This paper focuses on improving the robustness of spatiotemporal long-term prediction using a variational mode graph convolutional network (VMGCN) by introducing 3D channel attention. The deep learning network for this task relies on historical data inputs, yet real-time data can be corrupted by sensor noise, altering its distribution. We model this noise as independent and identically distributed (i.i.d.) Gaussian noise and incorporate it into the LargeST traffic volume dataset, resulting in data with both inherent and additive noise components. Our approach involves decomposing the corrupted signal into modes using variational mode decomposition, followed by feeding the data into a learning pipeline for prediction. We integrate a 3D attention mechanism encompassing spatial, temporal, and channel attention. The spatial and temporal attention modules learn their respective correlations, while the channel attention mechanism is used to suppress noise and highlight the significant modes in the spatiotemporal signals. Additionally, a learnable soft thresholding method is implemented to exclude unimportant modes from the feature vector, and a feature reduction method based on the signal-to-noise ratio (SNR) is applied. We compare the performance of our approach against baseline models, demonstrating that our method achieves superior long-term prediction accuracy, robustness to noise, and improved performance with mode truncation compared to the baseline models. The code of the paper is available at https://github.com/OsamaAhmad369/VMGCN.

In this work, we propose an efficient sampling design on the sphere for sampling order-limited spherical signals. The current sampling schemes on the sphere do not take into account the plausible rotational symmetry that is naturally exhibited by signals in many applications (e.g., climate data and antenna radiation pattern). These nearly rotationally symmetric signals have negligible dependence on the spherical harmonics of higher orders, and therefore can be modeled as order-limited signals. For sampling of spherical signals band-limited at$L$and order-limited at$M$, we present a sampling design for accurate computation of spherical harmonic transform and accurate signal reconstruction. The proposed sampling scheme achieves optimal dimensionality, requiring$M^{2} + (L-M)(2M+1)$samples, which match the degrees of freedom in harmonic space.

We present the design and scalable implementation of an exascale climate emulator for addressing the escalating computational and storage requirements of high-resolution Earth System Model simulations. We utilize the spherical harmonic transform to stochastically model spatio-temporal variations in climate data. This provides tunable spatio-temporal resolution and significantly improves the fidelity and granularity of climate emulation, achieving an ultra-high spatial resolution of $0.034^{\circ}(\sim 3.5 \mathbf{~ k m})$ in space. Our emulator, trained on 318 billion hourly temperature data points from a 35 -year and 31 billion daily data points from an 83-year global simulation ensemble, generates statistically consistent climate emulations. We extend linear solver software to mixed-precision arithmetic GPUs, applying different precisions within a single solver to adapt to different correlation strengths. The PaRSEC runtime system supports efficient parallel matrix operations by optimizing the dynamic balance between computation, communication, and memory requirements. Our BLAS3-rich code is optimized for systems equipped with four different families and generations of GPUs, scaling well to achieve 0.976 EFlop/s on 9, 025 nodes (36,100 AMD MI250X multichip module (MCM) GPUs) of Frontier (nearly full system), 0.739 EFlop/s on 1,936 nodes (7,744 Grace-Hopper Superchips (GH200)) of Alps, 0.243 EFlop/s on 1,024 nodes (4,096 A100 GPUs) of Leonardo, and 0.375 EFlop/s on 3,072 nodes (18,432 V100 GPUs) of Summit.

We propose a fast and accurate spherical harmonic transform (SHT) to facilitate harmonic analysis of current and forthcoming high-resolution datasets acquired on a regular grid, commonly encountered in a variety of applications including but not limited to, medical imaging, geophysics, and climate studies. In contrast to other methods that take the points on the sphere on a Gaussian grid (non-uniform) or a pre-defined uniform grid parameterized by the band-limit (spectral truncation) of the data, the proposed method computes the SHT of the data available on a grid formed by the arbitrary number of equiangular latitudes and longitudes. Since the number of temporal observations in the spatio-temporal data can be in the millions, we also propose a pre-computation for SHT that does not alter the asymptotic computational complexity but results in a significant reduction in the computation time. Our analysis of accuracy and computation time on both synthetic and real datasets validates the proposed developments. To demonstrate the utility of the proposed method, we implement the spatial isotropy test using the largest eigenvalue of the correlation matrix of harmonic coefficients as a test statistic and demonstrate the superior performance of the proposed method in comparison to least-squares for computing SHT.

We formulate and generalize a sampling scheme on the sphere based on the Gauss-Legendre (GL) quadrature that permits the accurate computation of the spherical harmonic transform (SHT) and its inverse for the signals band-limited at L and order-limited at M using L(2M + 1) number of samples. Signals that are nearly rotationally symmetric appear in many applications such as geophysics and antenna theory due to natural symmetry of the underlying process. For example, temperature variations in the climate data exhibit symmetry for a fixed co-latitude. Such signals when expanded in spherical harmonic domain do not have a strong dependence on spherical harmonics of higher orders. In this work, we propose a sampling method and develop associated SHT to save both the required number of samples and the computation cost while maintaining the signal reconstruction accuracy for order-limited signals. We also analyse numerical accuracy and computational complexity of the proposed transform and highlight the significance of proposed developments in the context of application in antenna theory.

This paper proposes a multi-spectral random forest classifier with suitable feature selection and masking for tree cover estimation in urban areas. The key feature of the proposed classifier is filtering out the built-up region using spectral indices followed by random forest classification on the remaining mask with carefully selected features. Using Sentinel-2 satellite imagery, we evaluate the performance of the proposed technique on a specified area (approximately 82 acres) of Lahore University of Management Sciences (LUMS) and demonstrate that our method outperforms a conventional random forest classifier as well as state-of-the-art methods such as European Space Agency (ESA) WorldCover 10m 2020 product as well as a DeepLabv3 deep learning architecture.

A reliable yet inexpensive tool for the estimation of flood water spread is conducive for efficient disaster management. The application of optical and SAR imagery in tandem provides a means of extended availability and enhanced reliability of flood mapping. We propose a methodology to merge these two types of imagery into a common data space and demonstrate its use in the identification of affected populations and infrastructure for the 2022 floods in Pakistan. The merging of optical and SAR data provides us with improved observations in cloud-prone regions; that is then used to gain additional insights into flood mapping applications. The use of open source datasets from WorldPop1and OSM2for population and roads respectively makes the exercise globally replicable. The integration of flood maps with spatial data on population and infrastructure facilitates informed policy design. We have shown that within the top five flood-affected districts in Sindh province, Pakistan, the affected population accounts for 31%, while the length of affected roads measures 1410.25 km out of a total of 7537.96 km.

Any policy-level decision-making procedure and academic research involving the optimum use of resources for development and planning initiatives depends on accurate population density statistics. The current cutting-edge datasets offered by WorldPop and Meta do not succeed in achieving this aim for developing nations like Pakistan; the inputs to their algorithms provide flawed estimates that fail to capture the spatial and land-use dynamics. In order to precisely estimate population counts at a resolution of 30 meters by 30 meters, we use an accurate built settlement mask obtained using deep segmentation networks and satellite imagery. The Points of Interest (POI) data is also used to exclude non-residential areas.

We present an operator formulation for linear transformations in the joint spatial-Slepian domain, which is enabled by the spatial-Slepian transform. The operator is an integral type and is specified by the spatial-Slepian transformation kernel, which is also utilized in finding a matrix representation for the operator using the orthogonal basis of Wigner-D functions. Conditions for the compactness and self-adjointness of the operator are presented and spectral analysis is carried out. The formulation is illustrated by choosing a Gaussian form for the spectral representation of the kernel to smooth out the noise in a bandlimited Earth topography map.

Clouds, together with their shadows, usually occlude ground-cover features in optical remote sensing images. This hinders the utilization of these images for a range of applications such as earth observation, land-cover classification and urban planning. In this work, we propose a deep unfolded and prior-aided robust principal component analysis (DUPA-RPCA) network for removing clouds and recovering ground-cover information in multi-temporal satellite images. We model these cloud-contaminated images as a sum of low rank and sparse elements and then unfold an iterative RPCA algorithm that has been designed for reweighted$\ell _{1}$minimization. As a result, the activation function in DUPA-RPCA adapts for every input at each layer of the network. Our experimental results on both Landsat and Sentinel images indicate that our method gives better accuracy and efficiency when compared with existing state of the art methods.

The depletion and variations of groundwater storage (GWS) are of critical importance for sustainable groundwater management. In this work, we use Gravity Recovery and Climate Experiment (GRACE) to estimate variations in the terrestrial water storage (TWS) and use it in conjunction with the Global Land Data Assimilation System (GLDAS) data to extract GWS variations over time for Indus river basin (IRB). We present a data processing framework that processes and combines these data-sets to provide an estimate of GWS changes. We also present the design of a band-limited optimally concentrated window function for spatial localization of the data in the region of interest. We construct the so-called optimal window for the IRB region and use it in our processing framework to analyze the GWS variations from 2005 to 2015. Our analysis reveals the expected seasonal variations in GWS and signifies groundwater depletion on average over the time period. Our proposed processing framework can be used to analyze spatio-temporal variations in TWS and GWS for any region of interest.

We develop a framework for generalized linear transformations of the joint spatial-Slepian domain representation of signals on the sphere. Such a representation is enabled by the spatial-Slepian transform on the sphere. We formulate a least-square signal estimation framework for reconstruction of the spherical signal from the modified (transformed) spatial-Slepian representation specified by the spatial-Slepian transformation kernel. We specialize the form of the kernel to present analytical expressions for the multiplicative and convolutive transformations, and use the latter to present illustrations on a Mars topography map.

We present a framework for the optimal filtering of spherical signals contaminated by realizations of an additive, zero-mean, uncorrelated and anisotropic noise process on the sphere. Filtering is performed in the wavelet domain given by the scale-discretized wavelet transform on the sphere. The proposed filter is optimal in the sense that it minimizes the mean square error between the filtered wavelet representation and wavelet representation of the noise-free signal. We also present a simplified formulation of the filter for the case when azimuthally symmetric wavelet functions are used. We demonstrate the use of the proposed optimal filter for denoising of an Earth topography map in the presence of additive, zero-mean, uncorrelated and white Gaussian noise, and show that the proposed filter performs better than the hard thresholding method and weighted spherical harmonic (weighted-SPHARM) signal estimation framework.

We present the optimal design of an azimuthally symmetric window signal for carrying out joint spatial-spectral domain filtering of a spherical (source) signal contaminated by a realization of an anisotropic noise process. The resulting window is used in the computation of spatially localized spherical harmonic transform of the noise-contaminated signal. We formulate the window design problem using the joint spatial-spectral domain filtering framework and choose the optimality criterion which minimizes the mean square error between the (noise-free) source signal and its filtered estimate. The azimuthally symmetric optimal window signal is shown to be specified by the statistics of the source and noise processes. We illustrate the capability of the proposed window signal by applying the joint spatial-spectral domain filtering framework to the bandlimited Mars topography map and demonstrate improvements in the output signal to noise ratio (SNR) for different values of input SNR.

We present a joint SO(3)-spectral domain filtering framework using the directional spatially localized spherical harmonic transform (DSLSHT), for the estimation and enhancement of random anisotropic signals on the sphere contaminated by random anisotropic noise. We design an optimal filter for filtering the DSLSHT representation of the noise-contaminated signal in the joint SO(3)-spectral domain. The filter is optimal in the sense that the filtered representation in the joint domain is the minimum mean square error estimate of the DSLSHT representation of the underlying (noise-free) source signal. We also derive a least square solution for the estimate of the source signal from the filtered representation in the joint domain. We demonstrate the capability of the proposed filtering framework using the Earth topography map in the presence of anisotropic, zero-mean, uncorrelated Gaussian noise, and compare its performance with the joint spatial-spectral domain filtering framework.

In practice, the finite number of samples of the spherical radiation pattern or antenna gain are taken on the sphere for both the reconstruction of the antenna radiation pattern and the computation of mobile handset performance measures such as directivity and mean effective gain (MEG). The acquisition of samples is time consuming as the measurements are required to be collected over the range of frequencies and in multiple spatial directions. It is therefore desired to have a sampling strategy that takes fewer number of samples for the accurate reconstruction of radiation pattern and incoming signal power distribution. In this work, we propose to use equiangular sampling, Gauss-Legendre sampling and optimal dimensionality sampling schemes on the sphere for the acquisition of measurements of spherical radiation pattern of the antenna for its reconstruction, analysis and evaluation of performance parameters of the antenna. By appropriately choosing the spherical harmonic degree band-limits of the gain and the power distribution model of the incoming signal, we demonstrate that the proposed sampling strategies require significantly fewer points for the accurate evaluation of MEG than the existing methods that rely on the approximate evaluation of the surface integral on the sphere.

We construct an overcomplete and multiscale dictionary of bandlimited Slepian functions on the sphere. Slepian functions are the bandlimited eigenfunctions obtained by solving spatial-spectral concentration problem on the sphere. To this end, we develop the hierarchical equal area iso-latitude iso-longitude pixelization (HEALLPix) scheme for hierarchical partitioning of the sphere into equal area sub-regions called pixels and present its quaternary tree structure. We then solve the concentration problem of finding bandlimited functions with maximal energy concentration in the given spatial region for each pixel and use these spatially concentrated bandlimited functions as dictionary elements. We analyze the span of the dictionary elements and their mutual coherence and show that the dictionary spans the space of bandlimited functions which are optimally (energy) concentrated within a pixel on the sphere with most of its elements exhibiting negligibly small mutual coherence. Hence, the proposed dictionary is a significant tool for use in multi-resolution analysis and sparse reconstruction of signals on the sphere.

We propose an antipodally symmetric sampling scheme of optimal dimensionality for the sampling of band-limited signals. The proposed scheme takes ~L2number of samples for the sampling of spherical signal of band-limit L and the accurate computation of its spherical harmonic transform (SHT). Since the number of samples are asymptotically equal to the degrees of freedom of the signal in harmonic space, the proposed scheme attains optimal spatial dimensionality. We also formulate the SHT associated with proposed sampling scheme. We employ the antipodal symmetry of the sampling points that is exploited to separate the signal into antipodally symmetric and asymmetric signals due to which the signal splits in harmonic space into the signals of even and odd spherical harmonic degrees. The exploitation of this splitting in the formulation of the SHT makes our method computationally efficient by a factor of four in comparison with the existing methods developed for sampling schemes that attain optimal spatial dimensionality. We also analyse the numerical accuracy of the proposed SHT by conducting numerical experiments and show that the proposed sampling and its associated SHT enable accurate signal reconstruction for band-limits in the range 15 ≤ L ≤ 127.

We propose a method for the accurate and robust reconstruction of the non-bandlimited finite rate of innovation signals on the sphere. For signals consisting of a finite number of Dirac functions on the sphere, we develop an annihilating filter based method for the accurate recovery of parameters of the Dirac functions using a finite number of observations of the bandlimited signal. In comparison to existing techniques, the proposed method enables more accurate reconstruction primarily due to the better conditioning of systems involved in the recovery of parameters. In order to reconstruct K Diracs on the sphere, the proposed method requires samples of the signal bandlimited in the spherical harmonic (SH) domain at SH degree equal or greater than K + √K + 1/4 - 1/2. In comparison to the existing state-of-the-art technique, the required bandlimit, and consequently the number of samples, of the proposed method is (approximately) the same. We also conduct numerical experiments to demonstrate that the proposed technique is more accurate than the existing methods by a factor of 107or more for 2 ≤ K ≤ 20.

We propose an iso-latitude sampling scheme for the representation of band-limited signals on the sphere. The proposed scheme is designed as a variant of the widely used Hierarchical Equal Area iso-Latitude Pixelization (HEALPix) scheme on the sphere. We use HEALPix grid of resolution Nsideto represent a signal of bandlimit (spherical harmonic degree) L. To reduce the number of samples, the proposed algorithm takes only L iso-latitude rings out of 4Nside-1 rings of the HEALPix. This selection is carried out by ensuring that the spherical harmonic transform (SHT) of the signal is computed accurately from the samples. The number of samples required by the proposed sampling scheme is smaller than that required by HEALPix by at least a factor of 3/2. For the proposed sampling scheme, we also formulate the spherical harmonic transform and conduct numerical experiments to evaluate the number of samples required by the proposed sampling scheme and the accuracy of the associated SHT.

We develop an algorithm for the extrapolation of band-limited signals on the sphere. The proposed algorithm improves the accuracy of the extrapolation of band-limited signal by using the information contained in the out-of-band harmonic coefficients of the signal to update the extrapolated signal at each iteration. The estimation of signals on the sphere from incomplete measurements finds applications in acoustics, cosmology and geophysics. The proposed algorithm does not only exploit the band-limited property of the signal, that is, force the harmonic coefficients outside the band-limit to zero, at each iteration as carried out in the existing algorithms but also uses the harmonic coefficients outside the harmonic domain to improve the accuracy of signal extrapolation. To demonstrate the improvement in the accuracy enabled by the proposed algorithm, we conduct numerical experiments and compare the results of the proposed algorithm with the existing iterative conjugate gradient method.

To support the applications where the measurements can only be taken over spatially limited region on the sphere due to practical limitations, we design a spatially-limited sampling scheme on the sphere for the computation of spherical harmonic transform (SHT) of band-limited signals. By enclosing the inaccessible region with the (anisotropic) ellipsoidal region followed by the rotation of the region to the pole or the equator, we propose an iso-latitude sampling scheme on the sphere. We also present a method to place the samples over the spatially-limited region such that the SHT can be computed accurately. Moreover, we formulate the SHT associated with the proposed sampling scheme and analyse its accuracy through numerical experiments. We also provide an illustration where we reconstruct the head-related transfer function (HRTF) from spatially-limited measurements and demonstrate that the proposed sampling design enables more accurate computation than the existing sampling schemes.

For the representation of spin-s band-limited functions on the sphere, we propose a sampling scheme with optimal number of samples equal to the number of degrees of freedom of the function in harmonic space. In comparison to the existing sampling designs, which require ~2L2samples for the representation of spin-s functions band-limited at L, the proposed scheme requires No= L2- s2samples for the accurate computation of the spin-s spherical harmonic transform (s-SHT). For the proposed sampling scheme, we also develop a method to compute the s-SHT. We place the samples in our design scheme such that the matrices involved in the computation of s-SHT are well-conditioned. We also present a multipass s-SHT to improve the accuracy of the transform. We also show the proposed sampling design exhibits superior geometrical properties compared to existing equiangular and Gauss-Legendre sampling schemes, and enables accurate computation of the s-SHT corroborated through numerical experiments.

For the fast and exact computation of spherical harmonic transform (SHT) of a band-limited signal defined on the sphere from its samples, the Gauss-Legendre (GL) and equiangular sampling schemes on the sphere require asymptotically least number of samples. In comparison to the equiangular scheme, the GL scheme has larger spatial dimensionality, defined as the number of the samples required for the exact computation of SHT. In this work, we propose an efficient GL sampling scheme with spatial dimensionality equal to that of equiangular scheme. We also propose optimisation of samples along longitude to further reduce the spatial dimensionality of equiangular, GL and efficient GL sampling schemes. Furthermore, we demonstrate that the accuracy of the SHT is not affected with the proposed reduction in the spatial dimensionality.

We develop a robust method for the accurate reconstruction of non-bandlimited finite rate of innovation signals composed of finite number of Diracs. For the recovery of parameters of K Diracs defining the signal, the proposed method requires more than (K + √K)2samples of the signal band-limited in harmonic domain such that the spherical harmonic transform can be computed using the samples. In comparison with the existing methods, the proposed method is robust in a sense that it does not require all Diracs to have distinct colatitude parameter. We first estimate the N number of Diracs which do not have distinct colatitude parameter. Once N is determined, the proposed method requires, at most, N2+N/2 + 1 unique and intelligently chosen rotations of the signal to recover all parameters accurately. We also provide illustrations to demonstrate the accurate reconstruction using the proposed method.

We design a sampling scheme on the sphere and a corresponding spherical harmonic transform (SHT) for the measurement and reconstruction of the diffusion signal in diffusion magnetic resonance imaging (dMRI). By exploiting the antipodal symmetry property of the diffusion signal in the spectral (spherical harmonic) domain, we design a sampling scheme that attains the optimal number of samples, equal to the degrees of freedom required to represent the antipodally symmetric band-limited diffusion signal in the spectral domain. Compared with other sampling schemes that can be used with the optimal number of samples, we demonstrate, through numerical experiments, that the proposed scheme enables more accurate computation of the SHT, and this accuracy is practically rotationally invariant. In addition, it results in more efficient computation of the SHT and storage of the diffusion signal.

We propose a Gauss-Legendre quadrature based sampling on the rotation group for the representation of a band-limited signal such that the Fourier transform (FT) of a signal can be exactly computed from its samples. Our figure of merit is the sampling efficiency, which is defined as a ratio of the degrees of freedom required to represent a band-limited signal in harmonic domain to the number of samples required to accurately compute the FT. The proposed sampling scheme is asymptotically as efficient as the most efficient scheme developed very recently. For the computation of FT and inverse FT, we also develop fast algorithms of complexity similar to the complexity attained by the fast algorithms for the existing sampling schemes. The developed algorithms are stable, accurate and do not have any pre-computation requirements. We also analyse the computation time and numerical accuracy of the proposed algorithms and show, through numerical experiments, that the proposed Fourier transforms are accurate with errors on the order of numerical precision.

We propose a sampling scheme on the sphere and develop a corresponding spherical harmonic transform (SHT) for the accurate reconstruction of the diffusion signal in diffusion magnetic resonance imaging (dMRI). By exploiting the antipodal symmetry, we design a sampling scheme that requires the optimal number of samples on the sphere, equal to the degrees of freedom required to represent the antipodally symmetric band-limited diffusion signal in the spectral (spherical harmonic) domain. Compared with existing sampling schemes on the sphere that allow for the accurate reconstruction of the diffusion signal, the proposed sampling scheme reduces the number of samples required by a factor of two or more. We analyse the numerical accuracy of the proposed SHT and show through experiments that the proposed sampling allows for the accurate and rotationally invariant computation of the SHT to near machine precision accuracy.

We develop a method to compute spherical harmonic transform (SHT) of a band-limited signal on the sphere discretized over a minimum dimensionality regular sampling grid on the sphere. For the computation of SHT of a signal band-limited at L, the proposed method requires L2 number of samples on a regular grid composed of L iso-latitude rings of samples with only L samples in each ring along longitude. Since a signal band-limited at L is represented by L2 degrees of freedom in the spectral (spherical harmonic) domain, the proposed method requires the minimal number of samples for the computation of SHT. In comparison to the other schemes that require 2L - 1 samples along each iso-latitude ring, we show that the SHT can be computed, by exploiting the structure of spectral domain, from only L samples in each iso-latitude ring. We also analyse the numerical accuracy and the computational complexity of our proposed SHT for a regular grid with equiangular sampling. We demonstrate, through numerical experiments, that the proposed SHT is sufficiently accurate for band-limits of interest in diffusion magnetic resonance imaging.

In this work, we design complete orthonormal basis functions, which are referred to as optimal basis functions, that span the vector sum of subspaces formed by band-limited spatially concentrated and space-limited spectrally concentrated functions. The optimal basis are shown to be a linear combination of band-limited functions with maximized energy concentration in some spatial region of interest and space-limited functions which maximize the energy concentration in some spectral region. The linear combination is designed with an optimality condition of maximizing the product of measures of energy concentration in the spatial and spectral domain. We also show that each optimal basis is an eigenfunction of a linear operator which maximizes the product of energy concentration measures in spatial and spectral domain. Finally, we discuss the properties of the proposed optimal basis functions and highlight their usefulness for the signal representation and data analysis due to the simultaneous concentration of the proposed basis functions in spatial and spectral domains.

This paper presents a novel sampling scheme on the sphere for obtaining head-related transfer function (HRTF) measurements and accurately computing the spherical harmonic transform (SHT). The scheme requires an optimal number of samples, given by the degrees of freedom in the spectral domain, for the accurate representation of the HRTF that is band-limited in the spherical harmonic domain. The proposed scheme allows for the samples to be easily taken over the sphere due to its iso-latitude structure and non-dense sampling near the poles. In addition, the scheme can be used when samples are not taken from the south polar cap region of the sphere as the HRTF measurements are not reliable in south polar cap region due to reflections from the ground. Furthermore, the scheme has a hierarchical structure, which enables the HRTF to be analyzed at different audible frequencies using the same sampling configuration. In comparison to the proposed scheme, none of the other sampling schemes on the sphere simultaneously possess all these properties. We conduct several numerical experiments to determine the accuracy of the SHT associated with the proposed sampling scheme. We show that the SHT attains accuracy on the order of numerical precision (10-14) when samples are taken over the whole sphere, both in the optimal sample placement and hierarchical configurations, and achieves an acceptable level of accuracy (10-5) when samples are not taken over the south polar cap region of the sphere for the band-limits of interest. Simulations are used to show the accurate reconstruction of the HRTF over the whole sphere, including unmeasured locations.

We investigate the problem of extrapolation of band-limited signals on the 2-sphere in the presence of noise. Specifically, given incomplete or spatially limited measurements subject to noise, find the unique extrapolation to the complete 2-sphere. We present an analytic solution to the extrapolation problem based on the expansion of a signal in Slepian basis corresponding to an orthogonal set of eigenfunctions of an associated energy concentration problem. An alternative equivalent iterative algorithm is also developed for practical implementation and guidelines are proposed to choose the parameters of the iterative algorithm. The capability of the proposed extrapolation is compared and demonstrated with the help of an illustration example.

In this paper we show that the spatially localized spherical harmonic transform (SLSHT), which represents a signal on the 2-sphere in the spatio-spectral domain, can be efficiently computed using new kernel-based formulations. In addition to the standard spatio-spectral domain, we show there are three other related transforms that provide alternative representations in the spatio-spatial, spectro-spatial and spectro-spectral domains. We provide inversion results that extend available results for the SLSHT. We show that for signals on the 2-sphere band-limited to degree L, the computational complexity using our class of kernel-based SLSHT transforms is O(L4) and outperforms the previous best known fast methods, which have complexity O(L5).

This paper investigates spectral filtering using isotropic spectral windows, which is a computationally efficient method of spatial smoothing on the sphere. We propose a Slepian eigenfunction window, which is obtained as a solution of the concentration problem on the sphere, as a good choice of the window function. We also unify a comprehensive set of quantitative tools, both spatial and spectral, to assess and compare the performance of different smoothing windows (i.e., smoothers). We analyze and compare the performance of the proposed window against the two best available candidates in the literature: von-Hann window and von Mises-Fisher distribution window. We establish that the latter window includes the popular Gauss window as a subcase. We show that the Slepian eigenfunction window has the smallest spatial variance (better spatial localization) and the smallest side-lobe level.

The ambiguity function and the Wigner distribution are fundamental tools in the time-frequency analysis. In this paper, we present an analog of the ambiguity function and the Wigner distribution for signals on the sphere. First, we formulate the ambiguity function for signals on the sphere which represents the signals in joint spatio-spectral domain and derive an inversion operation to obtain the signal from its ambiguity function. Next, we formulate the Wigner distribution for azimuthally symmetric signals on the sphere as a two dimensional spherical harmonics transform of the ambiguity function. We provide the matrix formulation of the Wigner distribution and discuss some of its useful properties. Finally, we illustrate the use of Wigner distribution for spatial and/or spectral localization of a signal in joint spatio-spectral domain. The obtained results provide the first step in designing more sophisticated transforms on the sphere.

The uncertainty principle is an important and powerful tool, with many applications in signal processing. This paper presents two concentration uncertainty principles for signals on the sphere which relate the localization of the concentration of a signal in spatial and spectral domains, as an analogue of the general Donoho and Stark uncertainty principles in time-frequency analysis. Using the spherical and spectral truncation operators, we derive the L1-norm and L2-norm uncertainty principles which respectively relate the signal concentration in spatial and spectral domains as absolute value and the energy of a signal. We also analyze the sharpness of the bound imposed by the derived L2-norm uncertainty principle. The proposed uncertainty measures can be applied to signal processing problems on the sphere.

In this paper, we consider the problem of signal extrapolation for discrete (i.e., sampled) signals on the sphere. We propose conjugate gradient based algorithm for estimating a signal on the sphere from limited or incomplete measurements in a spatial domain. We prove that the proposed algorithm is guaranteed to converge and show that it has faster convergence compared to the Papoulis algorithm. The results also show that the incomplete measurements distributed in different non-connected spatial regions yield better extrapolation results, compared to the connected region case.

This paper considers the problem of construction of low-pass filters on the unit sphere, which has wide ranging applications in the processing of signals on the unit sphere. We propose a design criterion for the construction of strictly bandlimited low-pass filters in the spectral domain with optimal concentration in the specified polar cap region in the spatial domain. Our approach uses the weighted sum of the first optimally concentrated eigenfunctions from appropriately formulated Slepian concentration problems on the sphere. Furthermore, in order to reduce the computational complexity of the proposed algorithm, we develop a closed-form expression to accurately model these eigenfunctions. We illustrate the construction of low-pass filters using the proposed approach and demonstrate the advantage of our method approach compared to a diffusion based approach in the literature in terms of control over both bandwidth in the spectral domain and concentration in the spatial domain.