| Summary: | Sampling theorems describe which types of signals can be reconstructed and under which conditions. In the 64 years since Shannon’s sampling theorem for bandlimited functions, we have learned to sample many other classes of signals, some of them nonbandlimited. However, for signals defined on the sphere, the vast majority of sampling theorems still deal with bandlimited signals. In many applications, the assumption of bandlimitedness is not optimal and hampers the accuracy of the reconstruction. A notable example is diffusion magnetic resonance imaging, the application considered in this thesis. In this work, we produce new sampling theorems that prove that certain nonbandlimited signals on the sphere can be sampled using a finite number of samples. We use them to improve the reconstruction of fiber orientations in diffusion magnetic resonance imaging. We proceed by first showing that certain non-bandlimited signals defined on the unit sphere in three dimensions can be sampled and reconstructed using a finite number of samples. When the sample locations are equiangular, we prove that the reconstruction is exact. We also argue and validate using numerical simulations that fewer samples can be used if they are uniformly distributed. In both cases, the number of samples required depends on the number of degrees of freedom, or rate of innovation, of the signals. Next, we consider different sampling kernels and signal models that can also be sampled at their rate of innovation. We develop an optimal kernel that allows us to sample and reconstruct signals at the critical rate, the number of degrees of freedom. We also consider antipodally symmetric kernels, a case which is not covered by the previous theory. Finally, we prove that orientations integrated along the azimuth and great circles can also be sampled at their rate of innovation. Because signals are often observed in noisy environments, we then investigate how to accurately recover the signals previously described from noisy measurements. We propose two major changes to the reconstruction algorithm: one that improves the results when we do not oversample and one that uses the additional information provided by oversampling. Finally, we apply our results to the recovery of fiber orientations in diffusion magnetic resonance imaging. The main advantage of our method over existing ones is that the angular resolution does not depend on the number of samples that is acquired. In theory, our reconstruction algorithm is able to recover fiber orientations regardless of their crossing angle, given an appropriate sampling scheme. We show, through numerical simulations, that our method is indeed able to distinguish fibers that cross at a very narrow angle.
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