tvd (version 0.1.0)

tvd1d: Perform Total Variation Denoising on a 1-Dimensional Signal

Description

When supplied a noisy sequential signal in vector y, performs TVD with regularization parameter lambda, and returns a denoised version of y.

Usage

tvd1d(y, lambda, method = "Condat")

Arguments

y

a numeric vector of sequential noisy data values

lambda

the total variation penalty coefficient

method

a string indicating the algorithm to use for denoising. Currently only supports method "Condat"

Value

• a numeric vector of the same length as y, containing denoised data.

Details

1D TVD is a filtering technique for a sequential univariate signal that attempts to find a vector x_tvd that approximates a noisy vector y, as: $$x_{tvd} = argmin_{x}(E(x, y) + \lambda V(x))$$ where E(x, y) is a loss function measuring the error in approximating y with x, and V(x) is the total variation of x: $$V(x) = sum(|x_{i+1} - x_{i}|)$$

TVD is particularly well-suited to recovering piecewise constant signals. The degree of approximation is controlled by the parameter lambda: for lambda = 0, x_tvd = y, and as lambda increases, x_tvd contains increasingly fewer value transitions, until, for a high enough value, it is constant.

Currently only implements Condat's fast squared-error loss TVD algorithm (method "Condat"), which is restricted to vectors of length 2^32 - 1 and shorter.

References

Condat, L. (2013) A Direct Algorithm for 1-D Total Variation Denoising IEEE Signal Processing Letters 20(11): 1054-1057. doi:10.1109/LSP.2013.2278339

Examples

## Generate a stepped signal
x = rep(c(1, 2, 3, 4, 2, 4, 3, 2, 1), each = 100)

## Create a noisy version of the signal
y = x + rnorm(length(x), sd = 0.5)

## Denoise the signal by Condat's methodlines(x.denoised, col = "red", lwd = 1)
x.denoised = tvd1d(y, lambda = 10, method = "Condat")

## Plot the original signal, the noisy signal, and the denoised signal
plot(y, col = "black", pch = 19, cex = 0.3)
lines(x, col = "blue", lwd = 3)
lines(x.denoised, col = "red", lwd = 3)
legend("topleft", legend = c("Original", "Noisy", "Denoised"),
  col = c("blue", "black", "red"), lty = c("solid", "solid", "solid"),
  lwd = c(2, 0, 1), pch = c(NA, 19, NA), pt.cex = c(NA, 0.3, NA), inset = 0.05)