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 ddescription
ximjinput–output value difference; used in likelihood terms
mimjoutput–output value difference; used in regularization terms
xixjinput–input value difference; used in both likelihood and regularization terms
ijsequence difference; used in both likelihood and regularization terms
(b) kernel functiondescription
1global
I(|d|≤W)hard (local in either value or sequence)
I(|d|2/2≤W)
Embedded Imagesoft (semi-local in either value or sequence)
Embedded Image
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
(c) loss functioninfluence function (derivative of loss function) kernel×directioncomposition
L0(d)=|d|0simple
L1(d)=|d|1L1(d)=1×sgn(d)
L2(d)=|d|2/2L2(d)=1×d
Embedded ImageLW,1(d)=I(|d|≤W)×sgn(d)composite
Embedded ImageLW,2(d)=I(|d|2Wd
Embedded ImageEmbedded Imagecomposite
Embedded ImageEmbedded Image