### Abstract

This paper describes a basic difference between multiwavelets and scalar wavelets that explains, without using zero moment properties, why certain complications arise in the implementation of discrete multiwavelet transforms. Assuming one wishes to avoid the use of prefilters in implementing the discrete multiwavelet transform, it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r^{2} scaling functions φ_{i,j}. This paper also introduces new K-balanced orthogonal multiwavelet bases based on symmetric FIR filters. The nonlinear design equations arising in this work are solved using Grobner bases. The K-balanced multiwavelet bases based on even-length symmetric FIR filters are shown to be particularly well behaved, as illustrated by special relations they satisfy and by the examples constructed.

Original language | English (US) |
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Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Publisher | Society of Photo-Optical Instrumentation Engineers |

Pages | 122-131 |

Number of pages | 10 |

Volume | 3813 |

State | Published - 1999 |

Event | Proceedings of the 1999 Wavelet Applications in Signal and Image Processing VII - Denver, CO, USA Duration: Jul 19 1999 → Jul 23 1999 |

### Other

Other | Proceedings of the 1999 Wavelet Applications in Signal and Image Processing VII |
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City | Denver, CO, USA |

Period | 7/19/99 → 7/23/99 |

### Fingerprint

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Condensed Matter Physics

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*(Vol. 3813, pp. 122-131). Society of Photo-Optical Instrumentation Engineers.

**Balanced multiwavelet bases based on symmetric FIR filters.** / Selesnick, Ivan.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of SPIE - The International Society for Optical Engineering.*vol. 3813, Society of Photo-Optical Instrumentation Engineers, pp. 122-131, Proceedings of the 1999 Wavelet Applications in Signal and Image Processing VII, Denver, CO, USA, 7/19/99.

}

TY - GEN

T1 - Balanced multiwavelet bases based on symmetric FIR filters

AU - Selesnick, Ivan

PY - 1999

Y1 - 1999

N2 - This paper describes a basic difference between multiwavelets and scalar wavelets that explains, without using zero moment properties, why certain complications arise in the implementation of discrete multiwavelet transforms. Assuming one wishes to avoid the use of prefilters in implementing the discrete multiwavelet transform, it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r2 scaling functions φi,j. This paper also introduces new K-balanced orthogonal multiwavelet bases based on symmetric FIR filters. The nonlinear design equations arising in this work are solved using Grobner bases. The K-balanced multiwavelet bases based on even-length symmetric FIR filters are shown to be particularly well behaved, as illustrated by special relations they satisfy and by the examples constructed.

AB - This paper describes a basic difference between multiwavelets and scalar wavelets that explains, without using zero moment properties, why certain complications arise in the implementation of discrete multiwavelet transforms. Assuming one wishes to avoid the use of prefilters in implementing the discrete multiwavelet transform, it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r2 scaling functions φi,j. This paper also introduces new K-balanced orthogonal multiwavelet bases based on symmetric FIR filters. The nonlinear design equations arising in this work are solved using Grobner bases. The K-balanced multiwavelet bases based on even-length symmetric FIR filters are shown to be particularly well behaved, as illustrated by special relations they satisfy and by the examples constructed.

UR - http://www.scopus.com/inward/record.url?scp=0033348291&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033348291&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0033348291

VL - 3813

SP - 122

EP - 131

BT - Proceedings of SPIE - The International Society for Optical Engineering

PB - Society of Photo-Optical Instrumentation Engineers

ER -