Motivation and Background Functional Analysis

Constrained spherical deconvolution (CSD) of diffusion‐weighted MRI (DW‐MRI) is a popular analysis method that extracts the full white matter (WM) fiber orientation density function (fODF) in the living human brain, noninvasively. It assumes that the DW‐MRI signal on the sphere can be represented as the spherical convolution of a single‐fiber response function (RF) and the fODF, and recovers the fODF through the inverse operation. CSD approaches typically require that the DW‐MRI data is sampled shell‐wise, and estimate the RF in a purely spherical manner using spherical basis functions, such as spherical harmonics (SH), disregarding any radial dependencies. This precludes analysis of data acquired with nonspherical sampling schemes, for example, Cartesian sampling. Additionally, nonspherical sampling can also arise due to technical issues, for example, gradient nonlinearities, resulting in a spatially dependent bias of the apparent tissue densities and connectivity information. Here, we adopt a compact model for the RFs that also describes their radial dependency. We demonstrate that the proposed model can accurately predict the tissue response for a wide range of b‐values. On shell‐wise data, our approach provides fODFs and tissue densities indistinguishable from those estimated using SH. On Cartesian data, fODF estimates and apparent tissue densities are on par with those obtained from shell‐wise data, significantly broadening the range of data sets that can be analyzed using CSD. In addition, gradient nonlinearities can be accounted for using the proposed model, resulting in much more accurate apparent tissue densities and connectivity metrics.

The Ingenuity time-of-flight (TF) PET/MR is a recently developed hybrid scanner combining the molecular imaging capabilities of PET with the excellent soft tissue contrast of MRI. It is becoming common practice to characterize the system's point spread function (PSF) and understand its variation under spatial transformations to guide clinical studies and potentially use it within resolution recovery image reconstruction algorithms. Furthermore, due to the system's utilization of overlapping and spherical symmetric Kaiser-Bessel basis functions during image reconstruction, its image space PSF and reconstructed spatial resolution could be affected by the selection of the basis function parameters. Hence, a detailed investigation into the multidimensional basis function parameter space is needed to evaluate the impact of these parameters on spatial resolution. Using an array of 12 × 7 printed point sources, along with a custom made phantom, and with the MR magnet on, the system's spatially variant image-based PSF was characterized in detail. Moreover, basis function parameters were systematically varied during reconstruction (list-mode TF OSEM) to evaluate their impact on the reconstructed resolution and the image space PSF. Following the spatial resolution optimization, phantom, and clinical studies were subsequently reconstructed using representative basis function parameters. Based on the analysis and under standard basis function parameters, the axial and tangential components of the PSF were found to be almost invariant under spatial transformations (~4 mm) while the radial component varied modestly from 4 to 6.7 mm. Using a systematic investigation into the basis function parameter space, the spatial resolution was found to degrade for basis functions with a large radius and small shape parameter. However, it was found that optimizing the spatial resolution in the reconstructed PET images, while having a good basis function superposition and keeping the image representation error to a minimum, is feasible, with the parameter combination range depending upon the scanner's intrinsic resolution characteristics. Using the printed point source array as a MR compatible methodology for experimentally measuring the scanner's PSF, the system's spatially variant resolution properties were successfully evaluated in image space. Overall the PET subsystem exhibits excellent resolution characteristics mainly due to the fact that the raw data are not under-sampled/rebinned, enabling the spatial resolution to be dictated by the scanner's intrinsic resolution and the image reconstruction parameters. Due to the impact of these parameters on the resolution properties of the reconstructed images, the image space PSF varies both under spatial transformations and due to basis function parameter selection. Nonetheless, for a range of basis function parameters, the image space PSF remains unaffected, with the range depending on the scanner's intrinsic resolution properties.

We analyse a discretisation method for solving (systems of) semilinear parabolic equations on Euclidean spheres. The approximation method is based upon a discretisation in space using spherical basis functions and can hence be of arbitrary order in space, provided that the true solution is sufficiently smooth, which is, at least locally in time, true for semilinear parabolic problems. We rigorously prove stability and convergence of the semi-discrete approximation, if basis functions with a prescribed Sobolev regularity are employed. For discretisation in time, we give two examples and prove the expected convergence orders in space and time for the fully discretised system.

While many geological and geophysical processes such as the melting of icecaps, the magnetic expression of bodies emplaced in the Earth's crust, or the surface displacement remaining after large earthquakes are spatially localized, many of these naturally admit spectral representations, or they may need to be extracted from data collected globally, e.g. by satellites that circumnavigate the Earth. Wavelets are often used to study such nonstationary processes. On the sphere, however, many of the known constructions are somewhat limited. And in particular, the notion of `dilation' is hard to reconcile with the concept of a geological region with fixed boundaries being responsible for generating the signals to be analyzed. Here, we build on our previous work on localized spherical analysis using an approach that is firmly rooted in spherical harmonics. We construct, by quadratic optimization, a set of bandlimited functions that have the majority of their energy concentrated in an arbitrary subdomain of the unit sphere. The `spherical Slepian basis' that results provides a convenient way for the analysis and representation of geophysical signals, as we show by example. We highlight the connections to sparsity by showing that many geophysical processes are sparse in the Slepian basis.

The local analysis of signals arising on the sphere is a common task in earth sciences. On the real line the analytic signal turned out to be an important representation in local one-dimensional signal processing. Its generalization to two dimensions is the monogenic signal, and the properties of the analytic and the monogenic signal in the Fourier domain are well known. A generalization to the sphere is given by the Hilbert transform on the sphere known from Clifford analysis. To obtain a spectral characterization, the transform has to be decomposed into spherical harmonic functions. In this paper, we derive the spherical harmonic coefficients of the Hilbert transform on the sphere and give a series expansion. This will show that it acts as a differential operator on the spherical harmonic basis functions of the Laplace equation solution, analogously to the Riesz transform in two dimensions. This allows an interpretation of the Hilbert transform suitable for signal processing of signals naturally arising on the two-sphere. We show that the scale space naturally arising is a Poisson scale space in the unit ball. In addition, the obtained interpretation of the Hilbert transform is used for orientation analysis of plane waves. This representation is justified as a novel signal model on the sphere which can be used to construct intensity and rotation-invariant operators for local signal analysis in a scale-space concept.

Image processing continues to be a challenging topic in many scientific fields such as medicine, computational physics and informatics especially with the discovery and development of 3D cases. Therefore, development of suitable tools that guarantee a best treatment is a necessity. Spherical shapes are a big class of 3D images whom processing necessitates adoptable tools. This encourages researchers to develop special mathematical bases suitable for 3D spherical shapes. The present work lies in this whole topic with the application of special spherical harmonics bases. In Jallouli et al. (Soft Comput 2018. https://doi.org/10.1007/s00500-018-3596-9), theoretical framework of spherical harmonics filters adapted to image processing has been developed. In the present paper, new approach based on Jallouli et al. (Soft Comput 2018. https://doi.org/10.1007/s00500-018-3596-9) is proposed for the reconstruction of images provided with spherical harmonics Shannon-type entropy to evaluate the order/disorder of the reconstructed image. Efficiency and accuracy of the approach are demonstrated by a simulation study on several spherical models.

Electrostatic T-matrix for a torus on bases of toroidal and spherical harmonics

The mean dynamic topography (MDT) can be computed as the difference between the mean sea level (MSL) and a gravimetric geoid. This requires that both data sets are spectrally consistent. In practice, it is quite common that the resolution of the geoid data is less than the resolution of the MSL data, hence, the latter need to be low-pass filtered before the MDT is computed. For this purpose conventional low-pass filters are inadequate, failing in coastal regions where they run into the undefined MSL signal on the conti- nents. In this paper, we consider the use of a bandlimited, spa- tially concentrated Slepian basis to obtain a low-resolution approximation of the MSL signal. We compute Slepian func- tions for the oceans and parts of the oceans and compare the performance of calculating the MDT via this approach with other methods, in particular the iterative spherical harmonic approach in combination with Gaussian low-pass filtering, and various modifications. Based on the numerical experi- ments, we conclude that none of these methods provide a low- resolution MSL approximation at the sub-decimetre level. In particular, we show that Slepian functions are not appropriate basis functions for this problem, and a Slepian representation of the low-resolution MSL signal suffers from broadband leakage. We also show that a meaningful definition of a low- resolution MSL over incomplete spherical domains involves orthogonal basis functions with additional properties that Slepian functions do not possess. A low-resolution MSL signal, spectrally consistent with a given geoid model, is obtained by a suitable truncation of the expansions of the MSL signal in terms of these orthogonal basis functions. We compute one of these sets of orthogonal basis functions using the Gram-Schmidt orthogonalization for spherical harmon- ics. For the oceans, we could construct an orthogonal basis only for resolutions equivalent to a spherical harmonic degree 36. The computation of a basis with a higher resolution fails due to inherent instabilities. Regularization reduces the insta- bilities but destroys the orthogonality and, therefore, pro- vides unrealistic low-resolution MSL approximations. More research is needed to solve the instability problem, per- haps by finding a different orthogonal basis that avoids it altogether.

Modeling a very broad spectrum of the Earth's gravity field needs observations from various measurement techniques with different spectral sensitivities. Typically, high‐resolution regional gravity data are combined with low‐resolution global observations. To exploit the gravitational information as optimally as possible, we set up a regional modeling approach using radial spherical basis functions, emphasizing the strengths of various data sets by the flexible combination of high‐ and middle‐resolution terrestrial, airborne, shipborne, and altimetry measurements. The basis functions are defined and located in the region of interest in such a manner, which the highest measure of information of the input data is captured. Any functional of the Earth's gravity field can be derived, as, e.g., quasi‐geoid heights or gravity anomalies. Here we present results of a study area in Northern Germany. A comprehensive cross validation to external observation data delivers standard deviations less than 5 cm. Differences to an existing regional quasi‐geoid model count on average ±6 cm and proof the plausibility of our solution. The comparison with existing global models reaches higher standard deviations for the more sensitive gravity anomalies as for quasi‐geoid heights, showing the additional value of our solution in the high frequency domain. Covering a broad frequency spectrum, our regional models can be used as basis for various applications, such as refinement of global models, national geoid determination, and detection of mass anomalies in the Earth's interior.

The operator-equivalent method was introduced by Stevens in 1952. This method enabled him to determine the quantum mechanical equivalent of a given spherical harmonic Ckq-the so-called Racah tensor operator Ckq-as an explicit function of the total angular momentum operator J within a constant J manifold. The method itself uses spherical harmonics in Cartesian coordinates and from each spherical harmonic the corresponding Racah tensor operator is calculated. This paper shows that it is useful to fix tau k- mod q mod as all spherical harmonics Cmod q mod + tau , mod q mod possess the same polynomial structure with coefficients showing a simple dependence on the absolute value of q. For the following Racah operators Cmod q mod + tau , mod q mod , which hold for all mod q mod , Stevens' operator-equivalent method has to be generalized. By an application of the Wigner-Eckart theorem to the operator-equivalents Cmod q mod + tau ,mod q, mod (J), the 3j-symbols with two identical js disclose its innermost functional dependence on the variables. To give an example the calculations are explicitly stated for tau =0, 1, 2, . . .5.

<p>Indirect illumination is an important part of realistic images, and accurately simulating the complex effects of indirect illumination in real-time applications has long been a challenge for the industry. One popular approach is to use offline precomputed solutions such as lightmaps (textures containing the precomputed lighting in a scene) to efficiently approximate these effects. Unfortunately, these offline solutions have historically enforced long iteration times that come at a cost to artist productivity. These solutions have additionally either supported only the low-frequency diffuse component of indirect lighting, yielding poor visual results for glossy or metallic materials, or have used overly expensive approximations. In recent years, the state of the art lightmap precomputation pipeline has shifted to using highly vectorised path tracing, often on GPU hardware, to compute the indirect illumination effects. The use of path tracing enables progressive rendering, wherein an approximation to the full solution is found and then refined as opposed to solving for the final result in a single step. Progressive rendering through path tracing thereby helps to provide rapid iteration for artists. This thesis describes a system that can progressively path-trace indirect illumination lightmaps on the GPU.Contributing to this system, itintroduces a new gather-based method for sample accumulation, enhances algorithms from prior work, and presents a range of encoding methods, including a novel progressive method for non-negative least-squares encoding of spherical basis functions. In addition, it presents a novel, efficient solution for high-quality precomputed diffuse and low-frequency specular indirect illumination that extends the Ambient Dice family of spherical basis functions. This solution provides comparable or better specular reconstruction to prior work at lower runtime cost and has potential for widespread use in real-time applications.</p>

A 3D shape descriptor based on spherical harmonics through evolutionary optimization

Nonstationary Wavelets on the m-Sphere for Scattered Data

<p>In this study, we investigate the optimal combination of local gravity observations and their contributions to the regional quasi-geoid model. The study area is located in Colorado, USA, with two types of regional data sets, namely terrestrial gravity data and airborne gravity data, available within the “1 cm geoid experiment”. The approach based on series expansions in terms of spherical radial basis functions (SRBF) is applied, which has been developed at DGFI-TUM in the last two decades. We use two different types of basis functions covering the same spectral domain separately for the terrestrial and the airborne measurements. The Shannon function is applied to the terrestrial data, and the Cubic Polynomial (CuP) function which has smoothing features is applied to the airborne data for filtering their high-frequency noise.</p><p>To assess the contributions of the regional terrestrial and airborne gravity data to the final quasi-geoid model, four solutions are compared, namely the combined solution, the terrestrial only, the airborne only, and finally the model only solution, i.e., only the global gravity model and the topographic model are used without any gravity data from regional measurements. By adding the terrestrial data to the GGM and the topographic model, the RMS error of the quasi-geoid model w.r.t the validation data (the mean solution of independent computations delivered by fourteen institutions from all over the world) drops from 4 to 1.8 cm, and it is further reduced to 1 cm by including the airborne data.</p>

Diagonalization of the fast multipole method (FMM) for the Helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves. The resulting k-space integral over the Ewald sphere is numerically evaluated. Storing the k-space quadrature samples of the method of moments (MoM) basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems. In this paper, it is proposed to expand the k-space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules. Aggregations, plane wave translations, and disaggregations in the realized multilevel fast multipole method (MLFMM) are carried out using the k-space samples of a numerical quadrature rule. However, the incoming plane waves on the finest MLFMM level are expanded in spherical harmonics again. Thus, due to the orthonormality of spherical harmonics, the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients. Overall, the resulting MLFMM can save a considerable amount of memory without compromising accuracy and numerical speed.