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

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")

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

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.

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

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)

Hi-Pass Downsampling operator (periodized)

Description

Hi-Pass Downsampling operator (periodized)

Usage

DownDyadHi(x, qmf)

Arguments

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

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)

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

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)

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

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

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

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)

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

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

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)

Generation of Gaussian White Noise

Description

Generation of Gaussian White Noise

Usage

GWN(n, sigma)

Arguments

Value

epsilon resulting noise.

Examples

GWN(10, 0.1)

Apply Hard Threshold

Description

Apply Hard Threshold

Usage

HardThresh(y, t)

Arguments

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)

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

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 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

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

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

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)

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

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

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)

Apply James-Stein Threshold

Description

(also called the nonnegative garrote)

Usage

JSThresh(y, t)

Arguments

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)

Median Absolute Deviation

Description

Compute the median absolute deviation.

Usage

MAD(x)

Arguments

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

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

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

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

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

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

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

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

Value

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

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

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

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)

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.

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

Value

Signal/Noise ratio.

Examples

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

Adaptive Threshold Selection Using Principle of SURE

Description

SURE referes to Stein's Unbiased Risk Estimate.

Usage

SUREThresh(y)

Arguments

Value

x Estimate of mean vector

thresh Threshold used.

Make signal a row vector

Description

Make signal a row vector

Usage

ShapeAsRow(sig)

Arguments

Value

row a row vector.

Examples

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

Apply Soft Threshold

Description

Apply Soft Threshold

Usage

SoftThresh(y, t)

Arguments

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)

Hi-Pass Upsampling operator; periodized

Description

Hi-Pass Upsampling operator; periodized

Usage

UpDyadHi(x, qmf)

Arguments

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

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

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

Value

thresh Value of Threshold.

Visually calibrated Adaptive Smoothing

Description

Visually calibrated Adaptive Smoothing

Usage

VisuThresh(y, thresh = "soft")

Arguments

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.

Convolution tool for two-scale transform

Description

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

Usage

aconv(f, x)

Arguments

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

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

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)

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

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)))

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

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

Value

n length(x).

J least power of two greater than n.

See Also

quadlength, dyad

Examples

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

Convolution tool for two-scale transform

Description

Filtering by periodic convolution of x with f.

Usage

iconvv(f, x)

Arguments

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

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

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)

Circular left shift of 1-d signal

Description

Circular left shift of 1-d signal

Usage

lshift(a)

Arguments

Value

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

Examples

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

Packet table indexing

Description

Packet table indexing

Usage

packet(d, b, n)

Arguments

Value

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

Examples

packet(1, 1, 8)

Find length and dyadic length of square matrix

Description

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

Usage

quadlength(x)

Arguments

Value

n length(x).

J least power of two greater than n.

Examples

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

Replicate and tile an array

Description

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

Usage

repmat(a, n, m)

Arguments

Examples

repmat(10,3,2)

Circular right shift of 1-d signal

Description

Circular right shift of 1-d signal

Usage

rshift(a)

Arguments

Value

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

Examples

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