Alice P. Bates

Novel sampling scheme on the sphere for head-related transfer function measurements

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.

An Optimal Dimensionality Sampling Scheme on the Sphere with Accurate and Efficient Spherical Harmonic Transform for Diffusion MRI

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.

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.

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.

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.

On the Use of Antipodal Optimal Dimensionality Sampling Scheme on the Sphere for Recovering Intra-Voxel Fibre Structure in Diffusion MRI

In diffusion magnetic resonance imaging (dMRI), the diffusion signal can be reconstructed from measurements collected on single or multiple spheres in \(\boldsymbol{q}\)-space using a spherical harmonic expansion. The number of measurements that can be acquired is severely limited and should be as small as possible. Previous sampling schemes have focused on using antipodal symmetry to reduce the number of samples and uniform sampling to achieve rotationally invariant reconstruction accuracy, but do not allow for an accurate or computationally efficient spherical harmonic transform (SHT). The recently proposed antipodal optimal dimensionality sampling scheme on the sphere requires the minimum number of samples, equal to the number of degrees of freedom for the representation of the antipodal symmetric band-limited diffusion signal in the spherical harmonic domain. In addition, it allows for the accurate and efficient computation of the SHT. In this work, we evaluate the use of this recently proposed scheme for the reconstruction of the diffusion signal and subsequent intra-voxel fibre structure estimation in dMRI. We show, through numerical experiments, that the use of this sampling scheme allows accurate and computationally efficient reconstruction of the diffusion signal, and improved estimation of intra-voxel fibre structure, in comparison to the antipodal electrostatic repulsion and spherical code sampling schemes with the same number of samples. We also demonstrate that it achieves rotationally invariant reconstruction accuracy to the same extent as the other two sampling schemes.

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 scheme 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 scheme.

On the use of Slepian functions for the reconstruction of the head-related transfer function on the sphere

In this work we investigate using the Slepian basis for the reconstruction of the head-related transfer function (HRTF) on the sphere. Measurements of the HRTF are unavailable over the south polar cap which tends to result in a large reconstruction error when reconstruction is performed in the traditionally used spherical harmonic basis. While the spherical harmonic basis is well-suited to applications where data is taken over the whole sphere, it is not a natural basis when considering a region on the sphere. The Slepian basis is a set of functions which are optimally concentrated and orthogonal within a region, unlike the spherical harmonic basis. We demonstrate through numerical experiments on randomly generated data and synthetic HRTF measurements that reconstruction of the HRTF in the Slepian basis is significantly more accurate at sample locations; the reconstruction error is up to 11 orders of magnitude smaller. The reconstruction error obtained using the Slepian basis is also smaller at other locations on the sphere, both within and outside of the region where measurements are taken, than in the spherical harmonic basis. We also briefly investigate truncation of the Slepian basis as a means of denoising the HRTF measurements and find that this reduces the reconstruction error. Our analysis suggests that the Slepian basis allows more accurate reconstruction than the spherical harmonic basis.

Unsupervised image analysis for zebrafish embryogenesis using lab-on-a-chip embryo arrays

The zebrafish is a popular vertebrate model organism that is widely deployed with lab-on-a-chip (LoC) technology for in-situ experiments in drug discovery and eco-toxicity assays. Although LoC devices enable easy manipulation and trapping of embryos for such experiments, constant human attention is required during and after the experiments for online and offline monitoring and analysis. The throughput and turnaround time are both limited from lack of automated image analysis. In this paper, a novel image analysis algorithm is developed to automatically recognise dead embryos and the first two stages of the zebrafish embryo development in order to detect anomalies caused by an applied chemical agent during embryogenesis. The algorithm has been examined using 55 zebrafish embryo images and has achieved a success rate of 94.5% in recognising the correct embryo development stage.