User's manual for SLIM programs in Madagascar

sfthr: Threshold float/complex inputs given a constant/varying threshold level.

sfthr < in.rsf > out.rsf fthr=fthr.rsf thr= mode=

Methods available:
- soft
- hard
- non-negative Garrote (nng)

Written by: Gilles Hennenfent & Colin Russell, UBC
Created: February 2006

string	fthr=		varying threshold level (positive number) (auxiliary input file name)
string	mode=		'soft', 'hard', 'nng' (default: soft)
float	thr=		threshold level (positive number)

Consider the vector $\mathbf{x}:=\{x_i\}_{0\leq i <m}\in\mathbb{R}^m$ . Soft thresholding is defined as

$$\mathcal{S}_\gamma(\mathbf{x}):=\{ (x_i)\cdot \max(\vert x_i\vert-\gamma,0)\}_{0\leq i <m},$$ (1)

with $\gamma$ a positive threshold level. Hard thresholding is defined as

$$\mathcal{H}_\gamma(\mathbf{x}):=\{\max(\vert x_i\vert-\gamma,0)\cdot x_i\}_{0\leq i <m}.$$ (2)

Finally, non-negative Garrote (nng) thresholding is defined as

$$\mathcal{T}^{nng}_\gamma(\mathbf{x}):=\{\max(\vert x_i\vert-\gamma,0)\cdot (x_i-\gamma^2/x_i)\}_{0\leq i <m}.$$ (3)

data,soft,hard,nng
Figure 1. Thresholding example. Line whose range is symmetric about the origin (a) thresholded using soft (b), hard (c), and NNG (d) methods.

The extension to positive varying threshold level is straightforward by replacing $\gamma$ by $\gamma_i$ in Eq.'s (1),(2), and (3).

In Madagascar, to soft threshold a dataset with a constant, use e.g.

bash$ sfmath n1=100 n2=1 output='1' | sfnoise rep=y >data.rsf
bash$ sfthr <data.rsf thr=2 >res1.rsf

or replace the last command by

bash$ sfthr <data.rsf thr=2 method=soft >res2.rsf

This is also equivalent to soft thresholding data.rsf with a vector of same size mythr.rsf whose entries are all set to 2.

bash$ sfmath n1=100 n2=1 output='2' >mythr.rsf
bash$ sfthr <data.rsf fthr=mythr.rsf method=soft >res3.rsf

If thr=.5 and fthr=mythr.rsf are specified at the same time, the effective threshold level is 1, obtained by multiplying mythr.rsf entries by 0.5

bash$ sfthr <data.rsf thr=.5 fthr=mythr.rsf method=soft >res4.rsf

data1,soft1,hard1,nng1
Figure 2. Random vector thresholding example.

Note that thresholding is an element-wise operation. sfthr can thus deal with arbitrarily large datasets.

User's manual for SLIM programs in Madagascar