## Abstract

We consider the sampling of band-limited spatio-temporal signals subject to the time-sequential (TS) constraint that only one spatial sample can be taken at a given time. This problem arises in acquisition systems where, owing to physical limitations, only one probe or sensor is available to scan an entire spatial region to sample a time-varying signal. Examples include feed-through image scanning (one spatial and one temporal dimension), video scanning (two spatial and one temporal dimension), electromechanical or holographic laser spot deflection for range imaging, active phased array radar or sonar beamformers (three spatial and one temporal dimension) and magnetic resonance spectroscopic functional imaging (three spatial, one spectral, and one temporal dimension). The question of interest, is to design an efficient sampling strategy, that will minimize the required sampling rate. We consider signals with a finite spatial support, or equivalently, infinite support signals that are sampled over a finite field of view. We assume that the time interval between samples within this spatial support is independent of the spatial positions at which they need to be taken. We call the minimum value of this time interval the response time of the sensor. The complete set of TS samples can be represented as an ordered sequence of spatial points indexed by their sampling instants. We call this sequence the sampling schedule. Now, in many practical applications the cost of the sensor is inversely proportional to its response time, and often physical limitations impose a hard lower bound on this time. The TS sampling design problem is therefore to determine an optimal sampling schedule minimizing the distortion of reconstructions from samples with a given time interval, or to maximize the intersample temporal interval, subject to constraints on the fidelity of the reconstruction of the signal from its samples. Allebach [1], the first to consider the TS sampling problem, provided a rather complete analysis of the spectral content of time-sequentially sampled signals for a given sampling schedule. However, with this theory, the only design strategy was exhaustive combinatorial search, which is infeasible in practical problems. Using the powerful techniques of lattice theory, we have recently developed a new unifying theory linking TS sampling with generalized multidimensional sampling [2, 3]. The results have a geometric nature, involving simultaneous lattice packing of the spectral and spatial supports in their respective domains. The theory, which applies to arbitrary spatial and spectral supports in any dimension, includes tight fundamental performance bounds, which are asypmtotically achievable. One of the surprising results is that, in the limit of large space-spatial bandwidth product, optimal time-sequential lattice patterns achieve the same rates as unconstrained (e.g., frame-instantenous) multidimensional sampling. Simple graphical design algorithms are provided for lattice sampling patterns minimizing the sampling rates. An application to tomography of time-varying objects is shown, providing a four-fold reduction in required scan rate.

Original language | English (US) |
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Title of host publication | Proceedings - International Conference on Pattern Recognition |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 47 |

Number of pages | 1 |

ISBN (Electronic) | 0818662751 |

DOIs | |

State | Published - 1994 |

Event | 12th IAPR International Conference on Pattern Recognition - Conference C: Signal Processing - Conference D: Parallel Computing, ICPR 1994 - Jerusalem, Israel Duration: Oct 9 1994 → Oct 13 1994 |

### Publication series

Name | Proceedings - International Conference on Pattern Recognition |
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Volume | 3 |

ISSN (Print) | 1051-4651 |

### Conference

Conference | 12th IAPR International Conference on Pattern Recognition - Conference C: Signal Processing - Conference D: Parallel Computing, ICPR 1994 |
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Country/Territory | Israel |

City | Jerusalem |

Period | 10/9/94 → 10/13/94 |

## ASJC Scopus subject areas

- Computer Vision and Pattern Recognition