Hemispherical Harmonics Research Articles

Hemispherical harmonic illumination and reflectance angular spectrum.

Light scattering plays an important role in physics, with wide applications in science and engineering. However, accurate and effective modeling of scattering remains a great challenge. In this study, we exploited the rendering equation using hemispherical harmonics to demonstrate an angular frequency representation that directly depicts scattering in a two-dimensional spectrum, free from any underlying assumptions. This representation offers a compact and intuitive characterization of mirror reflection, isotropic scattering, and anisotropic emission. The robust support of theoretical proofs and data-driven experimental results establishes the broad applicability of our computational model in conducting scattering analyses across diffuse, specular, and glossy materials. With the capability to characterize the scattering in angular frequency domain, we expect our proposed model to emerge as an essential tool in various domains, including surface feature recognition, reflectance data compression, and computer rendering.

A successive-order-scattering solution of the hemispherical harmonic decomposed radiative transfer equations

In this work, we applies the successive-order-scattering (SOS) method to the generally decomposed radiative transfer equations (GD-RTEs). The basic equations is presented and a numerical model is developed specifically to solve the hemispherical harmonic components of radiance. The numerical results are checked against those of the hemispherical harmonics method to validate the model and to investigate the factors influencing the accuracies of the new algorithm, including the optical depth of the sublayers and the criterion for convergence. We find that the trend to more isotropic of the scattered radiance and the additivity of the hemispherical harmonic components of radiance lead to a more efficient criterion of convergence, which allows the order of the decomposed equations reduced with the increasing order of scattering, while limiting the error caused by the reduction below the prescribed level. With the new criterion, the computation time can be saved by typically from a half of to an order of magnitude less than that with the conventional criterion, in presence of multiple scattering.

Universal global compact representation of head-related transfer functions with hemi-spherical harmonics and conformal mapping

The head-related transfer function(HRTF) is an essential part of spatial auditory display systems. In recent years, numerous HRTF databases are established to include human subjects' measurements, which enabled data-driven research projects such as HRTF prediction and personalization. However, in most cases, each HRTF database has its own unique measuring standard and source direction grid set. It's difficult to merge and efficiently represent the global HRTF information across multiple HRTF databases. Current presentation methods, such as principle component analysis (PCA) and spherical harmonics transform (SHT), are constrained by the source grid layout and regularization errors, which lays burdens for common feature extraction from different sets of HRTF measurements. In this work, we propose a novel approach for the global compact representation of HRTFs across different measuring standards, using hemispherical harmonics (HSH) and conformal mapping. The method takes into account the spatial domain covered by the typical HRTF measurements and proves to be less contained by the source direction grids. Both numerical and auditory model experiments are performed, to examine the representation error and consistency when including multiple HRTF databases with different measuring standards and grid layouts. In continuing work, we are using this method as a pre-processing approach for HRTF personalization utilizing multiple HRTF databases.

Appearance segmentation and documentation applied to cultural heritage surfaces

This paper describes the development and application of a novel supervised segmentation technique used for conservation documentation based on visible appearance changes of Cultural Heritage (CH) metal surfaces. The technique is based on employing a linear discriminant analysis model to classify Reflectance Transformation Imaging (RTI) reconstruction coefficients. The Hemispherical Harmonics (HSH) reconstruction coefficients for each pixel are first calculated and then normalized. This normalization increases the robustness and invariance of the application making it possible to apply it for documenting different surfaces and at different time intervals. In this paper, we presented three case studies related to corrosion assessment of CH objects through detection of corrosion and monitoring the degree of silver tarnishing. For each case study, a supervised data set is constructed, teaching the algorithm to recognize as distinct a specified appearance characteristic (such as corrosion, metal etc.) by comparing it to the reconstruction coefficients of each pixel. The segmented information is visualized by a simplified colormap. The calculated results are afterwards verified by visible inspection from conservation-restoration experts. The method can segment surfaces with changes in micro-geometry, but it reaches its limitation on surfaces with minimal topography and high specularity.

Objective evaluation of relighting models on translucent materials from multispectral RTI images

In this paper, we evaluate the quality of reconstruction i.e. relighting from images obtained by a newly developed multispectral reflectance transformation imaging (MS-RTI) system. The captured MS-RTI images are of objects with different translucency and color. We use the most common methods for relighting the objects: polynomial texture mapping (PTM) and hemispherical harmonics (HSH), as well as the recent discrete model decomposition (DMD). The results show that all three models can reconstruct the images of translucent materials, with the reconstruction error varying with translucency but still in the range of what has been reported for other non-translucent materials. DMD relighted images are marginally better for the most transparent objects, while HSH- and PTM- relighted images appear to be better for the opaquer objects. The estimation of the surface normals of highly translucent objects using photometric stereo is not very accurate. Utilizing the peak of the fitted angular reflectance field, the relighting models, especially PTM, can provide more accurate estimation of the surface normals.

Open and closed anatomical surface description via hemispherical area-preserving map

Spherical harmonics, one of the most widely used basis functions for shape description, rely heavily on a spherical parameterization of the given surface. Additionally, spherical harmonics based 3D modeling works well only for closed surfaces, while many anatomical structures are hemisphere-like open objects. Therefore, it is more natural to have a hemisphere-based approach for their shape description. In this work, we propose a novel framework for the shape description of open and closed hemisphere-like surfaces. We first develop two hemispherical area-preserving parameterization methods for simply-connected open and closed surfaces respectively, and then utilize the hemispherical harmonics basis functions to yield an accurate representation of hemisphere-like anatomical surfaces. We assess the performance of the proposed framework for the shape description of human head. In particular, 60 hemispherical anatomical surfaces (20 closed brain surfaces, 20 closed skull surfaces, and 20 open scalp surfaces) constructed from human head MRI scans are utilized for this purpose. For the three types of surfaces, our framework achieves a significant improvement in the surface reconstruction accuracy by 75%, 80% and 50% respectively when compared to the spherical harmonics based approach. This suggests that our new shape description framework can facilitate the biomedical analysis of hemisphere-like anatomical objects.

Calibration of spatial distribution of light sources in reflectance transformation imaging based on adaptive local density estimation

Reflectance transformation imaging (RTI) is a multilight-based imaging technique that can provide relevant information on both local microgeometry and visual appearance of a studied surface. The pixelwise angular reflectance is modeled to allow the relighting of the surface appearance under any arbitrary light direction. The primary methods used to model this local reflectance in each pixel are polynomial texture mapping, hemispherical harmonics, and, more recently, discrete modal decomposition. For all these methods, a uniform distribution of the light positions over the hemisphere is an implicit hypothesis. However, it is not always possible to satisfy this condition in practice. As a result of this nonuniform distribution, several artifacts can affect the reconstruction and alter the quality of the visual appearance assessment. To address this issue, we propose a methodology consisting of the estimation of the spatial distribution of the lighting directions used during RTI acquisitions based on a local density estimation. These local density values are used afterward to weight the least square regression and thus correct the contributions of each image to the RTI acquisition. This methodology is applied on three surfaces with visual singularities, which present different reflectance responses. From the presented results, it can be concluded that it is necessary to take into account this nonuniformity in order not to alter the quality of reconstruction/relighting from RTI data and the subsequent inspection task.

Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering

Zernike circular polynomials (ZCP) are widely used in optical testing, fabrication and adaptive optics. However, their ability to characterize information on spherical cap is often limited by aperture angle, with highly curved surface being particularly challenging. In this study, we propose a simple and systematic process to derive Zernike like functions that are applicable to all types of spherical cap. The analytical expressions of three function sets are calculated using Gram-Schmidt algorithm. They are hemispherical harmonics (HSH), Zernike spherical function (ZSF) and longitudinal spherical function (LSF). HSH satisfies Laplacian equation and composes a subset of spherical harmonics (SH). ZSF and LSF can be applied to arbitrary spherical cap and their orthogonality is invariant to aperture. The achieved functions, with their complete and orthogonal performance, can serve as essential tools for surface characterization required for a wide range of applications like large-angle lenses description in illumination design, aberration analysis in high aperture systems, human cornea measurement fitting, geomagnetic field modelling, etc. Moreover, they are important for graphics rendering in virtual reality and games by solving the reflectance equation efficiently.

The potential of Reflectance Transformation Imaging in Architectural Paint Research and the study of historic interiors: a case study from Stowe House, England

Reflectance Transformation Imaging (RTI) has been increasingly used for the recording, documentation and analysis of cultural heritage. A series of images of a static object, taken in raking light at constant exposure, is synthetised using Polynomial Texture Mapping or the Hemispherical Harmonics fitter, resulting in polynomial texture maps or RTI images. These visualisations capture colour and texture information, reveal low relief detail and emphasise surface topography. RTI has found several applications in the study of painted surfaces, yet it has not been utilised for the study of historic interiors, which are more commonly evaluated through Architectural Paint Research, a process of empirical examination of the painted finishes to an historic structure to establish its decorative history. Typically, this is achieved through a combination of archival research, paint and substrate cross-section analysis, and through the careful layer-by-layer exposure of the accumulated paint stratigraphy so that the texture, sheen and colour of each paint layer can be observed. This study evaluates the use of RTI as an alternative means for the study of historic wall paintings, with an emphasis on the revelation of overpainted designs. Considering the efficiency of RTI in recording and documentation of subtle surface variations, this research explores its use in texture comparisons between exposed, restored and covered areas of the wall painting as a diagnostic tool and for appraising condition.

On the evaluation of spatial–angular distributions of polarization characteristics of scattered radiation

This paper is focused on a Monte Carlo based projective algorithm for the estimation of bidirectional angular characteristics of scattered polarized radiation in the context of different sets of basic functions, normalized with certain weights. We consider hemispherical harmonics designed on the basis of associated shifted Jacobi polynomials in comparison with those designed as a factorization of modified Jacobi and Legendre polynomials. We provide a review of results of numerical simulation of two-dimensional angular distributions of the radiation intensity and degree of polarization, transmitted through and reflected by optically thick layers of the scattering and absorbing media.

Two-dimensional projection Monte Carlo estimators for the study of angular characteristics of polarized radiation

Abstract The paper presents a Monte Carlo algorithm for the study of bidirectional angular characteristics of a scattered polarized radiation based on projection expansion of the density of the corresponding angular distribution over hemispherical harmonics. The results of numerical estimation of two-dimensional angular distributions of the intensity and the polarization degree of the radiation passed through and reflected from optically thick layers of scattering and absorbing substance are presented.

A Unified Formulation of Radiative Transfer in Plane-Parallel Atmospheres Based on General Decomposition of Radiance. Part II: An Exemplifying Application to the Hemispherical Harmonics Method with Four Components

Abstract Based on the unified formulation, the so-called hemispherical harmonics method with four components (HSHM4) is derived following the general procedure described in the first paper of this series. The numerical results of this method, including the flux reflections, transmissions, and absorptions for different optical depths, incident beam angles, and single-scattering albedos and the emissions for different optical depths and single-scattering albedos, are reported, in comparison with the results of DISORT with 128 streams as a benchmark. The application of the method to the radiation scheme for climate models is illustrated with the numerical results, including the heating-rate profiles for the shortwave and longwave radiation, radiative fluxes at the top and the bottom boundaries of the atmosphere, and the actinic flux profiles, for typical model atmospheres of Earth and different clear or cloudy conditions. The comparison of results with those of the discrete ordinate methods (DOMs) and the spherical harmonic method (SHM) with four components shows that the HSHM4 has comparable accuracy with the four-stream DOM with double-Gaussian quadrature and is more accurate than the four-stream SHM and the four-stream DOM with full-range Gaussian quadrature in general, especially in the cases of longwave radiation when the multiple-scattering effect should be accounted for. The HSHM4 can be deployed consistently to all of the bands without artificial division between the longwave and shortwave. The development and validation of the HSHM4 exhibit the usefulness of the unified formulation in the study of atmospheric radiative transfer.

Discrete Modal Decomposition: a new approach for the reflectance modeling and rendering of real surfaces

Reflectance Transformation Imaging is a recent technique allowing for the measurement and the modeling of one of the most influential parameters on the appearance of a surface, namely the angular reflectance, thanks to the change in the direction of the lighting during acquisition. From these photometric stereo images (discrete data), the angular reflectance is modeled to allow both interactive and continuous relighting of the inspected surface. Two families of functions, based on polynomials and on hemispherical harmonics, are cited and used in the literature at this aim, respectively, associated to the PTM and HSH techniques. In this paper, we propose a novel method called Discrete Modal Decomposition (DMD) based on a particular and appropriate Eigen basis derived from a structural dynamic problem. The performance of the proposed method is compared with the PTM and HSH results on three real surfaces showing different reflection behaviors. Comparisons are made in terms of both visual rendering and of statistical error (local and global). The obtained results show that the DMD is more efficient in that it allows for a more accurate modeling of the angular reflectance when light–matter interaction is complex such as the presence of shadows, specularities and inter-reflections.

Helmholtz HSH-Based Basis: A Compact Phenomenological Representation of Arbitrary Reflectance

Visual appearance can be phenomenologically modeled through an integral equation, known as reflectance equation. It describes the surface radiance which depends on the interaction between incident light field and surface Bidirectional Reflectance Distribution Function (BRDF). Being defined on the Cartesian product of the incident and outgoing hemispheres, hemispherical basis is the natural way to represent surface BRDFs. Nonetheless, due to their compactness in the frequency space, spherical harmonics have been extensively used for this purpose. Addressing the geometrical compliance of hemispherical basis, this paper proposes a Cartesian product of the hemispherical harmonics to provide a compact representation of plausible BRDFs, while satisfying the Helmholtz reciprocity property. We provide an analytical analysis and experimental justification that our basis provides better approximation accuracy when compared to similar bases in literature.

Efficient robust image interpolation and surface properties using polynomial texture mapping

Abstract Polynomial texture mapping (PTM) uses simple polynomial regression to interpolate and re-light image sets taken from a fixed camera but under different illumination directions. PTM is an extension of the classical photometric stereo (PST), replacing the simple Lambertian model employed by the latter with a polynomial one. The advantage and hence wide use of PTM is that it provides some effectiveness in interpolating appearance including more complex phenomena such as interreflections, specularities and shadowing. In addition, PTM provides estimates of surface properties, i.e., chromaticity, albedo and surface normals. The most accurate model to date utilizes multivariate Least Median of Squares (LMS) robust regression to generate a basic matte model, followed by radial basis function (RBF) interpolation to give accurate interpolants of appearance. However, robust multivariate modelling is slow. Here we show that the robust regression can find acceptably accurate inlier sets using a much less burdensome 1D LMS robust regression (or ‘mode-finder’). We also show that one can produce good quality appearance interpolants, plus accurate surface properties using PTM before the additional RBF stage, provided one increases the dimensionality beyond 6D and still uses robust regression. Moreover, we model luminance and chromaticity separately, with dimensions 16 and 9 respectively. It is this separation of colour channels that allows us to maintain a relatively low dimensionality for the modelling. Another observation we show here is that in contrast to current thinking, using the original idea of polynomial terms in the lighting direction outperforms the use of hemispherical harmonics (HSH) for matte appearance modelling. For the RBF stage, we use Tikhonov regularization, which makes a substantial difference in performance. The radial functions used here are Gaussians; however, to date the Gaussian dispersion width and the value of the Tikhonov parameter have been fixed. Here we show that one can extend a theorem from graphics that generates a very fast error measure for an otherwise difficult leave-one-out error analysis. Using our extension of the theorem, we can optimize on both the Gaussian width and the Tikhonov parameter.

Towards efficient image irradiance modelling of convex Lambertian surfaces under single viewpoint and frontal illumination

Under local illumination assumption, phenomenological appearance models capture surface appearance through the mathematical modelling of the reflection process. Theoretically, due to the arbitrariness of the lighting function, the space of all possible images of a fixed-pose object under all possible illumination conditions is infinite dimensional. Nonetheless, due to their low- frequency nature, irradiance signals can be represented using low-order basis functions, where spherical harmonics (SH) has been extensively adopted. When capturing image irradiance from a single viewpoint, the visible part of the object's surface constructs the upper hemisphere of the surface normals where the SH is no longer orthonormal. In this paper, we propose the use of hemispherical harmonics (HSH) to model image irradiance of convex Lambertian objects perceived from single viewpoint under unknown distant as well as near illumination. We prove analytically, and validate experimentally, that the Lambertian reflectance kernel has a more compact harmonic expansion in the hemispherical domain when compared to its spherical counterpart. Our experiments illustrate that, despite of having poor approximation accuracy under very close lights, such behavior improves exponentially with little increase in the distance to the light source relative to the object size.

The Belgammel Ram, a Hellenistic-Roman BronzeProembolionFound off the Coast of Libya: test analysis of function, date and metallurgy, with a digital reference archive

The Belgammel Ram was found off the coast of Libya in 1964, and examined during 2008–9. The following techniques were used: surface non-contact digitizing using a laser scanner, reflectance transformation imaging using polynomial texture mapping and hemi-spherical harmonics, digital photogrammetry with dense surface modelling, structured light optical scanning, and X-ray fluorescence analysis. For internal structure the ram was examined by X-radiography and 3-D X-ray tomography. Metallurgical composition was studied by micro-drilling and subjecting the samples to scanning electron microscope X-ray micro-analysis, micro X-ray fluorescence and X-ray backscatter. The lead isotope composition was analysed. The alloy has average percentage composition Cu = 86.9, Sn = 6.3, Pb = 6.6, and Zn = < 0.10. The Belgammel Ram is probably a Hellenistic-Roman proembolion from a small military vessel or tesseraria. The archived data are at the Faculty of Engineering and Environment, Engineering Sciences, Material Data Centre, University of Southampton ([email protected]).

Analysis of radiative transfer between surfaces by hemispherical harmonics

A system of functions on the hemisphere is introduced for representation of incoming and outgoing surface radiation intensity. These functions are constructed from the polynomials J n ( x), which are orthogonal on the interval (0, 1) with the weight w( x) = x. The expansion of the radiation intensity in the series of hemispherical harmonics in combination with Galerkin's procedure transforms the integral equation of RHT between surfaces to a system of equations for angular moments. For the diffuse and linear-non-diffuse surfaces this system has a very simple form.