Sampling trajectories for mobile sensing

Classical sampling theory for sampling and reconstructing bandlimited fields in Rd addresses the problem of sampling on lattice points. We consider a generalization of this problem, in which one samples the field along 1-dimensional spatial trajectories in Rd rather than at points. The process of sampling records the value of the field at all points on the sampling trajectories. Such a sampling setup is relevant in the problem of spatial sampling using mobile sensors. We study various possible designs of sampling trajectories and discuss necessary and sufficient conditions for perfect reconstruction. We introduce a density metric for trajectories which we call path density, that quantifies the total length of these trajectories per unit volume in Rd. We formulate the problem of identifying optimal sampling trajectories that admit perfect reconstruction of bandlimited fields and are minimal in terms of the path density metric. We identify optimal sampling trajectories from a restricted class of trajectories.

[1]  Minh N. Do,et al.  Ieee Transactions on Image Processing on the Bandwidth of the Plenoptic Function , 2022 .

[2]  Ahmed H. Tewfik,et al.  Sampling theorems for two-dimensional isotropic random fields , 1988, IEEE Trans. Inf. Theory.

[3]  R.M. Mersereau,et al.  The processing of hexagonally sampled two-dimensional signals , 1979, Proceedings of the IEEE.

[4]  H. Landau.  Necessary density conditions for sampling and interpolation of certain entire functions , 1967 .

[5]  David Middleton,et al.  Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces , 1962, Inf. Control..

[6]  L. J. Cutrona,et al.  The Processing of Hexagonally Sampled Two-Dimensional Signals , 1979 .

[7]  K. Gröchenig.  RECONSTRUCTION ALGORITHMS IN IRREGULAR SAMPLING , 1992 .

[8]  Hamid Behmard,et al.  Sampling of Bandlimited Functions on Unions of Shifted Lattices , 2002 .

[9]  A. Papoulis,et al.  Generalized sampling expansion , 1977 .

[10]  Karlheinz Gröchenig,et al.  On Landau's Necessary Density Conditions for Sampling and Interpolation of Band-Limited Functions , 1996 .

[11]  Francesco Ripamonti,et al.  Moving microphone arrays to reduce spatial aliasing in the beamforming technique: theoretical background and numerical investigation. , 2008, The Journal of the Acoustical Society of America.

[12]  I. Bárány,et al.  convex sets , 2007 .

[13]  Minh N. Do,et al.  A Computable Fourier Condition Generating Alias-Free Sampling Lattices , 2009, IEEE Transactions on Signal Processing.

[14]  Christodoulos Chamzas,et al.  Sampling on Concentric Circles , 1998, IEEE Trans. Medical Imaging.

[15]  Martin Vetterli,et al.  Sampling and reconstructing spatial fields using mobile sensors , 2012, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).