Total variation denoising

A noisy signal \(y = x + n\) can be denoised, under the assumption that \(x\) is piecewise constant, by solving the problem \[ \hat x_\lambda = \mathrm{argmin}_x \|y - x\|_2^2 + \lambda \|Dx\|_{1} \] where \(Dx\) denotes the discrete derivative of \(x\).

The bottom graph demonstrates the taut string interpretation of TV denoising, where \(Y\) is the cumulative sum of \(y\). \(\hat x_\lambda\) is then found as the discrete derivative of \(\hat X_\lambda\).