‘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

x_{i}−m_{j}

input–output value difference; used in likelihood terms

m_{i}−m_{j}

output–output value difference; used in regularization terms

x_{i}−x_{j}

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)

soft (semi-local in either value or sequence)

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 function

influence function (derivative of loss function) kernel×direction