knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )

For a given noisy signal $f$, total variation regularization (also known as denoising) aims at recovering a *cleaned* version of signal $u$ by solving an equation of the following form
$$
\min_u ~ E(u,f) + \lambda V(u)
$$
where $E(u,f)$ is a fidelity term that measures closeness of noisy signal $f$ to a desired solution $u$, and $V(u)$ a penalty term in pursuit of smoothness of a solution. For a differentiable function $u : \Omega\rightarrow \mathbb{R}$, total variation is defined as
$$
V(u) = \int_{\Omega} \| \nabla u(x) \| dx
$$
and $\lambda$ a regularization parameter that balances fitness and smoothness defined by two terms.

Our **tvR** package provides two functions

`denoise1`

for 1d signal (usually with time domain), and`denoise2`

for 2d signal such as image.

Let's see two examples in the below.

```
library(tvR)
```

`denoise1`

We aim to solve TV-L2 problem, where $$ E(u,f) = \frac{1}{2} \int |u(x) - f(x)|^2 dx $$ with a penalty $V(u) = \sum_i |u_{i+1}-u_{i}|$ with two algorithms, including 1) iterative clippling algorithm and 2) majorization-minorization method.

As an example, let's create a stepped signal and add gaussian white noise with $\sigma = 0.25$

set.seed(1) x = rep(sample(1:5,10,replace=TRUE), each=50) ## main signal xnoised = x + rnorm(length(x), sd=0.25) ## add noise

First, let's compare how two algorithms perform with $\lambda=1.0$.

## 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="compare two algorithms", xlab="time domain", ylab="signal value") lines(xproc1, col="blue", lwd=2) lines(xproc2, col="red", lwd=2) legend("topright",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)

which shows somewhat seemingly inconsistent results. However, this should be understood as induced by their internal algorithmic details such as stopping criterion. In such sense, let's compare whether a single method is consistent with respect to the degree of regularization by varying parameters $\lambda=10^{-3}, 10^{-2}, 10^{-1}, 1$. For this comparison, we will use iterative clipping (`TVL2.IC`

) algorithm.

compare = list() for (i in 1:4){ compare[[i]] = denoise1(xnoised, lambda = 10^(i-4), method="TVL2.IC") }

par(mfrow=c(2,2)) for (i in 1:4){ pm = paste("lambda=1e",i-4,sep="") plot(xnoised, pch=19, cex=0.2, main=pm, xlab="time domain", ylab="signal value") lines(compare[[i]], col=as.integer(i+2), lwd=1.2) }

An observation can be made that the larger the $\lambda$ is, the smoother the fitted solution becomes.

`denoise2`

For a 2d signal case, we support both *TV-L1* and *TV-L2* problem, where
$$
E_{L_1}(u,f) = \int_\Omega |u(x) - f(x)|*1 dx \
E*{L_2}(u,f) = \int_\Omega |u(x) - f(x)|_2^2 dx
$$
given a 2-dimensional domain $\Omega \subset \mathbb{R}^2$ and a penalty $V(u) = \sum (u_x^2 + u_y^2)^{1/2}$. For *TV-L1* problem, we provide primal-dual algorithm, whereas *TV-L2* brings primal-dual algorithm as well as finite-difference scheme with fixed point iteration.

A typical yet major example of 2-dimensional signal is **image**, considering each pixel's value as $f(x,y)$ at location $(x,y)$. We'll use the gold standard image of Lena. In our example, we will use a version of gray-scale Lena image stored as a matrix of size $128 \times 128$ and add some gaussian noise as before with $\sigma=10$.

data(lena128) xnoised <- lena128 + array(rnorm(128*128, sd=10), c(128,128))

Let's see how different algorithms perform with $\lambda=10$.

## apply denoising process xproc1 <- denoise2(xnoised, lambda=10, method="TVL1.PrimalDual") xproc2 <- denoise2(xnoised, lambda=10, method="TVL2.FiniteDifference") xproc3 <- denoise2(xnoised, lambda=10, method="TVL2.PrimalDual")

par(mfrow=c(2,2),pty="s") gcol = gray(0:128/128) image(xnoised, main="Noised", col=gcol) image(xproc1, main="L1-PrimalDual", col=gcol) image(xproc2, main="L2-FiniteDifference", col=gcol) image(xproc3, main="L2-PrimalDual", col=gcol)

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