Decoupling geometry and appearance in Gaussian splatting for reflective surface reconstruction: A glossy image prior-guided approach.

Rendering large-scale, unbounded scenes on AR/VR-class devices is constrained by the computation, bandwidth, and storage cost of 3D Gaussian Splatting (3DGS). We propose a low-power, low-cost 3DGS hardware accelerator that renders full-HD images in real time, together with a hardware-friendly compression pipeline that combines iterative Gaussian pruning and fine-tuning, progressive spherical harmonics (SH) degree reduction, and vector quantization of all SH coefficients and colors. The scheme achieves a $51.6\times$51.6× model-size reduction with a 0.743 dB PSNR loss. The accelerator uses a frame-level pipeline that integrates point-based culling and projection with tile-based sorting and rasterization, skips zero-Jacobian matrix multiplications (reducing processing elements by 63% and computation by 53%), and adopts comparison-free tile-based sorting with deterministic latency. Implemented in a TSMC 28-nm process at 800MHz, the design occupies $\text{0.66}\;\text{mm}^{2}$0.66mm2 with 1.1438 M gates and 120 kB SRAM, consumes 0.219 W, and delivers 1219 Mpixels/J at 267.5 Mpixels/s, enabling 1080p at 129 FPS. Overall, it is $5.98\times$5.98× smaller in area, $5.94\times$5.94× higher throughput, and delivers $7.5\times$7.5× higher energy efficiency than prior 3DGS accelerators.

Scalp event-related potentials (ERPs) measured with electroencephalography (EEG) are temporally precise but spatially bandwidth-limited: skull and scalp blur cortical activity, and event-related EEG is still usually represented in electrode coordinates or dataset-specific data-driven components rather than in a sensor-space coordinate system linked to cortical anatomy. We aimed to develop and evaluate a cortex-anchored sensor-space basis for event-related EEG.

Approach. We constructed a multiscale sensor-space dictionary by forward-projecting cortical Laplace-Beltrami (LB) eigenmodes through a realistic EEG head model, yielding basis functions ordered by cortical spatial frequency. Using ERP-CORE (7 paradigms, 39 participants), we benchmarked the forward-projected LB basis against (i) spherical harmonics defined on the same montage and (ii) group PCA/ICA bases learned from trial-averaged time-frequency (TF) maps. Across canonical components (N170, N2pc, N400, P3b, LRP, ERN), we quantified reconstruction efficiency (R2as a function of the number of modes), concentration of evoked TF energy across modes, split-half reliability of mode scores (ICC), and low-dimensional reconstruction of group-level ERP contrast topographies.

Main results. LB closely matched spherical harmonics in reconstruction efficiency, but concentrated evoked TF energy more strongly in low-order modes. For N170, N400, P3b, and ERN, the first 10 LB modes captured about 70% of normalized TF energy, whereas spherical harmonics typically required 15-18 modes. Low-to-mid LB mode scores showed moderate-to-excellent reliability, often comparable to or slightly exceeding spherical harmonics. In addition, 10-15 LB modes reconstructed canonical ERP contrast maps with high correlations while preserving the expected sensor-space organization, including posterior N170, centro-parietal N400/P3, and fronto-central ERN patterns.

Significance. Forward-mapped LB eigenmodes provide a compact, anatomy-linked sensor representation for event-related EEG that complements spherical and data-adaptive bases. The proposed sensor-space basis offers a reusable, geometry-informed coordinate system that captures whole-scalp ERP structure compactly while remaining linked to cortical spatial organization.

This paper proposes an effective Gaussian management framework for high-fidelity scene reconstruction of both appearance and geometry. Unlike recent Gaussian Splatting (GS) pipelines that treat all primitives uniformly during optimization, our framework explicitly manages the attribute activation, representation and pruning of Gaussian. Specifically, our framework first introduces GauSep, a novel densification strategy that selectively activates Gaussian color or normal attributes to alleviate destructive gradient conflicts arising from dual supervision. We further propose GauRep, an adaptive Gaussian representation that dynamically adjusts spherical harmonics (SHs) orders and performs task-decoupled pruning to reduce redundancy at both the individual and global levels. To provide reliable geometric supervision for above mangement process, we additionally introduce CoRe, an regularized surface reconstruction module that distills robust normal fields from an SDF branch to the Gaussian representation through a confidence mechanism. Notably, the proposed Gaussian management is compatible with various reconstruction architectures and can be seamlessly integrated to improve performance while reducing size of the model. Extensive experiments demonstrate that our approach achieves superior or comparable performance in appearance and geometry reconstruction compared with state-of-the-art methods, while using significantly fewer parameters.

The [Formula: see text] symmetric Pascal matrix [Formula: see text] is a generalized discrete time and band-limiting operator for the binomial transform and its eigenvectors are generalized discrete prolate spheroidal wave functions which we call binomial prolates. Their generating functions are also generalized prolate spheroidal functions in the sense that they are simultaneously eigenfunctions of a third-order differential operator and an integral operator over the line [Formula: see text]. For even, positive integers N, we obtain an explicit formula for the generating function of an eigenvector of the symmetric Pascal matrix with eigenvalue 1. When [Formula: see text] for an odd prime p, we show that the generating function is equivalent modulo p to [Formula: see text], where [Formula: see text] is the number of points on the Legendre elliptic curve [Formula: see text] over the finite field [Formula: see text]. Furthermore when [Formula: see text], our generating function is the square of a period of [Formula: see text] modulo [Formula: see text] in the open p-adic unit disk.

MR data quality depends on static magnetic field (B0) homogeneity. At the beginning of each session, a brief field map quantifies subject-specific B0 variation, and shim coils are then set to counteract it. Conventional spherical-harmonic (SPH) shims have limited shimming power, motivating localized multi-coil (AC/DC) systems. However, subject motion can perturb the optimized field, necessitating real-time shim updates that require rapid tracking of B0 changes. We simulated real-time shimming under motion using jointly first-order SPH and a 31-channel AC/DC matrix coil. Measured B0 data initially shimmed with SPH were augmented with AC/DC terms in simulation, and real-time control was evaluated. Shimming with AC/DC coils added to the SPH coils improved field homogeneity, but motion eroded these gains. With simulated real-time updates informed by deep learning, B0 homogeneity was effectively maintained even during substantial motion. Performance matched simulated navigator-like real-time shimming with gradient-echo and echo-planar imaging, while adding no extra scan time in main imaging sequences. Multi-coil shimming offers clear benefits, but the gains may be reduced if shim terms are not updated in real time. Deep-learning-driven prediction of B0 changes provides a practical path to sequence-agnostic, motion-robust shimming across a broad range of MR protocols.

Virtual design of multilayered SOFC microstructures based on particle packing and spherical harmonics

When a molecule or sub-molecular entity is ordered on a surface in a manner that can be described by a tilt- and azimuthal-angle distribution, the projection of the molecular hyperpolarizability into the laboratory frame can be described using spherical harmonics. We illustrate that this lends itself to the construction of ten achiral order parameters associated with vibrational sum-frequency generation. We describe how these order parameters can be extracted from spectral data, first ignoring and then including the dispersion of the local electric fields. We then use these order parameters to determine the most probable orientation distribution without any assumptions about the surface alignment and in the absence of any other type of experimental data. This constitutes a flexible framework for describing a wide array of surface orientation distributions.

The connection between the spherical harmonics and the stationary orbital states is theoretically established using the rotational transformation and the SU(2) algebra. Through the inverse quantum Fourier transform, the spherical harmonics can be decomposed into the superposition of the stationary orbital states. Using the partial sums of the decompositions, the gradual change of superpositions during the generation of spherical harmonics can be visualized. Furthermore, the expansion coefficients of hyperbolic sine functions are utilized to analytically derive the time-varying wave packet state corresponding to the inclined circular orbit. The orthogonality of Euler functions is further used to show that a periodic integration of the wave packet state can exactly generate the stationary orbital states. These derived formulas are expected to be used to explore the resonant modes of optical microsphere cavities.

One of the main challenges in medical microwave imaging is to provide meaningful images when only minimal a priori information is available. Concerning the case of brain stroke imaging, this paper proposes a novel method for generating a low-resolution patient-specific approximation of the head by processing the same data used for the diagnosis. The method assumes knowledge of the head shape and a rough approximation of the fat and skin tissue above the skull. The approach consists of two steps. First, the measured data are processed using the first-order Born approximation scattering model to generate a qualitative head image. The underlying linear inverse problem is regularized by representing the unknown complex permittivity in terms of Chebyshev polynomials to improve its stability and effectiveness. Then, the voxels belonging to the skull are identified in the obtained image, and the skull boundaries are approximated from the discrete data using spherical harmonics. The proposed method is tested against an anthropomorphic phantom, demonstrating its capability to derive a patient-specific approximation. Moreover, the obtained background model enables stroke detection using real-time qualitative imaging.

We study, for 1 ≤ p ≤ ∞ , the Hardy space h e p ( B ) , the elastic analogue of the classical Hardy spaces of harmonic functions in the unit ball of R 3 . The space consists of vector-field solutions of the Lamé system satisfying the standard integrability condition on concentric spheres centered at the origin. Using the elastic Poisson kernel, we establish a Fatou-type theorem and show that h e p ( B ) is isomorphic to the R 3 -valued Lebesgue space L p on the unit sphere for 1 < p ≤ ∞ , while h e 1 ( B ) corresponds to the space of R 3 -valued Borel measures on the unit sphere. For 1 < p < ∞ , we prove that h e p ( B ) decomposes as the direct sum of three subspaces. The main contribution of this paper is to describe each of these subspaces along with the corresponding spaces of boundary values. In particular, two of these spaces consist of solutions of the Lamé equation for all eligible choices of the Lamé constants: one of them is the space of Riesz fields (solutions of the generalized Cauchy–Riemann equations) in h e p ( B ) ; the second is the space of fields given by the cross product of x with such Riesz fields. The results rely on the classical decomposition of L 2 vector fields on the sphere into the direct sum of three spaces of vector spherical harmonics, which we extend to L p .

We present a concise methodology to analyze structural response to the applied stress in amorphous solids, including metallic glasses (MG), glassy selenium, silica and polycarbonate, using high energy x-ray diffraction and atomic pair distribution function (PDF) analysis. To assess the structural anisotropy induced by applied axial stress, diffraction data were expanded into spherical harmonics. Using Bessel transformation, components of the structure function were converted into isotropic and anisotropic PDFs. The PDFs were compared to the expected model behavior for ideal elastic deformation to separate homogeneous affine strain from local non-affine strains. In metallic glass the range of non-affine deformation is limited to the nearest neighbor shell, suggesting local strain relaxation under stress that occurs even in the elastic regime. Beyond the second atomic shell strain is uniform. However, in glassy silica, polycarbonate and selenium strong local bonding inhibits local displacements and strain in short range order is accommodated by rotation of local units. Interestingly, beyond a molecular unit, deformation in covalent systems is similar to MG, and response of the medium range order scales with the macroscopic stress.

Hominin forelimbs have evolved from primarily locomotive to manipulative appendages over approximately 6 million years. As such, hand functions in fossil hominins and the Pan-Homo last common ancestor (LCA) are intensely debated, with carpal morphology central to this debate. However, owing to their irregular and challenging shapes, few studies have comprehensively quantified carpal morphology. We analyse the overall carpal morphology of anthropoids, including fossil hominins, using spherical harmonics and use classification methods to characterize fossil hominins within the context of extant taxa. Results show that hominins share with African apes derived carpal morphology possibly related to knuckle walking. Furthermore, unique modern human carpal morphology appears to have evolved from these possible knuckle-walking features and in a piecemeal manner, causing some hominin capitates to resemble those of palmigrade monkeys. Striking variation in biomechanically relevant carpal morphology and retention of potentially ancestral features persists as late as Homo naledi, suggesting that most hominins probably neither knuckle walked nor extensively used stone tools. These results indicate that the hominin carpus evolved from an African ape-like wrist, with radial-side reorganization related to manipulation occurring only recently. Although it remains unclear whether the LCA knuckle walked, our results suggest that this is the most likely existing hypothesis.

Cloaking from environmental detection has been a central goal in thermal engineering. The transformation theory has predicted its existence, requiring graded and highly anisotropic material properties. Its realization with general geometry was achieved recently in the two-dimensions. However, engineering free-form omnidirectional thermal cloaks in three-dimensional (3D) space remains far from being understood. Here, we physically realize omnidirectional three-dimensional thermal cloaking with arbitrarily complex shapes via a lattice composite formed by a 3D de-homogenization approach. The composite accurately delivers graded and anisotropic 3D thermal conductivities required for cloaking and features concise, well-connected structures fabricable with additive manufacturing. Numerical and experimental investigations demonstrate the successful concealing of complex objects from multi-directional thermal detection, such as cloaking an apple inside a pear. Integration of the strategy with spherical harmonics enables creating 3D cloaks with record-breaking complexity. Our findings mark the physical realization of the transformation theory in arbitrary 3D shape and provide a general, efficient, and practical route to realizing 3D thermal meta-devices.

The growing use of digital technologies in artistic presentation has highlighted the limitations of traditional 2D documentation, which cannot capture view-dependent optical characteristics essential for experiencing textured oil paintings. We present what is believed to be a novel approach for interactive virtual exhibition of impasto artworks using image synthesis inspired by Gaussian splatting techniques. A spherical harmonics (SH)-based framework is proposed to encode the bidirectional reflectance distribution function of artwork surfaces, enabling efficient representation of complex physically based rendering textures under varying illumination conditions. To the best of our knowledge, this is the first application of image synthesis specifically designed for planar artwork playback that combines novel view synthesis with photorealistic rendering to faithfully reproduce oil painting characteristics. We use a few SH coefficients (SH degree = 4) to approximate incident illumination and surface reflectance, reducing computational complexity while maintaining rendering fidelity. The system uses off-axis multiview photography with structured sampling (9×5 density) to capture the artwork's appearance from discrete viewpoints. A real-time tracking mechanism implemented with a depth camera dynamically determines the viewer's position and updates the viewing vector, enabling interactive perspective rendering. Experiments show that our approach achieves a negligible root-mean-square error (0.0015-0.0062) within the recommended viewing range. The simplified orthographic projection variant further improves computational efficiency, achieving 698.9 frames per second (FPS) on RTX 5090 and 30.4 FPS on embedded platforms. We demonstrate that SH-based image synthesis provides a cost-effective, scalable solution for photorealistic digital artwork presentation, bridging the gap between traditional art appreciation and interactive digital experiences.

The purpose of our paper is to provide a family of bilinear orthogonal expansions all based upon the same general pattern that is valid for a wide class of special functions. Our first family involves Jacobi, Laguerre, and Hermite polynomials. We give a discrete analogue of these bilinear expansions, the three families of classical orthogonal polynomials being replaced by zonal spherical functions associated with regular distance graphs. Such expansions playing a key role in the field of mathematical statistics, we show how our results apply to this field. We provide generalizations of the well-known Cramér–von Mises and Watson’s statistics, based upon an interpretation of their kernel in terms of the circular Laplacian. The product formula, well-known for zonal functions on Lie groups, is stated for distance-regular graphs, providing an elegant tool for proofs. Examples involving Hahn, q-Hahn, and Krawtchouk polynomials are given.

We present a unified geometric perspective on the symmetries underlying the spin 0, 1 and 2 static perturbations around a Schwarzschild black hole. In all cases, the symmetries are exact, each forming an SO(3,1) group. They can be formulated at the level of the action, provided the appropriate field variables are chosen. For spin 1 and 2, the convenient variables are certain combinations of the gauge fields for even perturbations, and dual scalars for odd perturbations. The even and odd sectors each have its own SO(3,1) symmetry. In addition, there is an SO(2) symmetry connecting them, furnishing an economical description of Chandrasekhar's duality. When decomposed into spherical harmonics, the perturbations form a non-trivial representation of SO(3,1), giving rise to ladder symmetries which explain the vanishing of the tidal Love numbers. Our work builds on earlier discussions of ladder symmetries, which were formulated in terms of the Newman-Penrose scalar at the level of the Teukolsky equation. Our formulation makes it possible to state the symmetries responsible for the vanishing of the Wilson coefficients characterizing the spin 0, 1 and 2 static tidal response in the effective point particle description of a black hole.

In water-scarce regions where climate variability limits surface water, groundwater is a vital resource for domestic, agricultural and industrial use. Traditional monitoring through boreholes is constrained by incomplete data, inaccessibility and data sparsity, making it difficult to track regional trends. Satellite tools such as the Gravity Recovery and Climate Experiment (GRACE) and its Follow-On, GRACE-FO, provide valuable alternatives for observing large-scale changes in terrestrial water storage. This review evaluates GRACE (-FO) applications in groundwater research, with emphasis on water-scarce regions. GRACE (-FO) applications in hydrogeology have grown steadily, with most case studies concentrated in Asia and Africa, where groundwater stress and sparse monitoring increase reliance on satellite data. Data-rich regions are underrepresented, underscoring the need for benchmark studies to validate GRACE (-FO). Case studies show both the potential and limits of GRACE (-FO): depletion trends in Saudi Arabia and Iran are linked to agricultural abstraction, while replenishment in parts of Africa and South America reflects rainfall recharge or reservoirs recharge. These confirm GRACE (-FO)’s ability to diagnose depletion hotspots and compare how climate and management shape storage. Study area delineation is often guided by groundwater use, data availability, and monitoring networks, with performance further influenced by land use, aquifer type, and study area size. While best suited to large basins, downscaling has extended GRACE (-FO) to smaller regions. Machine learning supports this, with Random Forest widely used for its robustness, though newer approaches remain underexplored. Future work should refine downscaling, strengthen integration with regional models, and link storage with water quality, particularly in coastal aquifers vulnerable to depletion and salinisation. • Mascon solution has been the preferred over Spherical Harmonics since 2020. • Groundwater depletion is common worldwide, largely from agricultural abstraction. • Random Forest is the most common and effective method for downscaling GRACE data. • GRACE validation heavily relies on in-situ groundwater measurements. • A key gap is limited use of GRACE in operational groundwater models and management.

Determining the internal structure of planetary bodies from gravitational observations is a key challenge in planetary geophysics. Traditional gravity inversion methods suffer from non-uniqueness due to trade-offs between mass distribution and depth, limiting their ability to resolve internal layering. We present SynthGen , a forward-modelling code developed to simulate the gravitational response of planetary bodies using parametric, multi-layer interior models without any a priori assumption, like the hydrostatic equilibrium. SynthGen calculates gravitational potential, Free-Air, and Bouguer anomalies through spherical harmonic expansions, leveraging the SHTools library (Wieczorek and Meschede, 2018). It accommodates a wide variety of internal configurations, including homogeneous layers with user-defined densities, thicknesses, and topographic geometries of internal interfaces, such as spherical, ellipsoidal, random, or Bouguer anomaly-derived interfaces. The code can be used both predictively and diagnostically: about the latter, SynthGen performs parameter-space exploration constrained by total mass, moment of inertia, and shape, identifying best-fit interior models by minimising the misfit between observed and synthetic gravity fields using combined statistical metrics. We apply SynthGen to Mercury, using the HgM009 gravity model derived from MESSENGER data (Genova et al., 2023), and recover crustal thickness and core parameters consistent with recent independent geophysical estimates. In predictive mode, SynthGen generates synthetic gravity fields for planetary bodies where gravity data are not available or are still limited in resolution, such as Ganymede. These simulations can support the planning and optimisation of space missions. By integrating physical constraints, statistical validation, and flexibility in model design, SynthGen offers a robust platform for planetary interior studies, constraining interior structures from gravity measurements across a broad range of Solar System bodies. • SynthGen simulates gravity fields of planetary bodies using parametric interior models and spherical harmonics. • SynthGen retrieves best-fit internal structures by comparing simulated and real gravity data with statistical metrics. • SynthGen can predict gravitational signals for future missions like JUICE and supports diverse planetary scenarios.

This paper proposes a novel beamforming framework in the reproducing kernel domain, derived from a unified interpretation of directional response as spatial differentiation of the sound field. By representing directional response using polynomial differential operators, the proposed method enables the formulation of arbitrary beam patterns including non-axisymmetric. This is achieved by reinterpreting beamforming as a functional acting on the sound field, parameterized by directivity function and a virtual observation position. The derivation of the reproducing kernel associated with the interior fields is mathematically supported by Hobson's theorem, which allows concise analytical expressions. Furthermore, the proposed framework generalizes conventional spherical harmonic (SH) domain beamformers by reinterpreting them as spatial differential operators, thereby clarifying their theoretical structure and extensibility. Four numerical simulations conducted in two- and three- dimensional space confirm the validity of the method. The results demonstrate improved sound field reconstruction accuracy when microphone directivity is explicitly modeled and show that the proposed beamformer achieves performance comparable to SH domain beamforming across a wide range of conditions while providing slightly improved performance at higher frequencies.