Sampling Signals With a Finite Rate of Innovation on the Sphere

• Published in: IEEE transactions on signal processing Vol. 61; no. 18; pp. 4552 - 4561
• Main Authors: Deslauriers-Gauthier, Samuel, Marziliano, Pina
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: annihilating filter; Applied sciences; Band theory; Biological and medical sciences; Computerized, statistical medical data processing and models in biomedicine; Convolution; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; finite rate of innovation; Fourier transforms; Harmonic analysis; Image reconstruction; Information, signal and communications theory; Kernel; Kernels; Mathematical analysis; Mathematical models; Medical management aid. Diagnosis aid; Medical sciences; Reconstruction; Sampling; Sampling theorem; Sampling, quantization; Signal and communications theory; Signal, noise; spherical convolution; spherical harmonic; Technological innovation; Telecommunications and information theory; Theorems; Transaction processing; State of the art; Band limited signal; Nuclear magnetic resonance imaging; Kernel method; Innovation; Azimuth; Convolution; finite rate of innovation; spherical convolution; Signal processing; Spherical harmonic; Medical imagery; Numerical simulation; Sampling theorem; Sampling; annihilating filter
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2013.2272289

The state of the art in sampling theory now contains several theorems for signals that are non-bandlimited. For signals on the sphere however, most theorems still require the assumptions of bandlimitedness. In this work we show that a particular class of non-bandlimited signals, which have a finite rate of innovation, can be exactly recovered using a finite number of samples. We consider a sampling scheme where K weighted Diracs are convolved with a kernel on the rotation group. We prove that if the sampling kernel has a bandlimit L = 2K then (2K - 1)(4K - 1) + 1 equiangular samples are sufficient for exact reconstruction. If the samples are uniformly distributed on the sphere, we argue that the signal can be accurately reconstructed using 4K 2 samples and validate our claim through numerical simulations. To further reduce the number of samples required, we design an optimal sampling kernel that achieves accurate reconstruction of the signal using only 3K samples, the number of parameters of the weighted Diracs. In addition to weighted Diracs, we show that our results can be extended to sample Diracs integrated along the azimuth. Finally, we consider kernels with antipodal symmetry which are common in applications such as diffusion magnetic resonance imaging.