ℓ2 and ℓ1 Trend Filtering: A Kalman Filter Approach [Lecture Notes]

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.