arXiv subjects

Sunghwan Moon

Publications and source records attributed to Sunghwan Moon.

At least 19 recordsLinked to original sources

arXiv2026

Photoacoustic tomography with time-dependent damping: Theoretical and a convolutional neural network-guided numerical inversion procedure

In photoacoustic tomography (PAT), a hybrid imaging modality that is based on the acoustic detection of optical absorption from biological tissue exposed to a pulsed laser, a short pulse laser generates an initial pressure proportional to the absorbed optical energy, which then propagates acoustically and is measured on the boundary. To account for the significant signal distortion caused by acoustic attenuation in biological tissue, we model PAT in heterogeneous media using a damped wave equation featuring spatially varying sound speed and a time-dependent damping term. Under natural assumptions, we show that the initial pressure is uniquely determined by the boundary measurements using a harmonic extension of the boundary data with energy decay. For constant damping, an expansion in Dirichlet eigenfunctions of $-c^2(\xx)\Delta$ leads to an explicit series reconstruction formula for the initial pressure. Finally, we develop a gradient free numerical method based on the Pontryagin's maximum principle to provide a robust and computationally viable approach to image reconstruction in attenuating PAT.

math.NA↗

arXiv2025

Memory- and Latency-Constrained Inference of Large Language Models via Adaptive Split Computing

Large language models (LLMs) have achieved near-human performance across diverse reasoning tasks, yet their deployment on resource-constrained Internet-of-Things (IoT) devices remains impractical due to massive parameter footprints and memory-intensive autoregressive decoding. While split computing offers a promising solution by partitioning model execution between edge devices and cloud servers, existing approaches fail to address the unique challenges of autoregressive inference, particularly the iterative token generation process and expanding key-value (KV) cache requirements. This work introduces the first autoregressive-aware split computing framework designed explicitly for LLM deployment on edge devices. Our approach makes three key contributions. First, we develop one-point split compression (OPSC), a mixed-precision quantization scheme that prevents out-of-memory failures by strategically partitioning models into front-end and back-end segments with different precision levels. Second, we propose a two-stage intermediate compression pipeline that combines threshold splitting (TS) and token-wise adaptive bit quantization (TAB-Q) to preserve accuracy-critical activations while dramatically reducing communication overhead. Third, we formulate a unified optimization framework that jointly selects optimal split points, quantization settings, and sequence lengths to satisfy strict memory and latency constraints. Extensive evaluations across diverse LLMs and hardware platforms demonstrate superior performance compared to state-of-the-art quantization methods, including SmoothQuant, OmniQuant, and Atom. The framework achieves a 1.49 inference speedup and significant communication overhead reduction while maintaining or improving model accuracy.

cs.LG↗

arXiv2025

A Machine Learning and Finite Element Framework for Inverse Elliptic PDEs via Dirichlet-to-Neumann Mapping

Inverse problems for Partial Differential Equations (PDEs) are crucial in numerous applications such as geophysics, biomedical imaging, and material science, where unknown physical properties must be inferred from indirect measurements. In this work, we present a new approach to solving the inverse problem for elliptic PDEs, using only boundary data. Our method leverages the Dirichlet-to-Neumann (DtN) map, which captures the relationship between boundary inputs and flux responses. This enables the reconstruction of the unknown physical properties within the domain from boundary measurements alone. Our framework employs a self-supervised machine learning algorithm that integrates a Finite Element Method (FEM) in the inner loop for the forward problem, ensuring high accuracy. Moreover, our approach illustrates its effectiveness in challenging scenarios with only partial boundary observations, which is often the case in real-world scenarios. In addition, the proposed algorithm effectively handles discontinuities by incorporating carefully designed loss functions. This combined FEM and machine learning approach offers a robust, accurate solution strategy for a broad range of inverse problems, enabling improved estimation of critical parameters in applications from medical diagnostics to subsurface exploration.

math.NA↗

arXiv2024

SoccerNet 2024 Challenges Results

The SoccerNet 2024 challenges represent the fourth annual video understanding challenges organized by the SoccerNet team. These challenges aim to advance research across multiple themes in football, including broadcast video understanding, field understanding, and player understanding. This year, the challenges encompass four vision-based tasks. (1) Ball Action Spotting, focusing on precisely localizing when and which soccer actions related to the ball occur, (2) Dense Video Captioning, focusing on describing the broadcast with natural language and anchored timestamps, (3) Multi-View Foul Recognition, a novel task focusing on analyzing multiple viewpoints of a potential foul incident to classify whether a foul occurred and assess its severity, (4) Game State Reconstruction, another novel task focusing on reconstructing the game state from broadcast videos onto a 2D top-view map of the field. Detailed information about the tasks, challenges, and leaderboards can be found at https://www.soccer-net.org, with baselines and development kits available at https://github.com/SoccerNet.

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↗

arXiv2023

Explicit inversion for variable-speed wave equations on bounded domains

We study the reconstruction of the initial pressure $f(x)=p(x,0)$ for the wave model \[ \partial_t^2 p(x,t)=c(x)\Delta_{x}p(x,t)\qquad (x,t)\in\Omega\times[0,\infty), \] posed on a bounded domain $\Omega$ with variable sound speed $c(\cdot)$. From time-resolved boundary measurements, we consider two settings: (i) measurement of $p|_{\partial\Omega\times[0,\infty)}$ under a Robin boundary condition $p+\alpha\,\partial_\nu p=0$ on $\partial\Omega\times[0,\infty)$ with $\alpha\gneq 0$, and (ii) measurement of $\partial_\nu p|_{\partial\Omega\times[0,\infty)}$ under a Dirichlet boundary condition $p=0$ on $\partial\Omega\times[0,\infty)$. Within a unified framework, we present explicit formulas that recover the spectral coefficients $\langle f,\phi_k^B\rangle$ of $f$ with respect to the eigenfunction bases of the operator $-c(\cdot)\Delta_{x}$ for boundary types $B\in\{D,R\}$. The framework integrates variable sound speed with Dirichlet/Robin boundary conditions in a single setting, enabling direct coefficient-level recovery from boundary data.

math.AP↗

arXiv2023

Reconstruction of the initial data from the trace of the solutions on an infinite time cylinder of damped wave equations

In this paper, we consider two types of damped wave equations: the weakly damped equation and the strongly damped equation. We recover the initial velocity from the trace of the solution on a space-time cylinder. This inverse problem is related to Photoacoustic Tomography (PAT), a hybrid medical imaging technique. PAT is based on generating acoustic waves inside of an object of interest and one of the mathematical problem in PAT is reconstructing the initial velocity from the solution of the wave equation measured on the outside of object. Using the spherical harmonics and spectral theorem, we demonstrate a way to recover the initial velocity.

math.AP↗

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

Singular value decomposition of the wave forward operator with radial variable coefficients

Photoacoustic tomography (PAT) is a novel and promising technology in hybrid medical imaging that involves generating acoustic waves in the object of interest by stimulating electromagnetic energy. The acoustic wave is measured outside the object. One of the key mathematical problems in PAT is the reconstruction of the initial function that contains diagnostic information from the solution of the wave equation on the surface of the acoustic transducers. Herein, we propose a wave forward operator that assigns an initial function to obtain the solution of the wave equation on a unit sphere. Under the assumption of the radial variable speed of ultrasound, we obtain the singular value decomposition of this wave forward operator by determining the orthonormal basis of a certain Hilbert space comprising eigenfunctions. In addition, numerical simulation results obtained using the continuous Galerkin method are utilized to validate the inversion resulting from the singular value decomposition.

math.NA↗

arXiv2022

Reconstruction of the initial function from the solution of the fractional wave equation measured in two geometric settings

Photoacoustic tomography (PAT) is a novel and rapidly promising technique in the field of medical imaging, based on the generation of acoustic waves inside an object of interest by stimulating non-ionizing laser pulses. This acoustic wave is measured using the detector on the outside of the object and converted into an image of the human body by several inversions. Thus, one of mathematical problems in PAT is how to recover the initial function from the solution of the wave equation on the outside of the object. In this study we consider the fractional wave equation and assume that the point-like detectors are located on the sphere and hyperplane. We provide how to recover the initial function from the data, the solution of the fractional wave equation, measured on the sphere and hyperplane.

math.AP↗

arXiv2020

Error estimate for a universal function approximator of ReLU network with a local connection

Neural networks have shown high successful performance in a wide range of tasks, but further studies are needed to improve its performance. We analyze the approximation error of the specific neural network architecture with a local connection and higher application than one with the full connection because the local-connected network can be used to explain diverse neural networks such as CNNs. Our error estimate depends on two parameters: one controlling the depth of the hidden layer, and the other, the width of the hidden layers.

cs.LG↗

arXiv2020

The conical Radon transform with vertices on triple lines

We study the inversion of the conical Radon which integrates a function in three-dimensional space from integrals over circular cones. The conical Radon recently got significant attention due to its relevance in various imaging applications such as Compton camera imaging and single scattering optical tomography. The unrestricted conical Radon transform is over-determined because the manifold of all cones depends on six variables: the center position, the axis orientation and the opening angle of the cone. In this work, we consider a particular restricted transform using triple line sensors where integrals over a three-dimensional set of cones are collected, determined by a one-dimensional vertex set, a one-dimensional set of central axes, and the one-dimensional set of opening angle. As the main result in this paper, we derive an analytic inversion formula for the restricted conical Radon transform. Along that way we define a certain ray transform adapted to the triple line sensor for which we establish an analytic inversion formula.

math.NA↗

arXiv2018

Photoacoustic Tomography with Direction Dependent Data: An Exact Series Reconstruction Approach

Photoacoustic image reconstruction often assumes that the restriction of the acoustic pressure on the detection surface is given. However, commonly used detectors often have a certain directivity and frequency dependence, in which case the measured data are more accurately described as a linear combination of the acoustic pressure and its normal derivative on the detection surface. In this paper, we consider the inverse source problem for data that are a combination of an acoustic pressure of the wave equation and its normal derivative For the special case of a spherical detection geometry we derive exact frequency domain reconstruction formulas. We present numerical results showing the robustness and validity of the derived formulas. Moreover, we compare several different combinations of the pressure and its normal derivative showing that used measurement model significantly affects the recovered initial pressure.

math.NA↗

arXiv2018

Some uniqueness theorems for a conical Radon transform

The conical Radon transform, which assigns to a given function $f$ on $\mathbb R^3$ its integrals over conical surfaces, arises in several imaging techniques, e.g. in astronomy and homeland security, especially when the so-called Compton cameras are involved. In many practical situations we know this transform only on a subset of its domain. In these situations, it is a natural question what we can say about $f$ from partial information. In this paper, we investigate some uniqueness theorems regarding a conical Radon transform.

math-ph↗

arXiv2016

Inversion of the attenuated V-line transform for SPECT with Compton cameras

The Compton camera is a promising alternative to the Anger camera for imaging gamma radiation, with the potential to significantly increase the sensitivity of SPECT. Two-dimensional Compton camera image reconstruction can be implemented by inversion of the V-line transform, which integrates the emission distribution over V-lines (unions of two half-lines), that have vertices on a surrounding detector array. Inversion of the V-line transform without attenuation has recently been addressed by several authors. However, it is well known from standard SPECT that ignoring attenuation can significantly degrade the quality of the reconstructed image. In this paper we address this issue and study the attenuated V-line transform accounting for attenuation of photons in SPECT with Compton cameras. We derive an analytic inversion approach based on circular harmonics expansion, and show uniqueness of reconstruction for the attenuated V-line transform. We further develop a discrete image reconstruction algorithm based on our analytic studies, and present numerical results that demonstrate the effectiveness of our algorithm.

math.NA↗

arXiv2016

Analytic inversion of a conical Radon transform arising in application of Compton cameras on the cylinder

Single photon emission computed tomography (SPECT) is a well established clinical tool for functional imaging. A limitation of current SPECT systems is the use of mechanical collimation, where only a small fraction of the emitted photons is actually used for image reconstruction. This results in large noise level and finally in a limited spatial resolution. In order to decrease the noise level and to increase the imaging resolution, Compton cameras have been proposed as an alternative to mechanical collimators. Image reconstruction in SPECT with Compton cameras yields to the problem of recovering a marker distribution from integrals over conical surfaces. Due to this and other applications, such conical Radon transforms recently got significant attention. In the current paper we consider the case where the cones of integration have vertices on a circular cylinder and axis pointing to the symmetry axis of the cylinder. As main results we derive analytic reconstruction methods for the considered transform. We also investigate the V-line transform with vertices on a circle and symmetry axis orthogonal to the circle, which arises in the special case where the absorber distribution is located in a horizontal plane.

math.NA↗

arXiv2016

Compressed sensing and sparsity in photoacoustic tomography

Increasing the imaging speed is a central aim in photoacoustic tomography. This issue is especially important in the case of sequential scanning approaches as applied for most existing optical detection schemes. In this work we address this issue using techniques of compressed sensing. We demonstrate, that the number of measurements can significantly be reduced by allowing general linear measurements instead of point-wise pressure values. A main requirement in compressed sensing is the sparsity of the unknowns to be recovered. For that purpose we develop the concept of sparsifying temporal transforms for three-dimensional photoacoustic tomography. We establish a two-stage algorithm that recovers the complete pressure Signals in a first step and then applies a standard reconstruction algorithm such as back-projection. This yields a novel reconstruction method with much lower complexity than existing compressed sensing approaches for photoacoustic tomography. Reconstruction results for simulated and for experimental data verify that the proposed compressed sensing scheme allows to significantly reducing the number of spatial measurements without reducing the spatial resolution.

math.NA↗

arXiv2015

Inversion of the elliptical Radon transform arising in migration imaging using the regular Radon transform

In recent years, many types of elliptical Radon transforms that integrate functions over various sets of ellipses/ellipsoids have been considered, relating to studies in bistatic synthetic aperture radar, ultrasound reflection tomography, radio tomography, and migration imaging. In this article, we consider the transform that integrates a given function in $\mathbf R^n$ over a set of ellipses (when $n=2$) or ellipsoids of rotation (when $n\geq 3$) with foci restricted to a hyperplane. We show a relation between this elliptical Radon transform and the regular Radon transform, and provide the inversion formula for the elliptical Radon transform using this relation. Numerical simulations are performed to demonstrate the suggested algorithms in two dimensions, and these simulations are also provided in this article.

math.FA↗