arXiv subjects

Zubair Khalid

Publications and source records attributed to Zubair Khalid.

At least 37 records · Page 2Linked to original sources

arXiv2021

Multiscale Optimal Filtering on the Sphere

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.

eess.SP↗

arXiv2020

Estimation of Groundwater Storage Variations in Indus River Basin using GRACE Data

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.

eess.SP↗

arXiv2020

Identification and Quantification of Aerosol Hot-spots over Lahore Region using MODIS Data

The increased concentration of aerosols in the air caused by ever-rising urbanization and the development of various industries has horrendous consequences on human health, environment and climate. The first step to counter adverse effects of air pollution in any region is to identify locations with a high concentration of aerosols, termed aerosol hot-spots. We specifically focus on the region of Lahore, Pakistan, a city that has consistently been ranked among the top ten most polluted cities in the world. In order to identify aerosol hot-spots in the city, we utilize two-year (2017-18) Aerosol Optical Thickness (AOT) data from MODerate resolution Imaging Spectroradiometer (MODIS) on the Aqua satellite. We propose a method based on the Glowworm Swarm Optimization (GSO) algorithm that discovers several aerosol hot-spots over Lahore comprising of major highways and industrial areas. Furthermore, we formulate two quantification metrics to gauge the amount of aerosol content over each hot-spot. We also analyze the temporal variation of aerosol content that suggests the addition or suppression of pollution sources in the hot-spots. Unlike previous studies, our work provides novel insight into the regional aerosol concentration of Lahore and calls for maintenance of air quality of the regions falling under the identified aerosol hot-spots.

eess.SP↗

arXiv2020

Joint SO(3)-Spectral Domain Filtering of Spherical Signals in the Presence of Anisotropic Noise

We present a joint SO(3)-spectral domain filtering framework using 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.

eess.SP↗

arXiv2020

Spatial-Slepian Transform on the Sphere

We present spatial-Slepian transform~(SST) for the representation of signals on the sphere to support localized signal analysis. We use well-optimally concentrated Slepian functions, obtained by solving the Slepian spatial-spectral concentration problem of finding bandlimited and spatially optimally concentrated functions on the sphere, to formulate the proposed transform and obtain the joint spatial-Slepian domain representation of the signal. Due to the optimal energy concentration of the Slepian functions in the spatial domain, the proposed spatial-Slepian transform allows us to probe spatially localized content of the signal. Furthermore, we present an inverse transform to recover the signal from the spatial-Slepian coefficients, and show that well-optimally concentrated rotated Slepian functions form a tight frame on the sphere. We develop an algorithm for the fast computation of the spatial-Slepian transform and carry out computational complexity analysis. We present the formulation of SST for zonal Slepian functions, which are spatially optimally concentrated in the polar cap~(axisymmetric) region, and provide an illustration using the Earth topography map. To demonstrate the utility of the proposed transform, we carry out localized variation analysis; employing SST for detecting hidden localized variations in the signal.

eess.SP↗

arXiv2019

Efficient sampling and robust 3D diffusion magnetic resonance imaging signal reconstruction

This paper presents novel single and multi-shell sampling schemes for diffusion MRI. In diffusion MRI, it is paramount that the number of samples is as small as possible in order that scan times are practical in a clinical setting. The proposed schemes use an efficient number of measurements in that the number of samples is equal to the degrees of freedom in the orthonormal bases used for reconstruction. Novel reconstruction algorithms based on smaller subsystems of linear equations, as compared to the standard regularized least-squares method, are developed for both single and multi-shells sampling schemes. The smaller matrices used in these novel reconstruction algorithms are designed to be well-conditioned, leading to improved reconstruction accuracy. Accurate and robust reconstruction is also achieved through incorporation of regularization into the novel reconstruction algorithms and using a Rician or non-central Chi noise model. We quantitatively validate our single and multi-shell schemes against standard least-squares reconstruction methods to show that they enable more accurate reconstruction when the number of samples is equal to the degrees of freedom in the basis. Human brain data is also used to qualitatively evaluate reconstruction

eess.SP↗

arXiv2018

Sampling Schemes for Accurate Reconstruction and Computation of Performance Parameters of Antenna Radiation Pattern

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 less number of samples for the accurate evaluation of MEG than the existing methods that rely on the approximate evaluation of the surface integral on the sphere.

eess.SP↗

arXiv2018

An Optimal-Dimensionality Sampling for Spin-$s$ Functions on the Sphere

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 ${\sim}2L^2$ samples for the representation of spin-$s$ functions band-limited at $L$, the proposed scheme requires $N_o=L^2-s^2$ samples 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 multi-pass $s$-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.

astro-ph.IM↗

arXiv2017

Iterative Residual Fitting for Spherical Harmonic Transform of Band-Limited Signals on the Sphere: Generalization and Analysis

We present the generalized iterative residual fitting (IRF) for the computation of the spherical harmonic transform (SHT) of band-limited signals on the sphere. The proposed method is based on the partitioning of the subspace of band-limited signals into orthogonal subspaces. There exist sampling schemes on the sphere which support accurate computation of SHT. However, there are applications where samples~(or measurements) are not taken over the predefined grid due to nature of the signal and/or acquisition set-up. To support such applications, the proposed IRF method enables accurate computation of SHTs of signals with randomly distributed sufficient number of samples. In order to improve the accuracy of the computation of the SHT, we also present the so-called multi-pass IRF which adds multiple iterative passes to the IRF. We analyse the multi-pass IRF for different sampling schemes and for different size partitions. Furthermore, we conduct numerical experiments to illustrate that the multi-pass IRF allows sufficiently accurate computation of SHTs.

cs.IT↗

arXiv2017

Optimal-Dimensionality Sampling on the Sphere: Improvements and Variations

For the accurate representation and reconstruction of band-limited signals on the sphere, an optimal-dimensionality sampling scheme has been recently proposed which requires the optimal number of samples equal to the number of degrees of freedom of the signal in the spectral (harmonic) domain. The computation of the spherical harmonic transform (SHT) associated with the optimal-dimensionality sampling requires the inversion of a series of linear systems in an iterative manner. The stability of the inversion depends on the placement of iso-latitude rings of samples along co-latitude. In this work, we have developed a method to place these iso-latitude rings of samples with the objective of improving the well-conditioning of the linear systems involved in the computation of the SHT. We also propose a multi-pass SHT algorithm to iteratively improve the accuracy of the SHT of band-limited signals. Furthermore, we review the changes in the computational complexity and improvement in accuracy of the SHT with the embedding of the proposed methods. Through numerical experiments, we illustrate that the proposed variations and improvements in the SHT algorithm corresponding to the optimal-dimensionality sampling scheme significantly enhance the accuracy of the SHT.

cs.IT↗

arXiv2017

Slepian Spatial-Spectral Concentration Problem on the Sphere: Analytical Formulation for Limited Colatitude-Longitude Spatial Region

In this paper, we develop an analytical formulation for the Slepian spatial-spectral concentration problem on the sphere for a limited colatitude-longitude spatial region on the sphere, defined as the Cartesian product of a range of positive colatitudes and longitudes. The solution of the Slepian problem is a set of functions that are optimally concentrated and orthogonal within a spatial or spectral region. These properties make them useful for applications where measurements are taken within a spatially limited region of the sphere and/or a signal is only to be analyzed within a region of the sphere. To support localized spectral/spatial analysis, and estimation and sparse representation of localized data in these applications, we exploit the expansion of spherical harmonics in the complex exponential basis to develop an analytical formulation for the Slepian concentration problem for a limited colatitude-longitude spatial region. We also extend the analytical formulation for spatial regions that are comprised of a union of rotated limited colatitude-longitude subregions. By exploiting various symmetries of the proposed formulation, we design a computationally efficient algorithm for the implementation of the proposed analytical formulation. Such a reduction in computation time is demonstrated through numerical experiments. We present illustrations of our results with the help of numerical examples and show that the representation of a spatially concentrated signal is indeed sparse in the Slepian basis.

astro-ph.CO↗

arXiv2017

Efficient Computation of Slepian Functions for Arbitrary Regions on the Sphere

In this paper, we develop a new method for the fast and memory-efficient computation of Slepian functions on the sphere. Slepian functions, which arise as the solution of the Slepian concentration problem on the sphere, have desirable properties for applications where measurements are only available within a spatially limited region on the sphere and/or a function is required to be analyzed over the spatially limited region. Slepian functions are currently not easily computed for large band-limits for an arbitrary spatial region due to high computational and large memory storage requirements. For the special case of a polar cap, the symmetry of the region enables the decomposition of the Slepian concentration problem into smaller subproblems and consequently the efficient computation of Slepian functions for large band-limits. By exploiting the efficient computation of Slepian functions for the polar cap region on the sphere, we develop a formulation, supported by a fast algorithm, for the approximate computation of Slepian functions for an arbitrary spatial region to enable the analysis of modern datasets that support large band-limits. For the proposed algorithm, we carry out accuracy analysis of the approximation, computational complexity analysis, and review of memory storage requirements. We illustrate, through numerical experiments, that the proposed method enables faster computation, and has smaller storage requirements, while allowing for sufficiently accurate computation of the Slepian functions.

cs.DM↗

arXiv2017

An Optimal Dimensionality Multi-shell Sampling Scheme with Accurate and Efficient Transforms for Diffusion MRI

This paper proposes a multi-shell sampling scheme and corresponding transforms for the accurate reconstruction of the diffusion signal in diffusion MRI by expansion in the spherical polar Fourier (SPF) basis. The sampling scheme uses an optimal number of samples, equal to the degrees of freedom of the band-limited diffusion signal in the SPF domain, and allows for computationally efficient reconstruction. We use synthetic data sets to demonstrate that the proposed scheme allows for greater reconstruction accuracy of the diffusion signal than the multi-shell sampling schemes obtained using the generalised electrostatic energy minimisation (gEEM) method used in the Human Connectome Project. We also demonstrate that the proposed sampling scheme allows for increased angular discrimination and improved rotational invariance of reconstruction accuracy than the gEEM schemes.

cs.CV↗

arXiv2017

Multi-shell Sampling Scheme with Accurate and Efficient Transforms for Diffusion MRI

We propose a multi-shell sampling grid and develop corresponding transforms for the accurate reconstruction of the diffusion signal in diffusion MRI by expansion in the spherical polar Fourier (SPF) basis. The transform is exact in the radial direction and accurate, on the order of machine precision, in the angular direction. The sampling scheme uses an optimal number of samples equal to the degrees of freedom of the diffusion signal in the SPF domain.

cs.DM↗

arXiv2017

Gauss-Legendre Sampling on the Rotation Group

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.

cs.IT↗

arXiv2016

Accurate Reconstruction of Finite Rate of Innovation Signals on the Sphere

We develop a method for the accurate reconstruction of 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 better conditioning of systems involved in the recovery of parameters. For the recovery of $K$ 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 + \sqrt{K + \frac{1}{4}} - \frac{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 the same or less. We also conduct numerical experiments to demonstrate that the proposed technique is more accurate than the existing methods by a factor of $10^{7}$ or more for $2 \le K\le 20$.

cs.IT↗

arXiv2015

An Optimal Dimensionality Sampling Scheme on the Sphere for Antipodal Signals In Diffusion Magnetic Resonance Imaging

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.

physics.med-ph↗

arXiv2015

Spherical Harmonic Expansion of Fisher-Bingham Distribution and 3D Spatial Fading Correlation for Multiple-Antenna Systems

This paper considers the 3D spatial fading correlation (SFC) resulting from an angle-of-arrival (AoA) distribution that can be modelled by a mixture of Fisher-Bingham distributions on the sphere. By deriving a closed-form expression for the spherical harmonic transform for the component Fisher-Bingham distributions, with arbitrary parameter values, we obtain a closed-form expression of the 3D-SFC for the mixture case. The 3D-SFC expression is general and can be used in arbitrary multi-antenna array geometries and is demonstrated for the cases of a 2D uniform circular array in the horizontal plane and a 3D regular dodecahedral array. In computational aspects, we use recursions to compute the spherical harmonic coefficients and give pragmatic guidelines on the truncation size in the series representations to yield machine precision accuracy results. The results are further corroborated through numerical experiments to demonstrate that the closed-form expressions yield the same results as significantly more computationally expensive numerical integration methods.

cs.IT↗