Robust knot detection and spline approximation using wavelet transform extrema and multi-target tracking

Mallat and Hwang (1992) and Mallat and Zhong (1992) describe a novel algorithm both for signal compression and noise removal based on the extraction of the extrema of continuous wavelet transforms (CWTs) at a dyadic sequence of scales, the chaining of the extrema across scale that come from the same feature in the original signal, and the separation of noise from signal through the estimation of the Lipschitz exponents of the singularities corresponding to each such chain of extrema. The results in Mallat and Hwang, and Mallat Zhong and in other efforts based on this technique show the apparent promise of the approach and lead to the conjecture that it is robust, but a number of interesting and important questions and potential applications remain. Mallat and Hwang, and Mallat and Zhong use a simple extrema matching algorithm based on the comparison of the values and the positions of extrema at a pair of consecutive scales. This algorithm cannot work equally well for all input signals, because the trajectories of CWT extrema depend on the type of the corresponding singularity, as well as on the wavelet. The present authors restrict the approximation class of signals to splines. In this case, a more sophisticated and reliable extrema matching algorithm is possible. They arrive at such an algorithm by treating the extrema matching problem as a multi-target tracking problem. The final result is a spline approximation algorithm whose robustness is shown using Cauchy-contaminated Gaussian noise