### Abstract

We introduce a general framework for the efficient computation of the real continuous wavelet transform (CWT) using a filter bank. The method allows arbitrary sampling along the scale axis, and achieves O(N) complexity per scale where N is the length of the signal. Previous algorithms that calculated non-dyadic samples along the scale axis had O(N log (N)) computations per scale. Our approach approximates the analyzing wavelet by its orthogonal projection (least-squares solution) onto a space defined by a compactly supported scaling function. We discuss the theory which uses a duality principle and recursive digital filtering for rapid calculation of the CWT. We derive error bounds on the wavelet approximation and show how to obtain any desired level of accuracy through the use of longer filters. Finally, we present examples of implementation for real symmetric and anti-symmetric wavelets.

Original language | English |
---|---|

Pages (from-to) | 103-119 |

Number of pages | 17 |

Journal | Signal Processing |

Volume | 57 |

Issue number | 2 |

Publication status | Published - 1997 Mar 1 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Software
- Signal Processing
- Computer Vision and Pattern Recognition
- Electrical and Electronic Engineering

### Cite this

*Signal Processing*,

*57*(2), 103-119.

}

*Signal Processing*, vol. 57, no. 2, pp. 103-119.

**Fast continuous wavelet transform : A least-squares formulation.** / Vrhel, M. J.; Lee, C.; Unser, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Fast continuous wavelet transform

T2 - A least-squares formulation

AU - Vrhel, M. J.

AU - Lee, C.

AU - Unser, M.

PY - 1997/3/1

Y1 - 1997/3/1

N2 - We introduce a general framework for the efficient computation of the real continuous wavelet transform (CWT) using a filter bank. The method allows arbitrary sampling along the scale axis, and achieves O(N) complexity per scale where N is the length of the signal. Previous algorithms that calculated non-dyadic samples along the scale axis had O(N log (N)) computations per scale. Our approach approximates the analyzing wavelet by its orthogonal projection (least-squares solution) onto a space defined by a compactly supported scaling function. We discuss the theory which uses a duality principle and recursive digital filtering for rapid calculation of the CWT. We derive error bounds on the wavelet approximation and show how to obtain any desired level of accuracy through the use of longer filters. Finally, we present examples of implementation for real symmetric and anti-symmetric wavelets.

AB - We introduce a general framework for the efficient computation of the real continuous wavelet transform (CWT) using a filter bank. The method allows arbitrary sampling along the scale axis, and achieves O(N) complexity per scale where N is the length of the signal. Previous algorithms that calculated non-dyadic samples along the scale axis had O(N log (N)) computations per scale. Our approach approximates the analyzing wavelet by its orthogonal projection (least-squares solution) onto a space defined by a compactly supported scaling function. We discuss the theory which uses a duality principle and recursive digital filtering for rapid calculation of the CWT. We derive error bounds on the wavelet approximation and show how to obtain any desired level of accuracy through the use of longer filters. Finally, we present examples of implementation for real symmetric and anti-symmetric wavelets.

UR - http://www.scopus.com/inward/record.url?scp=0031099214&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031099214&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031099214

VL - 57

SP - 103

EP - 119

JO - Signal Processing

JF - Signal Processing

SN - 0165-1684

IS - 2

ER -