## Exact Interpolation and Iterative Subdivision Schemes (1995)

Venue: | IEEE Trans. Signal Processing |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Herley95exactinterpolation,

author = {Cormac Herley},

title = {Exact Interpolation and Iterative Subdivision Schemes},

journal = {IEEE Trans. Signal Processing},

year = {1995},

volume = {43},

pages = {1348--1359}

}

### OpenURL

### Abstract

In this paper we examine the circumstances under which a discrete-time signal can be exactly interpolated given only every M-th sample. After pointing out the connection between designing an M -fold interpolator and the construction of an M -channel perfect reconstruction filter bank, we derive necessary and sufficient conditions on the signal under which exact interpolation is possible. Bandlimited signals are one obvious example, but numerous others exist. We examine these and show how the interpolators may be constructed. A main application is to iterative interpolation schemes, used for the efficient generation of smooth curves. We show that conventional bandlimited interpolators are not suitable in this context. We illustrate that a better criterion is to use interpolators that are exact for polynomial functions. Further, we demonstrate that these interpolators converge when iterated. We show how these may be designed for any polynomial degree N and any interpolation factor M . Th...

### Citations

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Citation Context ...ed by mimicking the analysis of [16]. There a link was formed between the expansion of discrete-time signals using filter banks [31, 40, 5], and the expansion continuous-time functions using wavelets =-=[41, 42]-=-. 4.1 Interpolating continuous-time signals from evenly spaced samples Assume that x a (t) is a function of t, with Fourier transform X a(\Omega\Gamma9 We will denote by x(n) its sampled version x(n) ... |

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Citation Context ..., and this is the approach that we will follow. Hence we wish to investigate the condition where the interpolation is exact. That is, suppose that X(z) is written in terms of its polyphase components =-=[4, 5]-=- X(z) = X 0 (z M ) + z \Gamma1 X 1 (z M ) + z \Gamma2 X 2 (z M ) + \Delta \Delta \Delta z \Gamma(M \Gamma1) XM \Gamma1 (z M ); (1) then we say that the interpolation is exact for the signal x(n), if w... |

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235 |
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Citation Context ... example, but we shall explore others. While it is not possible to interpolate general signals from one of their M-phase components, a popular use of interpolation is in iterative subdivision schemes =-=[6, 7, 8, 9, 10, 11]-=- where the aim is to produce a smooth curve going through a given set of points. Here the original signal is upsampled and interpolated iteratively, so that we generate more and more samples. For a di... |

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Citation Context ... example, but we shall explore others. While it is not possible to interpolate general signals from one of their M-phase components, a popular use of interpolation is in iterative subdivision schemes =-=[6, 7, 8, 9, 10, 11]-=- where the aim is to produce a smooth curve going through a given set of points. Here the original signal is upsampled and interpolated iteratively, so that we generate more and more samples. For a di... |

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Citation Context ...in [12, 13, 5] filter banks are used to derive sampling results, but using more than one channel. The connection between interpolation schemes and filter banks appears to have been first mentioned in =-=[14, 15]-=- for the two-channel case, where it was shown that the minimum length symmetric interpolator which is exact for polynomials of degree ! 2N \Gamma 1 is identical to H(z)H(z \Gamma1 ), where H(z) is the... |

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Citation Context ... example, but we shall explore others. While it is not possible to interpolate general signals from one of their M-phase components, a popular use of interpolation is in iterative subdivision schemes =-=[6, 7, 8, 9, 10, 11]-=- where the aim is to produce a smooth curve going through a given set of points. Here the original signal is upsampled and interpolated iteratively, so that we generate more and more samples. For a di... |

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Citation Context ..., and this is the approach that we will follow. Hence we wish to investigate the condition where the interpolation is exact. That is, suppose that X(z) is written in terms of its polyphase components =-=[4, 5]-=- X(z) = X 0 (z M ) + z \Gamma1 X 1 (z M ) + z \Gamma2 X 2 (z M ) + \Delta \Delta \Delta z \Gamma(M \Gamma1) XM \Gamma1 (z M ); (1) then we say that the interpolation is exact for the signal x(n), if w... |

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