Package 'rwavelet'

Title: Wavelet Analysis
Description: Perform wavelet analysis (orthogonal,translation invariant, tensorial, 1-2-3d transforms, thresholding, block thresholding, linear,...) with applications to data compression, denoising/regression or clustering. The core of the code is a port of 'MATLAB' Wavelab toolbox written by D. Donoho, A. Maleki and M. Shahram (<https://statweb.stanford.edu/~wavelab/>).
Authors: F. Navarro and C. Chesneau
Maintainer: Navarro Fabien <[email protected]>
License: LGPL (>= 2)
Version: 0.4.1.99999
Built: 2025-01-16 04:38:11 UTC
Source: https://github.com/fabnavarro/rwavelet

Help Index

Convolution tool for two-scale transform

Description

Filtering by periodic convolution of x with the time-reverse of f.

Usage

aconv(f, x)

Arguments

f

filter.
x

1-d signal.

Value

y filtered result.

See Also

iconvv, UpDyadHi, UpDyadLo, DownDyadHi, DownDyadLo.

Examples

qmf <- MakeONFilter('Haar')
x <- MakeSignal('HeaviSine',2^3)
aconv(qmf,x)

Construct 1d block partition

Description

This function is used to group the coefficients into blocks (or groups) of size L.

Usage

block_partition(x, L)

Arguments

x

(noisy) wc at a given scale.
L

block size.

Value

out partition of coefficients by block.

See Also

invblock_partition, BlockThresh.

Examples

x <- MakeSignal('Ramp', 8)
j0 <- 0
qmf <- MakeONFilter('Haar')
wc <- FWT_PO(x, j0, qmf)
L <- 2
wcb <- block_partition(wc, L)

Construct 2d block partition

Description

Group the coefficients into blocks (or groups) of size L.

Usage

block_partition2d(x, L)

Arguments

x

(noisy) wc at a given scale.
L

block size.

Value

out partition of coefficients by block.

See Also

invblock_partition2d

Examples

x <- matrix(rnorm(2^2), ncol=2)
j0 <- 0
qmf <- MakeONFilter('Haar')
wc <- FWT2_PO(x, j0, qmf)
L <- 2
wcb <- block_partition2d(wc, L)

1d wavelet Block Thresholding

Description

This function is used for thresholding coefficients by group (or block) according to the hard or soft thresholding rule.

Usage

BlockThresh(wc, j0, hatsigma, L, qmf, thresh = "hard")

Arguments

wc

wavelet coefficients.
j0

coarsest decomposition scale.
hatsigma

estimator of noise variance.
L

Block size (n mod L must be 0).
qmf

Orthonormal quadrature mirror filter.
thresh

'hard' or 'soft'.

Value

wcb wavelet coefficient estimators.

See Also

invblock_partition, invblock_partition.

Examples

n <- 64
x <- MakeSignal('Ramp', n)
sig <- 0.01
y <- x + rnorm(n, sd=sig)
j0 <- 1
qmf <- MakeONFilter('Daubechies',8)
wc <- FWT_PO(y, j0, qmf)
L <- 2
wcb <- BlockThresh(wc, j0, sig, L, qmf, "hard")

2d Wavelet Block thresholding

Description

This function is used to threshold the coefficients by group (or block).

Usage

BlockThresh2d(
  wc,
  j0,
  hatsigma,
  L = 4,
  qmf,
  lamb = 4.50524,
  thresh = "JamesStein"
)

Arguments

wc

wavelet coefficients.
j0

coarsest decomposition scale.
hatsigma

estimator of noise variance.
L

Block size (n mod L must be 0).
qmf

Orthonormal quadrature mirror filter.
lamb

Threshold parameter.
thresh

Thresholding rule: 'hard', or 'JamesStein'.

Value

wcb Thresholded wavelet coefficients.

cameraman Image

Description

A dataset containing cameraman image.

Usage

data(cameraman)

Format

A numeric 512 x 512 matrix.

Circular Shifting of a matrix/image

Description

Pixels that get shifted off one side of the image are put back on the other side.

Usage

CircularShift(matrix, colshift = 0, rowshift = 0)

Arguments

matrix

2-d signal (matrix).
colshift

column shift index (integer).
rowshift

row shift index (integer).

Value

result 2-d shifted signal.

See Also

FWT2_TI, IWT2_TI.

Examples

A <- matrix(1:4, ncol=2, byrow=TRUE)
CircularShift(A, 0, -1)

Find length and dyadic length of square array

Description

3d counterpart of Donoho's quadlength utilized by the 2d pair. Original matlab code Vicki Yang and Brani Vidakovic.

Usage

cubelength(x)

Arguments

x

3-d array; dim(n,n,n), n = 2^J (hopefully).

Value

n length(x).

J least power of two greater than n.

See Also

FWT3_PO, IWT3_PO.

Examples

cubelength(array(1:3, c(2,2,2)))

2-Fold Cross Validation for linear estimator

Description

Selection of the number of wavelet coefficients to be maintained by the cross validation method proposed by Nason in the case of threshold selection. This method is adapted here to select among linear estimators.

Usage

CVlinear(Y, L, qmf, D, wc)

Arguments

Y

Noisy observations.
L

Level of coarsest scale.
qmf

Orthonormal quadrature mirror filter.
D

Dimension vector of the models considered.
wc

1-d wavelet coefficients.

Value

CritCV Cross validation criteria.

hat_f_m_2FCV

References

Nason, G. P. (1996). Wavelet shrinkage using cross-validation. Journal of the Royal Statistical Society: Series B, 58(2), 463–479.

Navarro, F. and Saumard, A. (2017). Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases. ESAIM: Probability and Statistics, 21, 412–451.

Hi-Pass Downsampling operator (periodized)

Description

Hi-Pass Downsampling operator (periodized)

Usage

DownDyadHi(x, qmf)

Arguments

x

1-d signal at fine scale.
qmf

filter.

Value

y 1-d signal at coarse scale.

See Also

DownDyadLo, UpDyadHi, UpDyadLo, FWT_PO, iconvv.

Examples

qmf <- MakeONFilter('Haar')
x <- MakeSignal('HeaviSine',2^3)
DownDyadHi(x, qmf)

Lo-Pass Downsampling operator (periodized)

Description

Lo-Pass Downsampling operator (periodized)

Usage

DownDyadLo(x, qmf)

Arguments

x

1-d signal at fine scale.
qmf

filter.

Value

d 1-d signal at coarse scale.

See Also

DownDyadHi, UpDyadHi, UpDyadLo, FWT_PO, aconv.

Examples

qmf <- MakeONFilter('Haar')
x <- MakeSignal('HeaviSine',2^3)
DownDyadLo(x,qmf)

Index entire j-th dyad of 1-d wavelet xform

Description

Index entire j-th dyad of 1-d wavelet xform

Usage

dyad(j)

Arguments

j

integer.

Value

ix list of all indices of wavelet coeffts at j-th level.

Examples

dyad(0)

Find length and dyadic length of array

Description

Find length and dyadic length of array

Usage

dyadlength(x)

Arguments

x

array of length n = 2^J (hopefully).

Value

n length(x).

J least power of two greater than n.

See Also

quadlength, dyad

Examples

x <- MakeSignal('Ramp', 8)
dyadlength(x)

2-d tensor wavelet transform (periodized, orthogonal).

Description

A two-dimensional Wavelet Transform is computed for the array x. qmf filter may be obtained from MakeONFilter. To reconstruct, use ITWT2_PO.

Usage

FTWT2_PO(x, L, qmf)

Arguments

x

2-d image (n by n array, n dyadic).
L

coarse level.
qmf

quadrature mirror filter.

Value

wc 2-d wavelet transform.

See Also

ITWT2_PO, MakeONFilter.

Examples

qmf <- MakeONFilter('Daubechies', 10)
L <- 0
x <- matrix(rnorm(2^2), ncol=2)
wc <- FTWT2_PO(x, L, qmf)

Forward Wavelet Transform (periodized, orthogonal)

Description

1. qmf filter may be obtained from MakeONFilter. 2. usually, length(qmf) < 2^(L+1). 3. To reconstruct use IWT_PO.

Usage

FWT_PO(x, L, qmf)

Arguments

x

1-d signal; length(x) = 2^J.
L

Coarsest Level of V_0; L << J.
qmf

quadrature mirror filter (orthonormal).

Value

wc 1-d wavelet transform of x.

See Also

IWT_PO, MakeONFilter.

Examples

x <- MakeSignal('Ramp', 8)
L <- 0
qmf <- MakeONFilter('Haar')
wc <- FWT_PO(x, L, qmf)

Translation Invariant Forward Wavelet Transform

Description

1. qmf filter may be obtained from MakeONFilter. 2. usually, length(qmf) < 2^(L+1). 3. To reconstruct use IWT_TI.

Usage

FWT_TI(x, L, qmf)

Arguments

x

array of dyadic length n=2^J.
L

degree of coarsest scale.
qmf

orthonormal quadrature mirror filter.

Value

TIWT stationary wavelet transform table.

See Also

IWT_TI, MakeONFilter.

Examples

x <- MakeSignal('Ramp', 8)
L <- 0
qmf <- MakeONFilter('Haar')
TIWT <- FWT_TI(x, L, qmf)

2-d MRA Forwad Wavelet Transform (periodized, orthogonal)

Description

A two-dimensional wavelet transform is computed for the array x. qmf filter may be obtained from MakeONFilter. To reconstruct, use IWT2_PO.

Usage

FWT2_PO(x, L, qmf)

Arguments

x

2-d image (n by n array, n dyadic).
L

coarse level.
qmf

quadrature mirror filter.

Value

wc 2-d wavelet transform.

See Also

IWT2_PO, MakeONFilter.

Examples

qmf <- MakeONFilter('Daubechies', 10)
L <- 3
x <- matrix(rnorm(128^2),ncol=128)
wc <- FWT2_PO(x, L, qmf)

2-d Translation Invariant Forward Wavelet Transform

Description

1. qmf filter may be obtained from MakeONFilter. 2. usually, length(qmf) < 2^(L+1). 3. To reconstruct use IWT_TI.

Usage

FWT2_TI(x, L, qmf)

Arguments

x

2-d image (n by n real array, n dyadic).
L

degree of coarsest scale.
qmf

orthonormal quadrature mirror filter.

Value

TIWT translation-invariant wavelet transform table, (3(J-L)+1)n by n.

Examples

x <- matrix(rnorm(2^2), ncol=2)
L <- 0
qmf <- MakeONFilter('Haar')
TIWT <- FWT2_TI(x, L, qmf)

3-d MRA Forward Wavelet Transform (periodized, orthogonal)

Description

A three-dimensional wavelet transform is computed for the array x. qmf filter may be obtained from MakeONFilter. To reconstruct, use IWT3_PO.

Usage

FWT3_PO(x, L, qmf)

Arguments

x

3-d array (n by n by n array, n dyadic).
L

coarse level.
qmf

quadrature mirror filter.

Details

3-D counterpart of Donoho's FWT2_PO, original matlab code Vicki Yang and Brani Vidakovic.

Value

wc 3-d wavelet transform.

See Also

IWT3_PO, MakeONFilter.

Examples

qmf <- MakeONFilter('Daubechies', 10)
L <- 3
x <- array(rnorm(32^3), c(32,32,32))
wc <- FWT3_PO(x, L, qmf)

Generation of Gaussian White Noise

Description

Generation of Gaussian White Noise

Usage

GWN(n, sigma)

Arguments

n

sample size.
sigma

standard deviation.

Value

epsilon resulting noise.

Examples

GWN(10, 0.1)

Apply Hard Threshold

Description

Apply Hard Threshold

Usage

HardThresh(y, t)

Arguments

y

Noisy Data.
t

Threshold.

Value

x filtered result (y 1_|y|>t).

See Also

SoftThresh.

Examples

f <- MakeSignal('HeaviSine',2^3)
qmf <- MakeONFilter('Daubechies', 10)
L <- 0
wc <- FWT_PO(f, L, qmf)
thr <- 2
wct <- HardThresh(wc, thr)
fhard <- IWT_PO(wct, L, qmf)

Convolution tool for two-scale transform

Description

Filtering by periodic convolution of x with f.

Usage

iconvv(f, x)

Arguments

f

filter.
x

1-d signal.

Value

y filtered result.

See Also

aconv, UpDyadHi, UpDyadLo, DownDyadHi, DownDyadLo.

Examples

qmf <- MakeONFilter('Haar')
x <- MakeSignal('HeaviSine',2^3)
iconvv(qmf,x)

Inversion of the 1d block partition

Description

Inversion of the 1d block partition

Usage

invblock_partition(x, n, L)

Arguments

x

partition of coefficients by block.
n

scale.
L

block size.

See Also

block_partition, BlockThresh.

Examples

n <- 8
x <- MakeSignal('Ramp', n)
j0 <- 1
qmf <- MakeONFilter('Haar')
wc <- FWT_PO(x, j0, qmf)
L <- 2
wcb <- block_partition(wc, L)
wcib <- invblock_partition(wcb, n, L)

Inversion of the 2d block partition

Description

Inversion of the 2d block partition

Usage

invblock_partition2d(x, n, L)

Arguments

x

partition of coefficients by block.
n

scale.
L

block size.

Value

out coefficients.

See Also

block_partition2d

Examples

n <- 2
x <- matrix(rnorm(n^2), ncol=2)
j0 <- 0
qmf <- MakeONFilter('Haar')
wc <- FWT2_PO(x, j0, qmf)
L <- 2
wcb <- block_partition2d(wc, L)
wcib <- invblock_partition2d(wcb, n, L)

Inverse 2-d Tensor Wavelet Transform (periodized, orthogonal)

Description

If wc is the result of a forward 2d wavelet transform, with wc <- FTWT2_PO(x,L,qmf), then x <- ITWT2_PO(wc,L,qmf) reconstructs x exactly. qmf is a nice qmf, e.g. one made by MakeONFilter.

Usage

ITWT2_PO(wc, L, qmf)

Arguments

wc

2-d wavelet transform (n by n array, n dyadic).
L

coarse level.
qmf

quadrature mirror filter.

Value

x 2-d signal reconstructed from wc.

See Also

FTWT2_PO, MakeONFilter.

Examples

qmf <- MakeONFilter('Daubechies', 10)
L <- 0
x <- matrix(rnorm(2^2), ncol=2)
wc <- FTWT2_PO(x, L, qmf)
xr <- ITWT2_PO(wc,L,qmf)

Inverse Wavelet Transform (periodized, orthogonal)

Description

Suppose wc <- FWT_PO(x,L,qmf) where qmf is an orthonormal quad. mirror filter, e.g. one made by MakeONFilter. Then x can be reconstructed by x <- IWT_PO(wc,L,qmf).

Usage

IWT_PO(wc, L, qmf)

Arguments

wc

1-d wavelet transform: length(wc) = 2^J.
L

Coarsest scale (2^(-L) = scale of V_0); L << J.
qmf

quadrature mirror filter (orthonormal).

Value

x 1-d signal reconstructed from wc.

See Also

FWT_PO, MakeONFilter.

Examples

x <- MakeSignal('Ramp', 8)
L <- 0
qmf <- MakeONFilter('Haar')
wc <- FWT_PO(x, L, qmf)
xr <- IWT_PO(wc,L,qmf)

Invert Translation Invariant Wavelet Transform

Description

Invert Translation Invariant Wavelet Transform

Usage

IWT_TI(pkt, qmf)

Arguments

pkt

translation-invariant wavelet transform table (TIWT).
qmf

orthonormal quadrature mirror filter.

Value

x 1-d signal reconstructed from translation-invariant transform TIWT.

See Also

FWT_TI, MakeONFilter.

Examples

x <- MakeSignal('Ramp', 8)
L <- 0
qmf <- MakeONFilter('Haar')
TIWT <- FWT_TI(x, L, qmf)
xr <- IWT_TI(TIWT,qmf)

Inverse 2-d MRA Wavelet Transform (periodized, orthogonal)

Description

If wc is the result of a forward 2d wavelet transform, with wc <- FWT2_PO(x,L,qmf). then x <- IWT2_PO(wc,L,qmf) reconstructs x exactly qmf is a nice qmf, e.g. one made by MakeONFilter.

Usage

IWT2_PO(wc, L, qmf)

Arguments

wc

2-d wavelet transform (n by n array, n dyadic).
L

coarse level.
qmf

quadrature mirror filter.

Value

x 2-d signal reconstructed from wc.

See Also

FWT2_PO, MakeONFilter.

Examples

qmf <- MakeONFilter('Daubechies', 10)
L <- 3
x <- matrix(rnorm(128^2),ncol=128)
wc <- FWT2_PO(x, L, qmf)
xr <- IWT2_PO(wc,L,qmf)

Invert 2-d Translation Invariant Wavelet Transform

Description

Invert 2-d Translation Invariant Wavelet Transform

Usage

IWT2_TI(tiwt, L, qmf)

Arguments

tiwt

translation-invariant wavelet transform table, (3(J-L)+1)n by n.
L

degree of coarsest scale.
qmf

orthonormal quadrature mirror filter.

Value

x 2-d image reconstructed from translation-invariant transform TIWT.

Examples

x <- matrix(rnorm(2^2), ncol=2)
L <- 0
qmf <- MakeONFilter('Haar')
TIWT <- FWT2_TI(x, L, qmf)
xr <- IWT2_TI(TIWT,L,qmf)

Inverse 3-d MRA Wavelet Transform (periodized, orthogonal)

Description

If wc is the result of a forward 3-d wavelet transform, with wc <- FWT3_PO(x, L, qmf). then x <- IWT3_PO(wc, L, qmf) reconstructs x exactly qmf is a nice qmf, e.g. one made by MakeONFilter.

Usage

IWT3_PO(wc, L, qmf)

Arguments

wc

3-d wavelet transform (n by n by n array, n dyadic).
L

coarse level.
qmf

quadrature mirror filter.

Details

3-d counterpart of Donoho's IWT2_PO, original matlab code by Vicki Yang and Brani Vidakovic.

Value

x 3-d signal reconstructed from wc.

See Also

FWT3_PO, MakeONFilter.

Examples

qmf <- MakeONFilter('Daubechies', 10)
L <- 3
x <- array(rnorm(32^3), c(32, 32, 32))
wc <- FWT3_PO(x, L, qmf)
xr <- IWT3_PO(wc, L, qmf)

Apply James-Stein Threshold

Description

(also called the nonnegative garrote)

Usage

JSThresh(y, t)

Arguments

y

Noisy Data.
t

Threshold.

Value

x filtered result.

See Also

HardThresh, SoftThresh

Examples

f <- MakeSignal('HeaviSine', 2^3)
qmf <- MakeONFilter('Daubechies', 10)
L <- 0
wc <- FWT_PO(f, L, qmf)
thr <- 2
wct <- JSThresh(wc, thr)
fsoft <- IWT_PO(wct, L, qmf)

Lena Image

Description

A dataset containing Lena image.

Usage

data(lena)

Format

A numeric 512 x 512 matrix.

Circular left shift of 1-d signal

Description

Circular left shift of 1-d signal

Usage

lshift(a)

Arguments

a

1-d signal.

Value

l 1-d signal l(i) = x(i+1) except l(n) = x(1).

Examples

x <- MakeSignal('HeaviSine',2^3)
lshift(x)

Median Absolute Deviation

Description

Compute the median absolute deviation.

Usage

MAD(x)

Arguments

x

1-d signal.

Examples

x <- c(1, 1, 2, 2, 4, 6, 9)
MAD(x)

Generate Orthonormal QMF Filter for Wavelet Transform

Description

The Haar filter (which could be considered a Daubechies-2) was the first wavelet, though not called as such, and is discontinuous.

Usage

MakeONFilter(Type, Par)

Arguments

Type

string, 'Haar', 'Beylkin', 'Coiflet', 'Daubechies' 'Symmlet', 'Vaidyanathan','Battle'.
Par

integer, it is a parameter related to the support and vanishing moments of the wavelets, explained below for each wavelet.

Details

The Beylkin filter places roots for the frequency response function close to the Nyquist frequency on the real axis.

The Coiflet filters are designed to give both the mother and father wavelets 2*Par vanishing moments; here Par may be one of 1,2,3,4 or 5.

The Daubechies filters are minimal phase filters that generate wavelets which have a minimal support for a given number of vanishing moments. They are indexed by their length, Par, which may be one of 4,6,8,10,12,14,16,18 or 20. The number of vanishing moments is par/2.

Symmlets are also wavelets within a minimum size support for a given number of vanishing moments, but they are as symmetrical as possible, as opposed to the Daubechies filters which are highly asymmetrical. They are indexed by Par, which specifies the number of vanishing moments and is equal to half the size of the support. It ranges from 4 to 10.

The Vaidyanathan filter gives an exact reconstruction, but does not satisfy any moment condition. The filter has been optimized for speech coding.

The Battle-Lemarie filter generate spline orthogonal wavelet basis. The parameter Par gives the degree of the spline. The number of vanishing moments is Par+1.

Value

qmf quadrature mirror filter.

See Also

FWT_PO, IWT_PO, FWT2_PO, IWT2_PO.

Examples

Type <- 'Coiflet'
Par <- 1
qmf <- MakeONFilter(Type, Par)

Make artificial signal

Description

Make artificial signal

Usage

MakeSignal(name, n)

Arguments

name

string, 'HeaviSine', 'Bumps', 'Blocks', 'Doppler', 'Ramp','Cusp', 'Sing', 'HiSine', 'LoSine', 'LinChirp', 'TwoChirp', 'QuadChirp', 'MishMash', 'WernerSorrows' (Heisenberg), 'Leopold' (Kronecker), 'Riemann', 'HypChirps', 'LinChirps', 'Chirps', 'Gabor', 'sineoneoverx', 'Cusp2', 'SmoothCusp', 'Piece-Regular' (Piece-Wise Smooth), 'Piece-Polynomial' (Piece-Wise 3rd degree polynomial).
n

desired signal length.

Value

sig 1-d signal.

See Also

FWT_PO, IWT_PO, FWT2_PO, IWT2_PO.

Examples

name <- 'Cusp'
n <- 2^5
sig <- MakeSignal(name,n)

Make artificial 1-d signal

Description

Make artificial 1-d signal

Usage

MakeSignalNewb(name, n)

Arguments

name

string, 'Cusp','Step','Wave','Blip','Blocks', 'Bumps','HeaviSine','Doppler','Angles', 'Parabolas','Time Shifted Sine','Spikes','Corner'
n

desired signal length.

Value

sig 1-d signal.

See Also

FWT_PO, IWT_PO, FWT2_PO, IWT2_PO.

Examples

name <- 'Cusp'
n <- 2^5
sig <- MakeSignalNewb(name,n)

Minimax Thresholding

Description

Minimax Thresholding

Usage

MinMaxThresh(y)

Arguments

y

signal upon which to perform thresholding.

Value

x result.

References

D.L. Donoho and I.M. Johnstone (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3), 425–455.

Apply (-1)^t modulation

Description

h(t) = (-1)^(t-1) * x(t), 1 <= t <= length(x)

Usage

MirrorFilt(x)

Arguments

x

1-d signal.

Value

h 1-d signal with DC frequency content shifted to Nyquist frequency

See Also

DownDyadHi.

Examples

x <- MakeSignal('HeaviSine',2^3)
h <- MirrorFilt(x)

Apply Shrinkage with level-dependent Noise level estimation

Description

Apply Shrinkage with level-dependent Noise level estimation

Usage

MultiMAD(wc, L)

Arguments

wc

Wavelet Transform of noisy sequence.
L

low-resolution cutoff for Wavelet Transform.

Value

ws result of applying VisuThresh to each wavelet level, after scaling so MAD of coefficienst at each level = .6745

Apply Shrinkage to Wavelet Coefficients

Description

SURE referes to Stein's Unbiased Risk Estimate.

Usage

MultiSURE(wc, L)

Arguments

wc

Wavelet Transform of noisy sequence with N(0,1) noise.
L

low-frequency cutoff for Wavelet Transform.

Value

ws result of applying SUREThresh to each dyadic block.

Apply Universal Thresholding to Wavelet Coefficients

Description

Apply Universal Thresholding to Wavelet Coefficients

Usage

MultiVisu(wc, L)

Arguments

wc

Wavelet Transform of noisy sequence with N(0,1) noise.
L

low-frequency cutoff for Wavelet Transform

Value

x result of applying VisuThresh to each High Frequency Dyadic Block.

Packet table indexing

Description

Packet table indexing

Usage

packet(d, b, n)

Arguments

d

depth of splitting in packet decomposition.
b

block index among 2^d possibilities at depth d.
n

length of signal.

Value

p linear indices of all coeff's in that block.

Examples

packet(1, 1, 8)

Plot 1-d signal as baseline with series of spikes

Description

Plot 1-d signal as baseline with series of spikes

Usage

PlotSpikes(base, t, x, L, J)

Arguments

base

number, baseline level.
t

ordinate values.
x

1-d signal, specifies spike deflections from baseline.
L

level of coarsest scale.
J

least power of two greater than n.

Value

A plot of spikes on a baseline.

See Also

PlotWaveCoeff.

Examples

## Not run: 
PlotSpikes(base, t, x, L, J)

## End(Not run)

Spike-plot display of wavelet coefficients

Description

Spike-plot display of wavelet coefficients

Usage

PlotWaveCoeff(wc, L, scal)

Arguments

wc

1-d wavelet transform.
L

level of coarsest scale.
scal

scale factor (0 ==> autoscale).

Value

A display of wavelet coefficients (coarsest level NOT included) by level and position.

See Also

FWT_PO, IWT_PO, PlotSpikes.

Examples

x <- MakeSignal('Ramp', 128)
qmf <- MakeONFilter('Daubechies', 10)
L <- 3
scal <- 1
wc <- FWT_PO(x, L, qmf)
PlotWaveCoeff(wc,L,scal)

Find length and dyadic length of square matrix

Description

h(t) = (-1)^(t-1) * x(t), 1 <= t <= length(x)

Usage

quadlength(x)

Arguments

x

2-d image; dim(n,n), n = 2^J (hopefully).

Value

n length(x).

J least power of two greater than n.

Examples

quadlength(matrix(1:16,ncol=4))

Nuclear magnetic resonance (NMR) signal

Description

A dataset containing a NMR signal.

Usage

data(RaphNMR)

Format

A numeric vector of length 1024.

Source

MRS Unit, VA Medical Center, San Francisco. Adrain Maudsley, Ph.D., Professor of Radiology. This NMR signal was obtained from Chris Raphael, then a postdoctoral fellow in the Department of Statistics at Stanford University who was working on Hidden Markov Models for restoring NMR Spectra.

Replicate and tile an array

Description

Repeat copies of array (equivalent of the repmat matlab function).

Usage

repmat(a, n, m)

Arguments

a

input array (scalar, vector, matrix).
n

number of time to repeat input array in row and column dimensions.
m

repetition factor.

Examples

repmat(10,3,2)

Circular right shift of 1-d signal

Description

Circular right shift of 1-d signal

Usage

rshift(a)

Arguments

a

1-d signal.

Value

r 1-d signal r(i) = x(i-1) except r(1) = x(n).

Examples

x <- MakeSignal('HeaviSine', 2^3)
rshift(x)

Make signal a row vector

Description

Make signal a row vector

Usage

ShapeAsRow(sig)

Arguments

sig

a row or column vector.

Value

row a row vector.

Examples

sig <- matrix(1:4)
row <- ShapeAsRow(sig)

3-d Shepp-Logan phantom

Description

A dataset containing a 3d head phantom that can be used to test 3-d reconstruction algorithms. Shepp-Logan phantom is well-known imitation of human cerebral.

Usage

data(SLphantom)

Format

A numeric array of size 64x64x64.

Signal/Noise ratio

Description

Signal/Noise ratio

Usage

SNR(x, y)

Arguments

x

Original reference signal.
y

Restored or noisy signal.

Value

Signal/Noise ratio.

Examples

n <- 2^4
x <- MakeSignal('HeaviSine', n)
y <- x + rnorm(n, mean=0, sd=1)
SNR(x, y)

Apply Soft Threshold

Description

Apply Soft Threshold

Usage

SoftThresh(y, t)

Arguments

y

Noisy Data.
t

Threshold.

Value

x filtered result (y 1_|y|>t).

See Also

HardThresh

Examples

f <- MakeSignal('HeaviSine', 2^3)
qmf <- MakeONFilter('Daubechies', 10)
L <- 0
wc <- FWT_PO(f, L, qmf)
thr <- 2
wct <- SoftThresh(wc, thr)
fsoft <- IWT_PO(wct, L, qmf)

Adaptive Threshold Selection Using Principle of SURE

Description

SURE refers to Stein's Unbiased Risk Estimate.

Usage

SUREThresh(y)

Arguments

y

Noisy Data with Std. Deviation = 1.

Value

x Estimate of mean vector

thresh Threshold used.

Hi-Pass Upsampling operator; periodized

Description

Hi-Pass Upsampling operator; periodized

Usage

UpDyadHi(x, qmf)

Arguments

x

1-d signal at coarser scale.
qmf

filter.

Value

u 1-d signal at finer scale.

See Also

DownDyadLo, DownDyadHi, UpDyadLo, IWT_PO, aconv.

Examples

qmf <- MakeONFilter('Haar')
x <- MakeSignal('HeaviSine',2^3)
UpDyadHi(x,qmf)

Lo-Pass Upsampling operator; periodized

Description

Lo-Pass Upsampling operator; periodized

Usage

UpDyadLo(x, qmf)

Arguments

x

1-d signal at coarser scale.
qmf

filter.

Value

y 1-d signal at finer scale.

See Also

DownDyadLo, DownDyadHi, UpDyadHi, IWT_PO, iconvv.

Examples

qmf <- MakeONFilter('Haar')
x <- MakeSignal('HeaviSine',2^3)
UpDyadLo(x,qmf)

Upsampling operator

Description

Upsampling operator

Usage

UpSampleN(x, s)

Arguments

x

1-d signal, of length n.
s

upsampling scale, default = 2.

Value

y 1-d signal, of length s*n with zeros interpolating alternate samples y(s*i-1) = x(i), i=1,...,n

Adaptive Threshold Selection Using Principle of SURE

Description

SURE referes to Stein's Unbiased Risk Estimate.

Usage

ValSUREThresh(x)

Arguments

x

Noisy Data with Std. Deviation = 1.

Value

thresh Value of Threshold.

Visually calibrated Adaptive Smoothing

Description

Visually calibrated Adaptive Smoothing

Usage

VisuThresh(y, thresh = "soft")

Arguments

y

Signal upon which to perform visually calibrated Adaptive Smoothing.
thresh

'hard' or 'soft'.

Value

x result of applying VisuThresh.

References

D.L. Donoho and I.M. Johnstone (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3), 425–455.

Wavelet-based Feature EXtraction

Description

Todo: add description

Usage

WaveFEX(type = "PO", obs, qmf, scale = log2(nrow(obs)), iwt = F, alpha = F)

Arguments

type

Orthogonal transform: PO or translation invariant transform:TI.
obs

A matrix of n x p functional data (n should be a power of 2).
qmf

Orthonormal quadrature mirror filter.
scale

Finest scale of the decomposition.
iwt

Logical to specify forward (FALSE) or inverse (TRUE) wavelet transform.
alpha

Logical TRUE to include scaling coefficients at the first scale.

Value

contj Matrix scale x p of Squared norm of the wavelet coefficient at each scale.

wc wavelet coefficients matrix.

References

Antoniadis, A., Brossat, X., Cugliari, J., & Poggi, J. M. (2013). Clustering functional data using wavelets. Int. J. Wavelets Multiresolution Inf. Process., 11(01), 1350003.

Cheam, A., Fredette, M., Marbac, M., & Navarro, F. (2023). Translation-invariant functional clustering on COVID-19 deaths adjusted on population risk factors. J. R. Stat. Soc., C: Appl. Stat., 72(2), 387-413.