denoise1 {tvR}

Total Variation Denoising for Signal

Description

Given a 1-dimensional signal f, it solves an optimization of the form,

u^* = argmin_u E(u,f)+\lambda V(u)

where E(u,f) is fidelity term and V(u) is total variation regularization term. The naming convention of a parameter method is <problem type> + <name of algorithm>. For more details, see the section below.

Usage

denoise1(signal, lambda = 1, niter = 100, method = c("TVL2.IC", "TVL2.MM"))

Arguments

signal

vector of noisy signal.

lambda

regularization parameter (positive real number).

niter

total number of iterations.

method

indicating problem and algorithm combination.

Value

a vector of same length as input signal.

Algorithms for TV-L2 problem

The cost function for TV-L2 problem is

min_u \frac{1}{2} |u-f|_2^2 + \lambda |\nabla u|

where for a given 1-dimensional vector, |\nabla u| = \sum |u_{i+1}-u_{i}|. Algorithms (in conjunction with model type) for this problems are

"TVL2.IC"

Iterative Clipping algorithm.

"TVL2.MM"

Majorization-Minorization algorithm.

The codes are translated from MATLAB scripts by Ivan Selesnick.

References

Rudin LI, Osher S, Fatemi E (1992). “Nonlinear total variation based noise removal algorithms.” Physica D: Nonlinear Phenomena, 60(1-4), 259–268. ISSN 01672789.

Selesnick IW, Parekh A, Bayram I (2015). “Convex 1-D Total Variation Denoising with Non-convex Regularization.” IEEE Signal Processing Letters, 22(2), 141–144. ISSN 1070-9908, 1558-2361.

Examples

## generate a stepped signal
x = rep(sample(1:5,10,replace=TRUE), each=50)

## add some additive white noise
xnoised = x + rnorm(length(x), sd=0.25)

## apply denoising process
xproc1 = denoise1(xnoised, method = "TVL2.IC")
xproc2 = denoise1(xnoised, method = "TVL2.MM")

## plot noisy and denoised signals
plot(xnoised, pch=19, cex=0.1, main="Noisy signal")
lines(xproc1, col="blue", lwd=2)
lines(xproc2, col="red", lwd=2)
legend("bottomleft",legend=c("Noisy","TVL2.IC","TVL2.MM"),
col=c("black","blue","red"),#' lty = c("solid", "solid", "solid"),
lwd = c(0, 2, 2), pch = c(19, NA, NA),
pt.cex = c(1, NA, NA), inset = 0.05)