l(2) and l(1) Trend Filtering: A Kalman Filter Approach

Two of the most popular denoising algorithms are l(2) and l(1) trend filtering, which are used in science, engineering, and statistical signal and image processing. 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 (e.g., piecewise constant, piecewise linear, piecewise quadratic, and so on) observed in additive white Gaussian noise.