Table 2.

A generalized functional for noise removal from piecewise constant (PWC) signals. The functional combines differences, losses and kernel functions described in table 1 into a function to be minimized over all samples, pairwise. Various solver algorithms are used to minimize this functional with respect to the solution; these are described in table 3.

generalized functional for piec ewise constant noise removal
$H\left[m\right]=\underset{i=1}{\overset{N}{\Sigma }}\underset{j=1}{\overset{N}{\Sigma }}\mathit{\Lambda }(x_i-m_j,m_i-m_j,x_i-x_j,i-j)$
existing methods	function Λ	notes
linear diffusion	(1/2)|mi − mj|2I(i − j = 1)	solved by weighted mean  filtering; cannot produce PWC  solutions; not PWC
step-fitting (Gill 1970;  Kerssemakers et al. 2006)	(1/2)|xi − mj |2I(i − j = 0)	termination criteria based on  number of jumps; PWC
objective step-fitting  (Kalafut & Visscher 2008)	(1/2)|xi − mj|2I(i − j = 0)  +λ|mi − mj|0I(i − j = 1)	likelihood term the same upto  log transformation;  regularization parameter λ  fixed by data; PWC
total variation regularization  (Rudin et al. 1992)	(1/2)|xi − mj|2I(i − j = 0)  + γ|mi − mj|I(i − j = 1)	convex; fused Lasso signal  approximator is the same;  PWC
total variation diffusion	|mi − mj|I(i − j = 1)	convex; partially minimized by  iterated 3-point median filter;  PWC
mean shift clustering	min((1/2)|mi − mj|2, W)	non-convex; PWC
likelihood mean shift  clustering	min((1/2)|xi − mj|2, W)	non-convex; K-means is similar  but not a direct special case  (see text); PWC
soft mean shift clustering	1 − exp(−β|mi − mj|2/2)/β	non-convex; PWC
soft likelihood mean shift  clustering	1 − exp(−β|xi − mj|2/2)/β	non-convex; soft-K-means is  similar but not a direct special  case (see text); PWC
convex clustering shrinkage  (Pelckmans et al. 2005)	(1/2)|xi − mj|2I(i − j = 0)  + γ|mi − mj|	convex; PWC
bilateral filter (Mrazek et al. 2006)	[1 − exp(−β|mi − mj|2/2)/β]  ×I(|i − j| ≤ W)	non-convex