-size detectors, these methods yield systematically blurred images. Although sharper images can be obtained by compensating for finite-detector effects, supervised learning approaches typically require ground-truth images that may not be available in practice. We propose a self-supervised reconstruction method based on Noisier2Inverse that addresses finite-size detector effects without requiring ground-truth data. Our approach operates directly on noisy measurements and learns to recover high-quality PAT images in a ground-truth-free manner. Its key components are: (i) PAT-specific modeling that recasts the problem as angular deblurring; (ii) a Noisier2Inverse formulation in the polar domain that leverages the known angular point-spread function; and (iii) a novel, statistically grounded early-stopping rule. In experiments, the proposed method consistently outperforms alternative approaches that do not use supervised data and achieves performance close to supervised benchmarks, while remaining practical for real acquisitions with finite-size detectors.

cs.CV↗

arXiv2026

Data-Consistent Learning of Inverse Problems

Inverse problems are inherently ill-posed, suffering from non-uniqueness and instability. Classical regularization methods provide mathematically well-founded solutions, ensuring stability and convergence, but often at the cost of reduced flexibility or visual quality. Learned reconstruction methods, such as convolutional neural networks, can produce visually compelling results, yet they typically lack rigorous theoretical guarantees. DC (DC) networks address this gap by enforcing the measurement model within the network architecture. In particular, null-space networks combined with a classical regularization method as an initial reconstruction define a convergent regularization method. This approach preserves the theoretical reliability of classical schemes while leveraging the expressive power of data-driven learning, yielding reconstructions that are both accurate and visually appealing.

math.NA↗

arXiv2026

HyDRA: Hybrid Denoising Regularization for Measurement-Only DEQ Training

Solving image reconstruction problems of the form $\mathbf{A} \mathbf{x} = \mathbf{y}$ remains challenging due to ill-posedness and the lack of large-scale supervised datasets. Deep Equilibrium (DEQ) models have been used successfully but typically require supervised pairs $(\mathbf{x},\mathbf{y})$. In many practical settings, only measurements $\mathbf{y}$ are available. We introduce HyDRA (Hybrid Denoising Regularization Adaptation), a measurement-only framework for DEQ training that combines measurement consistency with an adaptive denoising regularization term, together with a data-driven early stopping criterion. Experiments on sparse-view CT demonstrate competitive reconstruction quality and fast inference.

cs.CV↗

arXiv2025

Noisier2Inverse: Self-Supervised Learning for Image Reconstruction with Correlated Noise

We propose Noisier2Inverse, a correction-free self-supervised deep learning approach for general inverse problems. The proposed method learns a reconstruction function without the need for ground truth samples and is applicable in cases where measurement noise is statistically correlated. This includes computed tomography, where detector imperfections or photon scattering create correlated noise patterns, as well as microscopy and seismic imaging, where physical interactions during measurement introduce dependencies in the noise structure. Similar to Noisier2Noise, a key step in our approach is the generation of noisier data from which the reconstruction network learns. However, unlike Noisier2Noise, the proposed loss function operates in measurement space and is trained to recover an extrapolated image instead of the original noisy one. This eliminates the need for an extrapolation step during inference, which would otherwise suffer from ill-posedness. We numerically demonstrate that our method clearly outperforms previous self-supervised approaches that account for correlated noise.

cs.CV↗

arXiv2024

Implicit learning to determine variable sound speed and the reconstruction operator in photoacoustic tomography

Photoacoustic tomography (PAT) is a hybrid medical imaging technique that offer high contrast and a high spatial resolution. One challenging mathematical problem associated with PAT is reconstructing the initial pressure of the wave equation from data collected at the specific surface where the detectors are positioned. The study addresses this problem when PAT is modeled by a wave equation with unknown sound speed $c$, which is a function of spatial variables, and under the assumption that both the Dirichlet and Neumann boundary values on the detector surface are measured. In practical, we introduce a novel implicit learning framework to simultaneously estimate the unknown $c$ and the reconstruction operator using only Dirichlet and Neumann boundary measurement data. The experimental results confirm the success of our proposed framework, demonstrating its ability to accurately estimate variable sound speed and the reconstruction operator in PAT.

math.NA↗

arXiv2024

ECALL: Expectation-calibrated learning for unsupervised blind deconvolution

Blind deconvolution aims to recover an original image from a blurred version in the case where the blurring kernel is unknown. It has wide applications in diverse fields such as astronomy, microscopy, and medical imaging. Blind deconvolution is a challenging ill-posed problem that suffers from significant non-uniqueness. Solution methods therefore require the integration of appropriate prior information. Early approaches rely on hand-crafted priors for the original image and the kernel. Recently, deep learning methods have shown excellent performance in addressing this challenge. However, most existing learning methods for blind deconvolution require a paired dataset of original and blurred images, which is often difficult to obtain. In this paper, we present a novel unsupervised learning approach named ECALL (Expectation-CALibrated Learning) that uses separate unpaired collections of original and blurred images. Key features of the proposed loss function are cycle consistency involving the kernel and associated reconstruction operator, and terms that use expectation values of data distributions to obtain information about the kernel. Numerical results are used to support ECALL.

math.NA↗

arXiv2023

Mesh Density Adaptation for Template-based Shape Reconstruction

In 3D shape reconstruction based on template mesh deformation, a regularization, such as smoothness energy, is employed to guide the reconstruction into a desirable direction. In this paper, we highlight an often overlooked property in the regularization: the vertex density in the mesh. Without careful control on the density, the reconstruction may suffer from under-sampling of vertices near shape details. We propose a novel mesh density adaptation method to resolve the under-sampling problem. Our mesh density adaptation energy increases the density of vertices near complex structures via deformation to help reconstruction of shape details. We demonstrate the usability and performance of mesh density adaptation with two tasks, inverse rendering and non-rigid surface registration. Our method produces more accurate reconstruction results compared to the cases without mesh density adaptation.

cs.GR↗

arXiv2023

Reconstructing unknown inclusions for the biharmonic equation

Herein, we study an inverse problem for detecting unknown obstacles by the enclosure method using the Dirichlet--to--Neumann map for measurements. We justify the method for an penetrable obstacle case involving a biharmonic equation. We use complex geometrical optics solutions with a logarithmic phase to reconstruct some non--convex parts of the obstacle.

math.AP↗

arXiv2023

Convergence analysis of equilibrium methods for inverse problems

Solving inverse problems $Ax = y$ is central to a variety of practically important fields such as medical imaging, remote sensing, and non-destructive testing. The most successful and theoretically best-understood method is convex variational regularization, where approximate but stable solutions are defined as minimizers of $ \|A(\cdot) - y^\delta\|^2 / 2 + \alpha \mathcal{R}(\cdot)$, with $\mathcal{R}$ a regularization functional. Recent methods such as deep equilibrium models and plug-and-play approaches, however, go beyond variational regularization. Motivated by these innovations, we introduce implicit non-variational (INV) regularization, where approximate solutions are defined as solutions of $A^*(A x - y^\delta) + \alpha R(x) = 0$ for some regularization operator $R$. When the regularization operator is the gradient of a functional, INV reduces to classical variational regularization. However, in methods like DEQ and PnP, $R$ is not a gradient field, and the existing theoretical foundation remains incomplete. To address this, we establish stability and convergence results in this broader setting, including convergence rates and stability estimates measured via a absolute Bregman distance.

math.NA↗

arXiv2023

Self-supervised learning for a nonlinear inverse problem with forward operator involving an unknown function arising in photoacoustic tomography

In this article, we are concerned with a nonlinear inverse problem with a forward operator involving an unknown function. The problem arises in diverse applications and is challenging in the presence of an unknown function, which makes it ill-posed. Additionally, the nonlinear nature of the problem makes it difficult to use traditional methods, and thus, the study addresses a simplified version of the problem by either linearizing it or assuming knowledge of the unknown function. Here, we propose self-supervised learning to directly tackle a nonlinear inverse problem involving an unknown function. In particular, we focus on an inverse problem derived in photoacoustic tomograpy (PAT), which is a hybrid medical imaging with high resolution and contrast. PAT can be modeled based on the wave equation. The measured data provide the solution to an equation restricted to surface and initial pressure of an equation that contains biological information on the object of interest. The speed of a sound wave in the equation is unknown. Our goal is to determine the initial pressure and the speed of the sound wave simultaneously. Under a simple assumption that sound speed is a function of the initial pressure, the problem becomes a nonlinear inverse problem involving an unknown function. The experimental results demonstrate that the proposed framework performs successfully.

math.NA↗

arXiv2022

Global well-posedness and scattering of the two dimensional cubic focusing nonlinear Schrödinger system

In this article, we prove the global well-posedness and scattering of the cubic focusing infinite coupled nonlinear Schrödinger system on $\mathbb{R}^2$ below the threshold in $L_x^2h^1(\mathbb{R}^2\times \mathbb{Z})$. We first establish the variational characterization of the ground state, and derive the threshold of the global well-posedness and scattering. Then we show the global well-posedness and scattering below the threshold by the concentration-compactness/rigidity method, where the almost periodic solution is excluded by adapting the argument in the proof of the mass-critical nonlinear Schrödinger equations by B. Dodson. As a byproduct of the scattering of the cubic focusing infinite coupled nonlinear Schödinger system, we obtain the scattering of the cubic focusing nonlinear Schrödinger equation on the small cylinder, this is the first large data scattering result of the focusing nonlinear Schrödinger equations on the cylinders. In the article, we also show the global well-posedness and scattering of the two dimensional $N-$coupled focusing cubic nonlinear Schrödinger system in $\left(L^2(\mathbb{R}^2) \right)^N$.

math.AP↗

arXiv2017

On the modified scattering of $3$-d Hartree type fractional Schrödinger equations with Coulomb potential

In this paper we study 3-d Hartree type fractional Schrödin-ger equations: \begin{equation} i\partial_{t}u-|\nabla|^αu = λ\left(|x|^{-γ} *| u|^{2} \right)u,\;\;1 < α< 2,\;\;0 < γ< 3,\;\; λ\in \mathbb R \setminus \{0\}. \end{equation} In \cite{cho} it is known that no scattering occurs in $L^2$ for the long range ($0 < γ\le 1$). In \cite{c0, chooz2, cho1} the short-range scattering ($1 < γ< 3$) was treated for the scattering in $H^s$. In this paper we consider the critical case ($γ= 1$) and prove a modified scattering in $L^\infty$ on the frequency to the Cauchy problem with small initial data. For this purpose we investigate the global behavior of $x e^{it\nabla} u$, $x^2 e^{it\nabla} u$ and $\langleξ\rangle^5 \widehat{e^{it\nabla} u}$. Due to the non-smoothness of $\nabla$ near zero frequency the range of $α$ is restricted to $(\frac{17}{10}, 2)$.

math.AP↗

arXiv2017

On the Neumann Problem of Hardy-Sobolev critical equations with the multiple singularities

Let $N \geq 3$ and $Ω\subset \mathbb{R}^N$ be $C^2$ bounded domain. We study the existence of positive solution $u \in H^1(Ω)$ of \begin{align*} \left\{ \begin{array}{l} -Δu + λu = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} + \frac{|u|^{2^*(s)-2}u}{|x-x_2|^s}\text{ in }Ω\\ \frac{\partial u}{\partial ν} = 0 \text{ on }\partialΩ, \end{array}\right. \end{align*} where $0 < s <2$, $2^*(s) = \frac{2(N-s)}{N-2}$ and $x_1, x_2 \in \overlineΩ$ with $x_1 \neq x_2$. First, we show the existence of positive solutions to the equation provided the positive $λ$ is small enough. In case that one of the singularities locates on the boundary and the mean curvature of the boundary at this singularity is positive, the existence of positive solutions is always obtained for any $λ> 0$. Furthermore, we extend the existence theory of solutions to the equations for the case of the multiple singularities with different exponents.

math.AP↗

arXiv2015

Almost sure local wellposedness of energy critical fractional Schrödinger equations with hartree nonlinearity

We consider a Cauchy problem of energy-critical fractional Schrödinger equation with Hartree nonlinearity below the energy space. Using a method of randomization of functions on $\mathbb{R}^d$ associated with the Wiener decomposition, introduced by Á. Bényi, T. Oh, and O. Pocovnicu \cite{beohpo1,beohpo2}, we prove that the Cauchy problem is almost surely locally well-posed. Our result includes Hartree Schrödinger equation ($α= 2$).

math.AP↗

arXiv2015

Energy concentration of the focusing energy-critical FNLS

We consider the fractional nonlinear Schrödinger equation (FNLS) with general dispersion $|\nabla|^α$ and focusing energy-critical nonlinearities $-|u|^\frac{2α}{d-α}u$ and $-(|x|^{-2α} * |u|^2) u$. By adopting Kenig-Tsutsumi \cite{mets}, Kenig-Merle \cite{keme} and Killip-Visan \cite{kv} arguments, we show the energy concentration of radial solutions near the maximal existence time. For this purpose we use Sobolev inequalities for radial functions and establish strong energy decoupling of profiles. And we also show that when the kinetic energy is confined the maximal existence time is finite for some large class of initial data satisfying the initial energy $E(φ)$ is less than energy of ground state $E(W_α)$ but $\||\nabla|^\frac\alpha2 φ\|_{L^2} \ge \||\nabla|^\frac\alpha2 W_α\|_{L^2}$.

math.AP↗