@article{da39f668e5de4d1e9244c3bc8fe13780,

title = "Fast Implementation of the Continuous Wavelet Transform with Integer Scales",

abstract = "We describe a fast noniterative algorithm for the evaluation of continuous spline wavelet transforms at any integer scale m. In this approach, the input signal and the analyzing wavelet are both represented by polynomial splines. The algorithm uses a combination of moving sum and zero-padded filters, and its complexity per scale is O(N), where N is the signal length. The computation is exact, and the implementation is noniterative across scales. We also present examples of spline wavelets exhibiting properties that are desirable for either singularity detection (first and second derivative operators) or Gabor-like time-frequency signal analysis.",

author = "Michael Unser and Akram Aldroubi and Schiff, {Steven J.}",

note = "Funding Information: The continuous wavelet transform (CWT) of a continuous-time signal s(x) is defined by the convolution integral (cf. [l], [2]) where $!(x)i s the analyzing wavelet, and a and b are continuously varying shi@ and scale parameters, respectively. Because of its unique localization and scaling properties, this representation provides an attractive tool for the analysis of signals with nonstationary characteristics [l], [3]. For implementation purposes, the parameters a and b are usually discretized. In the dyadic case, the analysis is restricted to scales that are powers of two: a = 22 and i E 2'. If, in addition, the wavelet 4 is derived from a multiresolution analysis [4], it is possible to compute the wavelet transform in a very efficient fashion. In the critically sampled case, the discretization is a = 2', b = 2'k, k E 2, and one can use Mallat's algorithm, which relies on the use of two complementary low-and high-pass filters and has an overall O(N) complexity [3], [4]. A similar strategy is also applicable for the oversampled case (n = 2', b = k) using zero-padded (or {"}?I trous{"}) filters [5], [6] with an O(N) complexity per scale. All the procedures described above are computationally very efficient and therefore widely used in practice. Unfortunately, they are limited to the computation of dyadic wavelet transforms. Here, we will consider an alternative scheme that allows for a finer discretiza-tion of the CWT at the integers (a = m, b = k ) and has a complexity per scale that is comparable with that of Shensa's algorithm (O(M)). This greater freedom in the choice of n is possible only because we are restricting ourselves to the class of polynomial spline CWT's and exploiting some very special scaling properties of the corresponding basis functions (B-splines). In addition, for a fixed integer value of a, the corresponding CWT is itself a polynomial spline and is therefore continuously known. Consequently, the method provides an exact computation. The disadvantages are minimal because splines Manuscript received May 13, 1993; revised May 27, 1994. This work was supported, in part, by NIH grant IR29-MH 50006-01. The associate editor coordinating the review of this paper and approving it for publication was Prof. AB N. Akansu.",

year = "1994",

month = dec,

doi = "10.1109/78.340787",

language = "English (US)",

volume = "42",

pages = "3519--3523",

journal = "IEEE Transactions on Signal Processing",

issn = "1053-587X",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

number = "12",

}