Abstract

$\ell_2$ and $\ell_1$ trend filtering are two of the most popular denoising algorithms that are widely used in science, engineering, and statistical signal and image processing applications. They are typically treated as separate entities, with the former as a linear time invariant (LTI) filter which is commonly used for smoothing the noisy data and detrending the time-series signals while the latter is a nonlinear filtering method suited for the estimation of piecewise-polynomial signals (\eg, piecewise-constant, piecewise-linear, piecewise-quadratic and \etc) observed in additive white Gaussian noise. In this article, we propose a Kalman filtering approach to design and implement $\ell_2$ and $\ell_1$ trend filtering % (QV and TV regularization) with the aim of teaching these two approaches and explaining their differences and similarities. Hopefully the framework presented in this article will provide a straightforward and unifying platform for understanding the basis of these two approaches. In addition, the material may be useful in lecture courses in signal and image processing, or indeed, it could be useful to introduce our colleagues in signal processing to the application of Kalman filtering in the design of $\ell_2$ and $\ell_1$ trend filtering.