## 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...

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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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Citation Context ...ructure Of course much more work is necessary before we can guarantee exact interpolation. Our task is made simpler since we can use many of the tools developed in the field of multirate filter banks =-=[5, 30, 31]-=-. We will assume familiarity with the basics of the filter bank literature. Consider the structure shown in Figure 3. This is the most trivial example of an M-channel filter bank, since it merely sepa... |

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Citation Context ... i (z)X(z) to be zero. 2 Clearly then, unless X(e jw ) contains Fourier components only at isolated frequencies, the interpolator cannot be realizable. This is so, since by the Paley-Wiener criterion =-=[33]-=-, any filter whose spectrum is zero over any band, however narrow, cannot have a rational transfer function. If the filters H 0 i (z) are irrational, then so are the OE i (z M ) and hence, by (8), so ... |

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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 ...\Gamma1 ), where H(z) is the Daubechies filter [16] of length 2N . There are many parallels between discrete-time polynomial interpolation schemes, and continuous-time spline interpolation algorithms =-=[17, 18]-=-. Related work on continuous-time interpolation problems has appeared in [19, 20, 21, 22]. An application of the iterative subdivision scheme to signal expansions appears in [23]. An excellent text on... |

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Citation Context ...ther words the infinite iteration of the discrete-time interpolation gives a function that can be used to interpolate a continuous-time function from its evenly-spaced samples. Subdivision algorithms =-=[6, 7, 8, 9, 11]-=- have been extensively studied for curve generation applications, in particular where two-fold interpolation is used. This of course restricts the sampling density to be 2 p times as dense as that of ... |

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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 ...nce. In the limit, however, p !1 (26) will tend to a continuous-time function provided that the infinite product converges. Convergence of products of this kind were examined in the case for M = 2 in =-=[16, 44, 45], an-=-d for arbitrary M in [46, 27, 47]. A sufficient condition given for convergence in [27, 47] is that P (z) should have an adequate number of zeros at the M-th roots of unitysk = e j2��k=M ; k = 1; ... |

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Citation Context ...toring. Thus any two-channel perfect reconstruction filter bank can be used in an interpolation scheme; and the product function of a two-channel perfect reconstruction filter bank is an interpolator =-=[34, 35]-=-. The set of signals that is interpolated are those X(z) for which X(z)P (\Gammaz) = 0. The first work to draw attention to the connection between two-channel filter banks and interpolators appears to... |

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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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Citation Context ...lation algorithms [17, 18]. Related work on continuous-time interpolation problems has appeared in [19, 20, 21, 22]. An application of the iterative subdivision scheme to signal expansions appears in =-=[23]-=-. An excellent text on continuous-time interpolation is [24]. Relations between polynomial interpolation and wavelets are explored in [25, 26]. Starting from the point of view of the construction of M... |

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Citation Context ... and sampling of multiband and bandpass continuous-time signals. The constraint that the set (\Gamma��; ��) should not contain frequencies separated by 2��k=M closely echoes the requiremen=-=ts given in [38]-=- and [39] for sampling bandpass and multiband signals at the Nyquist-Landau rate. 3.3 Rational design examples 3.3.1 Polynomials Suppose that we wish to design an interpolator for signals that are sam... |

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Citation Context ... parallels between discrete-time polynomial interpolation schemes, and continuous-time spline interpolation algorithms [17, 18]. Related work on continuous-time interpolation problems has appeared in =-=[19, 20, 21, 22]-=-. An application of the iterative subdivision scheme to signal expansions appears in [23]. An excellent text on continuous-time interpolation is [24]. Relations between polynomial interpolation and wa... |

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Citation Context ... that extension to the multidimensional case follows similar lines. It is also possible to use the same kind of arguments to address the problem nonuniform periodic sampling and second order sampling =-=[36]-=-. 3 Designing an M-fold interpolator For most signals that can be interpolated the interpolator will have an irrational transfer function, and we examine this case next. The case where the interpolato... |

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Citation Context ...polynomial interpolation and wavelets are explored in [25, 26]. Starting from the point of view of the construction of M-channel filter banks with certain regularity properties, recent work by Heller =-=[27, 28, 29]-=- produces interpolators that are the same as those designed in Section 3.3.1. The outline of the paper is as follows. In Section 2 we derive the necessary and sufficient conditions under which a discr... |

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Citation Context ...amma 1 = p (the number of zeros in A(z)). We should observe that solving linear equations to find the filters in a filter bank, which is all that we are doing, has been used often before, for example =-=[37, 5, 30]-=-. Remarks: 1. In the limiting case where LK = 1 there are no equations to solve. This requires (LDsLN ): 1 = i + (LD \Gamma 1)(M \Gamma 1) \Gamma LN + 1: In the M = 2 case this gives LN = LD for the o... |

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