Continuous wavelet transform with arbitrary scales and O(N) complexity

Abstract

The continuous wavelet transform (CWT) is a common signal-processing tool for the analysis of nonstationary signals. We propose here a new B-spline-based method that allows the CWT computation at any scale. A nice property of the algorithm is that the computational cost is independent of the scale value. Its complexity is of the same order as that of the fastest published methods, without being restricted to dyadic or integer scales. The method reduces to the filtering of an auxiliary (pre-integrated) signal with an expanded mask that acts as a kind of modified ‘à trous’ filter. The algorithm is well-suited for a parallel implementation.

Introduction

The continuous wavelet transform (CWT) of a signal f with the wavelet ψ is defined as

$$\text{W}{}_{\mathrm{\psi }}\text{f(a,b)=}\text{1}\text{a}\text{∫}\text{−∞}\text{+∞}\text{f(x)ψ}\text{b−x}\text{a}\text{d}\text{x.}$$

It can be interpreted as the correlation of the input signal with a time-reversed version of ψ rescaled by a factor of a. For a 1-D input signal, the result is a 2-D description of the signal with respect to time b and scale a. The scale a is inversely proportional to the central frequency of the rescaled wavelet ψ_a(x)=ψ(x/a) which is typically a bandpass function; b represents the time location at which we analyze the signal. The larger the scale a, the wider the analyzing function ψ_a, and hence smaller the corresponding analyzed frequency. The output value is maximized when the frequency of the signal matches that of the corresponding dilated wavelet. The main advantage over the Fourier transform (FT) analysis is that the frequency description is localized in time. The advantage over the short-time Fourier transform (STFT) is that the window size varies; low frequencies are analyzed over wide time windows, and high frequencies over narrow time windows, which is more effective than to use a fixed-size analysis. Typical applications of the CWT are the detection and characterization of singularities [3], [14], pattern recognition [6], image processing [4], [15], fractal analysis [2], [12], [23], noise reduction [11] and the analysis of biomedical signals [7], [10], [25].

The main contribution of this paper is the development of a fast algorithm for the computation of the CWT at any real scale a and integer time localization b. Mallat's fast wavelet algorithm [12] uses the multiresolution properties of the wavelet to compute the CWT at dyadic scales a=2ⁱ and time shifts b=2ⁱk, k∈Z [17]; it achieves an overall O(N) complexity. Other techniques compute the wavelet transform at dyadic scales and integer time points with an ‘à trous’ approach. Their complexity per scale is O(N), the same as Mallat's algorithm, but with a larger leading constant [3], [5], [9], [16].

Despite their speed, these methods may not be precise enough for some applications, since a dyadic scale progression cannot be finer than an octave sub-band decomposition. To achieve a better scale resolution, other approaches have been proposed, either based on M-band decomposition inside an octave [17], [24] or on a generalization of the two-scale relation to general integer N-scale relations [8], [22]. However, none of these algorithms can handle arbitrary scales.

Our purpose here is to develop a novel and fast algorithm that works for any real value of a. It takes advantage of a B-spline decomposition of the input signal and of the mother wavelet. The method exploits the fact that B-splines are compactly supported and that the convolution of two B-splines can be expressed analytically [13].