Package 'tvR'

tvR : Total Variation Regularization

Description

tvR provides tools for denoising noisy signal and images via Total Variation Regularization. Reducing the total variation of the given signal is known to remove spurious detail while preserving essential structural details. For now, we provide tools for denoising only on 1-dimensional signals or 2-dimensional images, where the latter be represented as 2d or 3d array.

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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)

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Total Variation Denoising for Image

Description

Given an image 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

denoise2(
  data,
  lambda = 1,
  niter = 100,
  method = c("TVL1.PrimalDual", "TVL2.PrimalDual", "TVL2.FiniteDifference"),
  normalize = FALSE
)

Arguments

data	standard 2d or 3d array.
lambda	regularization parameter (positive real number).
niter	total number of iterations.
method	indicating problem and algorithm combination.
normalize	a logical; TRUE to make the range in $[0,1]$, or FALSE otherwise.

Value

denoised array as same size of data.

Data format

An input data can be either (1) 2-dimensional matrix representaing grayscale image, or (2) 3-dimensional array for color image.

Algorithms for TV-L1 problem

The cost function for TV-L2 problem is

$min_u |u-f|_1 + \lambda |\nabla u|$

where for a given 2-dimensional array, $|\nabla u| = \sum sqrt(u_x^2 + u_y^2)$ Algorithms (in conjunction with model type) for this problems are

"TVL1.PrimalDual"

Primal-Dual algorithm.

Algorithms for TV-L2 problem

The cost function for TV-L2 problem is

$min_u |u-f|_2^2 + \lambda |\nabla u|$

and algorithms (in conjunction with model type) for this problems are

"TVL2.PrimalDual"

Primal-Dual algorithm.

"TVL2.FiniteDifference"

Finite Difference scheme with fixed point iteration.

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.

Chambolle A, Pock T (2011). “A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging.” Journal of Mathematical Imaging and Vision, 40(1), 120–145. ISSN 0924-9907, 1573-7683.

Examples

## Not run: 
## Load grey-scale 'lena' data
data(lena128)

## Add white noise
sinfo   <- dim(lena128)   # get the size information
xnoised <- lena128 + array(rnorm(128*128, sd=10), sinfo)

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

## compare
gcol = gray(0:256/256)
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(lena128, main="original", col=gcol)
image(xnoised, main="noised", col=gcol)
image(xproc1, main="TVL2.FiniteDifference", col=gcol)
image(xproc2, main="TVL1.PrimalDual", col=gcol)
par(opar)

## End(Not run)

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lena image at size of $(128 \times 128)$

Description

Lena is probably one of the most well-known example in image processing and computer vision. Due to CRAN instability, history of this image can be found by googling the story of Lena.

Usage

data(lena128)

Format

matrix of size $(128\times 128)$

Source

USC SIPI Image Database.

References

Gonzalez, Rafael C. and Woods, Richard E. (2017) Digital Image Processing (4th ed.). ISBN 0133356728.

Examples

data(lena128)
image(lena128, col=gray((0:100)/100), axes=FALSE, main="lena128")

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