Table 1.

‘Components’ for PWC denoising methods. All the methods in this paper can be constructed using all pairwise differences between input samples, output samples and sequence indices. These differences are then used to define kernel and loss functions. Loss functions and kernels are combined to make the generalized functional to be minimized with respect to the output signal m. Function I(S) is the indicator function such that I(S) = 1 if the condition S is true, and I(S) = 0 otherwise.

(a) difference d	description
xi − mj	input–output value difference; used in likelihood terms
mi − mj	output–output value difference; used in regularization terms
xi − xj	input–input value difference; used in both likelihood and  regularization terms
i − j	sequence difference; used in both likelihood and regularization  terms

(b) kernel function	description
1	global
I(|d|≤W)	hard (local in either value or sequence)
I(|d|2/2≤W)
exp(−β|d|)	soft (semi-local in either value or sequence)
exp(−β|d|2/2)
I(d = 1)	isolates only sequentially adjacent terms when used as sequence  kernel
I(d = 0)	isolates only terms that have the same index when used as  sequence kernel
	influence function (derivative of loss function)

(c) loss function	kernel × direction	composition
L0(d) = |d|0		simple
L1(d) = |d|1	$L_1^{\prime }\left(d\right)=1\times \mathrm{sgn}\left(d\right)$
L2(d) = |d|2/2	$L_2^{\prime }\left(d\right)=1\times d$
LW,1(d) = min(|d|, W)	$L_{W,1}^{\prime }\left(d\right)=I(\mid d\mid \le W)\times \mathrm{sgn}\left(d\right)$	composite
LW,2(d) = min(|d|2/2, W)	$L_{W,2}^{\prime }\left(d\right)=I({\mid d\mid }^2∕2\le W)\times d$
Lβ,1(d) = 1 − exp(−β|d|)/β	$L_{\beta ,1}^{\prime }\left(d\right)=\exp (-\beta \mid d\mid )\times \mathrm{sgn}\left(d\right)$	composite
Lβ,2(d) = 1 − exp(−β|d|2/2)/β	$L_{\beta ,2}^{\prime }\left(d\right)=\exp (-\beta {\mid d\mid }^2∕2)\times d$