We propose a 4-dimensional (4D) basis and sampling scheme, along with a corresponding reconstruction algorithm, for the measurement and reconstruction of the b-tensor encoded diffusion signal in diffusion magnetic resonance imaging (MRI). This is only the second basis proposed for representing the b-tensor encoded diffusion signal and the first to allow for planar tensor measurements. We design a sampling scheme that attains an efficient number of samples, equal to the degrees of freedom required to represent the diffusion signal in the proposed 4D basis. The properties of the diffusion signal are studied to provide recommendations on how many b-tensor measurements to use. Evaluation of the proposed scheme using Monte Carlo simulations of the diffusion signal is done to show that the proposed scheme gives accurate interpolation of the signal.

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 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.

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.