# API Reference¶

## Main interface¶

class dtcwt.Transform1d(biort='near_sym_a', qshift='qshift_a')

An implementation of the 1D DT-CWT in NumPy.

Parameters: biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift().
forward(X, nlevels=3, include_scale=False)

Perform a n-level DTCWT decompostion on a 1D column vector X (or on the columns of a matrix X).

Parameters: X – 1D real array or 2D real array whose columns are to be transformed nlevels – Number of levels of wavelet decomposition A dtcwt.Pyramid-like object representing the transform result.

If biort or qshift are strings, they are used as an argument to the biort() or qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

inverse(pyramid, gain_mask=None)

Perform an n-level dual-tree complex wavelet (DTCWT) 1D reconstruction.

Parameters: pyramid – A dtcwt.Pyramid-like object containing the transformed signal. gain_mask – Gain to be applied to each subband. Reconstructed real array.

The l-th element of gain_mask is gain for wavelet subband at level l. If gain_mask[l] == 0, no computation is performed for band l. Default gain_mask is all ones. Note that l is 0-indexed.

class dtcwt.Transform2d(biort='near_sym_a', qshift='qshift_a')

An implementation of the 2D DT-CWT via NumPy. biort and qshift are the wavelets which parameterise the transform.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

forward(X, nlevels=3, include_scale=False)

Perform a n-level DTCWT-2D decompostion on a 2D matrix X.

Parameters: X – 2D real array nlevels – Number of levels of wavelet decomposition A dtcwt.Pyramid compatible object representing the transform-domain signal
inverse(pyramid, gain_mask=None)

Perform an n-level dual-tree complex wavelet (DTCWT) 2D reconstruction.

Parameters: pyramid – A dtcwt.Pyramid-like class holding the transform domain representation to invert. gain_mask – Gain to be applied to each subband. A numpy-array compatible instance with the reconstruction.

The (d, l)-th element of gain_mask is gain for subband with direction d at level l. If gain_mask[d,l] == 0, no computation is performed for band (d,l). Default gain_mask is all ones. Note that both d and l are zero-indexed.

class dtcwt.Transform3d(biort='near_sym_a', qshift='qshift_a', ext_mode=4)

An implementation of the 3D DT-CWT via NumPy. biort and qshift are the wavelets which parameterise the transform. Valid values are documented in dtcwt.coeffs.biort() and dtcwt.coeffs.qshift().

forward(X, nlevels=3, include_scale=False, discard_level_1=False)

Perform a n-level DTCWT-3D decompostion on a 3D matrix X.

Parameters: X – 3D real array-like object nlevels – Number of levels of wavelet decomposition biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift(). discard_level_1 – True if level 1 high-pass bands are to be discarded. a dtcwt.Pyramid instance

Each element of the Pyramid highpasses tuple is a 4D complex array with the 4th dimension having size 28. The 3D slice [l][:,:,:,d] corresponds to the complex higpass coefficients for direction d at level l where d and l are both 0-indexed.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

There are two values for ext_mode, either 4 or 8. If ext_mode = 4, check whether 1st level is divisible by 2 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coefs can be divided by 4. If any dimension size is not a multiple of 4, append extra coefs by repeating the edges. If ext_mode = 8, check whether 1st level is divisible by 4 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coeffs can be divided by 8. If any dimension size is not a multiple of 8, append extra coeffs by repeating the edges twice.

If discard_level_1 is True the highpass coefficients at level 1 will not be discarded. (And, in fact, will never be calculated.) This turns the transform from being 8:1 redundant to being 1:1 redundant at the cost of no-longer allowing perfect reconstruction. If this option is selected then the first element of the highpasses tuple will be None. Note that dtcwt.Transform3d.inverse() will accept the first element being None and will treat it as being zero.

inverse(pyramid)

Perform an n-level dual-tree complex wavelet (DTCWT) 3D reconstruction.

Parameters: pyramid – The dtcwt.Pyramid-like instance representing the transformed signal. biort – Level 1 wavelets to use. See biort(). qshift – Level >= 2 wavelets to use. See qshift(). ext_mode – Extension mode. See below. Reconstructed real image matrix.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

There are two values for ext_mode, either 4 or 8. If ext_mode = 4, check whether 1st level is divisible by 2 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coefs can be divided by 4. If any dimension size is not a multiple of 4, append extra coefs by repeating the edges. If ext_mode = 8, check whether 1st level is divisible by 4 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coeffs can be divided by 8. If any dimension size is not a multiple of 8, append extra coeffs by repeating the edges twice.

class dtcwt.Pyramid(lowpass, highpasses, scales=None)

A representation of a transform domain signal.

Backends are free to implement any class which respects this interface for storing transform-domain signals. The inverse transform may accept a backend-specific version of this class but should always accept any class which corresponds to this interface.

lowpass

A NumPy-compatible array containing the coarsest scale lowpass signal.

highpasses

A tuple where each element is the complex subband coefficients for corresponding scales finest to coarsest.

scales

(optional) A tuple where each element is a NumPy-compatible array containing the lowpass signal for corresponding scales finest to coarsest. This is not required for the inverse and may be None.

dtcwt.backend_name = 'numpy'

A string providing a short human-readable name for the DTCWT backend currently being used. This corresponds to the name parameter passed to dtcwt.push_backend(). The default backend is numpy but can be overridden by setting the DTCWT_BACKEND environment variable to a valid backend name.

dtcwt.push_backend(name)

Switch backend implementation to name. Push the previous backend onto the backend stack. The previous backend may be restored via dtcwt.pop_backend().

Parameters: name – string identifying which backend to switch to ValueError – if name does not correspond to a known backend

name may take one of the following values:

dtcwt.pop_backend()

Restore the backend after a call to push_backend(). Calls to pop_backend() and pop_backend() may be nested. This function will undo the most recent call to push_backend().

Raises: IndexError – if one attempts to pop more backends than one has pushed.
dtcwt.preserve_backend_stack()

Return a generator object which can be used to preserve the backend stack even when an exception has been raise. For example:

# current backend is NumPy
assert dtcwt.backend_name == 'numpy'

with dtcwt.preserve_backend_stack():
dtcwt.push_backend('opencl')
# ... things which may raise an exception

# current backend is NumPy even if an exception was thrown
assert dtcwt.backend_name == 'numpy'


Functions to load standard wavelet coefficients.

dtcwt.coeffs.biort(name)

Load level 1 wavelet by name.

Parameters: name – a string specifying the wavelet family name a tuple of vectors giving filter coefficients
Name Wavelet
antonini Antonini 9,7 tap filters.
legall LeGall 5,3 tap filters.
near_sym_a Near-Symmetric 5,7 tap filters.
near_sym_b Near-Symmetric 13,19 tap filters.
near_sym_b_bp Near-Symmetric 13,19 tap filters + BP filter

Return a tuple whose elements are a vector specifying the h0o, g0o, h1o and g1o coefficients.

See Rotational symmetry modified wavelet transform for an explanation of the near_sym_b_bp wavelet filters.

Raises: IOError – if name does not correspond to a set of wavelets known to the library. ValueError – if name specifies a dtcwt.coeffs.qshift() wavelet.
dtcwt.coeffs.qshift(name)

Load level >=2 wavelet by name,

Parameters: name – a string specifying the wavelet family name a tuple of vectors giving filter coefficients
Name Wavelet
qshift_06 Quarter Sample Shift Orthogonal (Q-Shift) 10,10 tap filters, (only 6,6 non-zero taps).
qshift_a Q-shift 10,10 tap filters, (with 10,10 non-zero taps, unlike qshift_06).
qshift_b Q-Shift 14,14 tap filters.
qshift_c Q-Shift 16,16 tap filters.
qshift_d Q-Shift 18,18 tap filters.
qshift_b_bp Q-Shift 18,18 tap filters + BP

Return a tuple whose elements are a vector specifying the h0a, h0b, g0a, g0b, h1a, h1b, g1a and g1b coefficients.

See Rotational symmetry modified wavelet transform for an explanation of the qshift_b_bp wavelet filters.

Raises: IOError – if name does not correspond to a set of wavelets known to the library. ValueError – if name specifies a dtcwt.coeffs.biort() wavelet.

## Keypoint analysis¶

dtcwt.keypoint.find_keypoints(highpass_highpasses, method=None, alpha=1.0, beta=0.4, kappa=0.16666666666666666, threshold=None, max_points=None, upsample_keypoint_energy=None, upsample_highpasses=None, refine_positions=True, skip_levels=1)
Parameters: highpass_highpasses – (NxMx6) matrix of highpass subband images method – (optional) string specifying which keypoint energy method to use alpha – (optional) scale parameter for 'fauqueur' method beta – (optional) shape parameter for 'fauqueur' method kappa – (optiona) suppression parameter for 'kingsbury' method threshold – (optional) minimum keypoint energy of returned keypoints max_points – (optional) maximum number of keypoints to return upsample_keypoint_energy – is non-None, a string specifying a method used to upscale the keypoint energy map before finding keypoints upsample_subands – is non-None, a string specifying a method used to upscale the subband image before finding keypoints refine_positions – (optional) should the keypoint positions be refined to sub-pixel accuracy skip_levels – (optional) number of levels of the transform to ignore before looking for keypoints (Px4) array of P keypoints in image co-ordinates

Warning

The interface and behaviour of this function is the subject of an open research project. It is provided in this release as a preview of forthcoming functionality but it is subject to change between releases.

The rows of the returned keypoint array give the x co-ordinate, y co-ordinate, scale and keypoint energy. The rows are sorted in order of decreasing keypoint energy.

If refine_positions is True then the positions (and energy) of the keypoints will be refined to sub-pixel accuracy by fitting a quadratic patch. If refine_positions is False then the keypoint locations will be those corresponding directly to pixel-wise maxima of the subband images.

The max_points and threshold parameters are cumulative: if both are specified then the max_points greatest energy keypoints with energy greater than threshold will be returned.

Usually the keypoint energies returned from the finest scale level are dominated by noise and so one usually wants to specify skip_levels to be 1 or 2. If skip_levels is 0 then all levels will be used to compute keypoint energy.

The upsample_highpasses and upsample_keypoint_energy parameters are used to control whether the individual subband coefficients and/org the keypoint energy map are upscaled by 2 before finding keypoints. If these parameters are None then no corresponding upscaling is performed. If non-None they specify the upscale method as outlined in dtcwt.sampling.upsample().

If method is None, the default 'fauqueur' method is used.

Name Description Parameters used
fauqueur Geometric mean of absolute values[1] alpha, beta
bendale Minimum absolute value[2] none
kingsbury Cross-product of orthogonal highpasses kappa

[1] Julien Fauqueur, Nick Kingsbury, and Ryan Anderson. Multiscale Keypoint Detection using the Dual-Tree Complex Wavelet Transform. 2006 International Conference on Image Processing, pages 1625-1628, October 2006. ISSN 1522-4880. doi: 10.1109/ICIP.2006.312656. http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4106857.

[2] Pashmina Bendale, Bill Triggs, and Nick Kingsbury. Multiscale Keypoint Analysis based on Complex Wavelets. In British Machine Vision Con-ference (BMVC), 2010. http://www-sigproc.eng.cam.ac.uk/~pb397/publications/BTK_BMVC_2010_abstract.pdf.

## Image sampling¶

This module contains function for rescaling and re-sampling high- and low-pass highpasses.

Note

All of these functions take an integer co-ordinate (x, y) to be the centre of the corresponding pixel. Therefore the upper-left pixel notionally covers the interval (-0.5, 0.5) in x and y. An image with N rows and M columns, therefore, has an extent (-0.5, M-0.5) on the x-axis and an extent of (-0.5, N-0.5) on the y-axis. The rescale and upsample functions in this module will use this region as the extent of the image.

dtcwt.sampling.sample(im, xs, ys, method=None)

Sample image at (x,y) given by elements of xs and ys. Both xs and ys must have identical shape and output will have this same shape. The location (x,y) refers to the centre of im[y,x]. Samples at fractional locations are calculated using the method specified by method (or 'lanczos' if method is None.)

Parameters: im – array to sample from xs – x co-ordinates to sample ys – y co-ordinates to sample method – one of ‘bilinear’, ‘lanczos’ or ‘nearest’ ValueError – if xs and ys have differing shapes
dtcwt.sampling.sample_highpass(im, xs, ys, method=None, sbs=None)

As sample() except that the highpass image is first phase shifted to be centred on approximately DC, and has additional ‘sbs’ input allowing selection and re-ordering of subbands. This is useful mainly when the exact locations one wishes to sample from vary by subband.

‘sbs’ should ordinarily be a numpy array of subband indices, in ascending order, e.g., np.array([0,2,3,5]) for just subbands 0, 2, 3 and 5; The returned array will be flattened to just 4 subbands. Pass [0,1,2,3,4,5] for all subbands.

Take care when re-ordering, preferably keeping the ‘sbs’ array outside the scope of this function to keep track of the new indices.

1. Forshaw, Feb 2014.
dtcwt.sampling.rescale(im, shape, method=None)

Return a resampled version of im scaled to shape.

Since the centre of pixel (x,y) has co-ordinate (x,y) the extent of im is actually $$x \in (-0.5, w-0.5]$$ and $$y \in (-0.5, h-0.5]$$ where (y,x) is im.shape. This returns a sampled version of im that has the same extent as a shape-sized array.

dtcwt.sampling.rescale_highpass(im, shape, method=None, sbs=None)

As rescale() except that the highpass image is first phase shifted to be centred on approximately DC, and has additional ‘sbs’ input allowing selection and re-ordering of subbands. This is useful mainly when the exact locations one wishes to sample from vary by subband.

‘sbs’ should ordinarily be a list of subband indices, in ascending order, e.g., np.array([0,2,3,5]) for just subbands 0, 2, 3 and 5; The returned array will be flattened to just 4 subbands. Pass [0,1,2,3,4,5] for all subbands.

Take care when re-ordering, preferably keeping the ‘sbs’ array outside the scope of this function to keep track of the new indices.

1. Forshaw, Feb 2014.
dtcwt.sampling.upsample(image, method=None)

Specialised function to upsample an image by a factor of two using a specified sampling method. If image is an array of shape (NxMx...) then the output will have shape (2Nx2Mx...). Only rows and columns are upsampled, depth axes and greater are interpolated but are not upsampled.

Parameters: image – an array containing the image to upsample method – if non-None, a string specifying the sampling method to use.

If method is None, the default sampling method 'lanczos' is used. The following sampling methods are supported:

Name Description
nearest Nearest-neighbour sampling
bilinear Bilinear sampling
lanczos Lanczos sampling with window radius of 3
dtcwt.sampling.upsample_highpass(im, method=None)

As upsample() except that the highpass image is first phase rolled so that the filter has approximate DC centre frequency. The upshot is that this is the function to use when re-sampling complex subband images.

## Image registration¶

Note

This module is experimental. It’s API may change between versions.

This module implements function for DTCWT-based image registration as outlined in [1]. These functions are 2D-only for the moment.

dtcwt.registration.estimatereg(source, reference, regshape=None, levels=None)

Estimate registration from which will map source to reference.

Parameters: source – transformed source image reference – transformed reference image

The reference and source parameters should support the same API as dtcwt.Pyramid.

The local affine distortion is estimated at at 8x8 pixel scales. Return a NxMx6 array where the 6-element vector at (N,M) corresponds to the affine distortion parameters for the 8x8 block with index (N,M).

Use the velocityfield() function to convert the return value from this function into a velocity field.

If not-None, levels is a sequence of sequences of 0-based level indices to use when calculating the registration. If None then a default set of levels are used.

dtcwt.registration.velocityfield(avecs, shape, method=None)

Given the affine distortion parameters returned from estimatereg(), return a tuple of 2D arrays giving the x- and y- components of the velocity field. The shape of the velocity component field is shape. The velocities are measured in terms of normalised units where the image has width and height of unity.

The method parameter is interpreted as in dtcwt.sampling.rescale() and is the sampling method used to resize avecs to shape.

dtcwt.registration.warp(I, avecs, method=None)

A convenience function to warp an image according to the velocity field implied by avecs.

dtcwt.registration.warptransform(t, avecs, levels, method=None)

Return a warped version of a transformed image acting only on specified levels.

Parameters: t – a transformed image avecs – an array of affine distortion parameters levels – a sequence of 0-based indices specifying which levels to act on

t should be a dtcwt.Pyramid-compatible instance.

The method parameter is interpreted as in dtcwt.sampling.rescale() and is the sampling method used to resize avecs to shape.

Note

This function will clone the transform t but it is a shallow clone where possible. Only the levels specified in levels will be deep-copied and warped.

## Plotting functions¶

Convenience functions for plotting DTCWT-related objects.

dtcwt.plotting.overlay_quiver(image, vectorField, level, offset)

Overlays nicely coloured quiver plot of complex coefficients over original full-size image, providing a useful phase visualisation.

Parameters: image – array holding grayscale values on the interval [0, 255] to display vectorField – a single [MxNx6] numpy array of DTCWT coefficients level – the transform level (1-indexed) of vectorField. offset – Offset for DTCWT coefficients (typically 0.5)

Note

The level parameter is 1-indexed meaning that the third level has index “3”. This is unusual in Python but is kept for compatibility with similar MATLAB routines.

Should also work with other types of complex arrays (e.g., SLP coefficients), as long as the format is the same.

Usage example:

import dtcwt import dtcwt.plotting as plotting

mandrill = datasets.mandrill()

transform2d = dtcwt.Transform2d() mandrill_t = transform2d.forward(mandrill, nlevels=5)

plotting.overlay_quiver(mandrill*255, mandrill_t.highpasses[-1], 5, 0.5)

## Miscellaneous and low-level support functions¶

Useful utilities for testing the 2-D DTCWT with synthetic images

dtcwt.utils.appropriate_complex_type_for(X)

Return an appropriate complex data type depending on the type of X. If X is already complex, return that, if it is floating point return a complex type of the appropriate size and if it is integer, choose an complex floating point type depending on the result of numpy.asfarray().

dtcwt.utils.as_column_vector(v)

Return v as a column vector with shape (N,1).

dtcwt.utils.asfarray(X)

Similar to numpy.asfarray() except that this function tries to preserve the original datatype of X if it is already a floating point type and will pass floating point arrays through directly without copying.

dtcwt.utils.drawcirc(r, w, du, dv, N)

Generate an image of size N*N pels, containing a circle radius r pels and centred at du,dv relative to the centre of the image. The edge of the circle is a cosine shaped edge of width w (from 10 to 90% points).

Python implementation by S. C. Forshaw, November 2013.

dtcwt.utils.drawedge(theta, r, w, N)

Generate an image of size N * N pels, of an edge going from 0 to 1 in height at theta degrees to the horizontal (top of image = 1 if angle = 0). r is a two-element vector, it is a coordinate in ij coords through which the step should pass. The shape of the intensity step is half a raised cosine w pels wide (w>=1).

T. E . Gale’s enhancement to drawedge() for MATLAB, transliterated to Python by S. C. Forshaw, Nov. 2013.

dtcwt.utils.reflect(x, minx, maxx)

Reflect the values in matrix x about the scalar values minx and maxx. Hence a vector x containing a long linearly increasing series is converted into a waveform which ramps linearly up and down between minx and maxx. If x contains integers and minx and maxx are (integers + 0.5), the ramps will have repeated max and min samples.

dtcwt.utils.stacked_2d_matrix_matrix_prod(mats1, mats2)

Interpret mats1 and mats2 as arrays of 2D matrices. I.e. mats1 has shape PxQxNxM and mats2 has shape PxQxMxR. The result is a PxQxNxR array equivalent to:

result[i,j,:,:] = mats1[i,j,:,:].dot(mats2[i,j,:,:])


for all valid row and column indices i and j.

dtcwt.utils.stacked_2d_matrix_vector_prod(mats, vecs)

Interpret mats and vecs as arrays of 2D matrices and vectors. I.e. mats has shape PxQxNxM and vecs has shape PxQxM. The result is a PxQxN array equivalent to:

result[i,j,:] = mats[i,j,:,:].dot(vecs[i,j,:])


for all valid row and column indices i and j.

dtcwt.utils.stacked_2d_vector_matrix_prod(vecs, mats)

Interpret mats and vecs as arrays of 2D matrices and vectors. I.e. mats has shape PxQxNxM and vecs has shape PxQxN. The result is a PxQxM array equivalent to:

result[i,j,:] = mats[i,j,:,:].T.dot(vecs[i,j,:])


for all valid row and column indices i and j.

## Compatibility with MATLAB¶

Functions for compatibility with MATLAB scripts. These functions are intentionally similar in name and behaviour to the original functions from the DTCWT MATLAB toolbox. They are included in the library to ease the porting of MATLAB scripts but shouldn’t be used in new projects.

Note

The functionality of dtwavexfm2b and dtwaveifm2b has been folded into dtwavexfm2 and dtwaveifm2. For convenience of porting MATLAB scripts, the original function names are available in the dtcwt module as aliases but they should not be used in new code.

dtcwt.compat.dtwavexfm(X, nlevels=3, biort='near_sym_a', qshift='qshift_a', include_scale=False)

Perform a n-level DTCWT decompostion on a 1D column vector X (or on the columns of a matrix X).

Parameters: X – 1D real array or 2D real array whose columns are to be transformed nlevels – Number of levels of wavelet decomposition biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift(). The real lowpass image from the final level A tuple containing the (N, M, 6) shape complex highpass subimages for each level. If include_scale is True, a tuple containing real lowpass coefficients for every scale.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

Example:

# Performs a 5-level transform on the real image X using the 13,19-tap
# filters for level 1 and the Q-shift 14-tap filters for levels >= 2.
Yl, Yh = dtwavexfm(X,5,'near_sym_b','qshift_b')

dtcwt.compat.dtwaveifm(Yl, Yh, biort='near_sym_a', qshift='qshift_a', gain_mask=None)

Perform an n-level dual-tree complex wavelet (DTCWT) 1D reconstruction.

Parameters: Yl – The real lowpass subband from the final level Yh – A sequence containing the complex highpass subband for each level. biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift(). gain_mask – Gain to be applied to each subband. Reconstructed real array.

The l-th element of gain_mask is gain for wavelet subband at level l. If gain_mask[l] == 0, no computation is performed for band l. Default gain_mask is all ones. Note that l is 0-indexed.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

Example:

# Performs a reconstruction from Yl,Yh using the 13,19-tap filters
# for level 1 and the Q-shift 14-tap filters for levels >= 2.
Z = dtwaveifm(Yl, Yh, 'near_sym_b', 'qshift_b')

dtcwt.compat.dtwavexfm2(X, nlevels=3, biort='near_sym_a', qshift='qshift_a', include_scale=False)

Perform a n-level DTCWT-2D decompostion on a 2D matrix X.

Parameters: X – 2D real array nlevels – Number of levels of wavelet decomposition biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift(). The real lowpass image from the final level A tuple containing the complex highpass subimages for each level. If include_scale is True, a tuple containing real lowpass coefficients for every scale.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

Example:

# Performs a 3-level transform on the real image X using the 13,19-tap
# filters for level 1 and the Q-shift 14-tap filters for levels >= 2.
Yl, Yh = dtwavexfm2(X, 3, 'near_sym_b', 'qshift_b')

dtcwt.compat.dtwaveifm2(Yl, Yh, biort='near_sym_a', qshift='qshift_a', gain_mask=None)

Perform an n-level dual-tree complex wavelet (DTCWT) 2D reconstruction.

Parameters: Yl – The real lowpass subband from the final level Yh – A sequence containing the complex highpass subband for each level. biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift(). gain_mask – Gain to be applied to each subband. Reconstructed real array

The (d, l)-th element of gain_mask is gain for subband with direction d at level l. If gain_mask[d,l] == 0, no computation is performed for band (d,l). Default gain_mask is all ones. Note that both d and l are zero-indexed.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

Example:

# Performs a 3-level reconstruction from Yl,Yh using the 13,19-tap
# filters for level 1 and the Q-shift 14-tap filters for levels >= 2.
Z = dtwaveifm2(Yl, Yh, 'near_sym_b', 'qshift_b')

dtcwt.compat.dtwavexfm2b(X, nlevels=3, biort='near_sym_a', qshift='qshift_a', include_scale=False)

Perform a n-level DTCWT-2D decompostion on a 2D matrix X.

Parameters: X – 2D real array nlevels – Number of levels of wavelet decomposition biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift(). The real lowpass image from the final level A tuple containing the complex highpass subimages for each level. If include_scale is True, a tuple containing real lowpass coefficients for every scale.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

Example:

# Performs a 3-level transform on the real image X using the 13,19-tap
# filters for level 1 and the Q-shift 14-tap filters for levels >= 2.
Yl, Yh = dtwavexfm2(X, 3, 'near_sym_b', 'qshift_b')

dtcwt.compat.dtwaveifm2b(Yl, Yh, biort='near_sym_a', qshift='qshift_a', gain_mask=None)

Perform an n-level dual-tree complex wavelet (DTCWT) 2D reconstruction.

Parameters: Yl – The real lowpass subband from the final level Yh – A sequence containing the complex highpass subband for each level. biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift(). gain_mask – Gain to be applied to each subband. Reconstructed real array

The (d, l)-th element of gain_mask is gain for subband with direction d at level l. If gain_mask[d,l] == 0, no computation is performed for band (d,l). Default gain_mask is all ones. Note that both d and l are zero-indexed.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

Example:

# Performs a 3-level reconstruction from Yl,Yh using the 13,19-tap
# filters for level 1 and the Q-shift 14-tap filters for levels >= 2.
Z = dtwaveifm2(Yl, Yh, 'near_sym_b', 'qshift_b')

dtcwt.compat.dtwavexfm3(X, nlevels=3, biort='near_sym_a', qshift='qshift_a', include_scale=False, ext_mode=4, discard_level_1=False)

Perform a n-level DTCWT-3D decompostion on a 3D matrix X.

Parameters: X – 3D real array-like object nlevels – Number of levels of wavelet decomposition biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift(). ext_mode – Extension mode. See below. discard_level_1 – True if level 1 high-pass bands are to be discarded. The real lowpass image from the final level A tuple containing the complex highpass subimages for each level.

Each element of Yh is a 4D complex array with the 4th dimension having size 28. The 3D slice Yh[l][:,:,:,d] corresponds to the complex higpass coefficients for direction d at level l where d and l are both 0-indexed.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

There are two values for ext_mode, either 4 or 8. If ext_mode = 4, check whether 1st level is divisible by 2 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coefs can be divided by 4. If any dimension size is not a multiple of 4, append extra coefs by repeating the edges. If ext_mode = 8, check whether 1st level is divisible by 4 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coeffs can be divided by 8. If any dimension size is not a multiple of 8, append extra coeffs by repeating the edges twice.

If discard_level_1 is True the highpass coefficients at level 1 will be discarded. (And, in fact, will never be calculated.) This turns the transform from being 8:1 redundant to being 1:1 redundant at the cost of no-longer allowing perfect reconstruction. If this option is selected then Yh[0] will be None. Note that dtwaveifm3() will accepts Yh[0] being None and will treat it as being zero.

Example:

# Performs a 3-level transform on the real 3D array X using the 13,19-tap
# filters for level 1 and the Q-shift 14-tap filters for levels >= 2.
Yl, Yh = dtwavexfm3(X, 3, 'near_sym_b', 'qshift_b')

dtcwt.compat.dtwaveifm3(Yl, Yh, biort='near_sym_a', qshift='qshift_a', ext_mode=4)

Perform an n-level dual-tree complex wavelet (DTCWT) 3D reconstruction.

Parameters: Yl – The real lowpass subband from the final level Yh – A sequence containing the complex highpass subband for each level. biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift(). ext_mode – Extension mode. See below. Reconstructed real image matrix.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

There are two values for ext_mode, either 4 or 8. If ext_mode = 4, check whether 1st level is divisible by 2 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coefs can be divided by 4. If any dimension size is not a multiple of 4, append extra coefs by repeating the edges. If ext_mode = 8, check whether 1st level is divisible by 4 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coeffs can be divided by 8. If any dimension size is not a multiple of 8, append extra coeffs by repeating the edges twice.

Example:

# Performs a 3-level reconstruction from Yl,Yh using the 13,19-tap
# filters for level 1 and the Q-shift 14-tap filters for levels >= 2.
Z = dtwaveifm3(Yl, Yh, 'near_sym_b', 'qshift_b')


## Backends¶

The following modules provide backend-specific implementations. Usually you won’t need to import these modules directly; the main API will use an appropriate implementation. Occasionally, however, you may want to benchmark one implementation against the other.

### NumPy¶

A backend which uses NumPy to perform the filtering. This backend should always be available.

class dtcwt.numpy.Pyramid(lowpass, highpasses, scales=None)

A representation of a transform domain signal.

Backends are free to implement any class which respects this interface for storing transform-domain signals. The inverse transform may accept a backend-specific version of this class but should always accept any class which corresponds to this interface.

lowpass

A NumPy-compatible array containing the coarsest scale lowpass signal.

highpasses

A tuple where each element is the complex subband coefficients for corresponding scales finest to coarsest.

scales

(optional) A tuple where each element is a NumPy-compatible array containing the lowpass signal for corresponding scales finest to coarsest. This is not required for the inverse and may be None.

class dtcwt.numpy.Transform1d(biort='near_sym_a', qshift='qshift_a')

An implementation of the 1D DT-CWT in NumPy.

Parameters: biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift().
forward(X, nlevels=3, include_scale=False)

Perform a n-level DTCWT decompostion on a 1D column vector X (or on the columns of a matrix X).

Parameters: X – 1D real array or 2D real array whose columns are to be transformed nlevels – Number of levels of wavelet decomposition A dtcwt.Pyramid-like object representing the transform result.

If biort or qshift are strings, they are used as an argument to the biort() or qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

inverse(pyramid, gain_mask=None)

Perform an n-level dual-tree complex wavelet (DTCWT) 1D reconstruction.

Parameters: pyramid – A dtcwt.Pyramid-like object containing the transformed signal. gain_mask – Gain to be applied to each subband. Reconstructed real array.

The l-th element of gain_mask is gain for wavelet subband at level l. If gain_mask[l] == 0, no computation is performed for band l. Default gain_mask is all ones. Note that l is 0-indexed.

class dtcwt.numpy.Transform2d(biort='near_sym_a', qshift='qshift_a')

An implementation of the 2D DT-CWT via NumPy. biort and qshift are the wavelets which parameterise the transform.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

forward(X, nlevels=3, include_scale=False)

Perform a n-level DTCWT-2D decompostion on a 2D matrix X.

Parameters: X – 2D real array nlevels – Number of levels of wavelet decomposition A dtcwt.Pyramid compatible object representing the transform-domain signal
inverse(pyramid, gain_mask=None)

Perform an n-level dual-tree complex wavelet (DTCWT) 2D reconstruction.

Parameters: pyramid – A dtcwt.Pyramid-like class holding the transform domain representation to invert. gain_mask – Gain to be applied to each subband. A numpy-array compatible instance with the reconstruction.

The (d, l)-th element of gain_mask is gain for subband with direction d at level l. If gain_mask[d,l] == 0, no computation is performed for band (d,l). Default gain_mask is all ones. Note that both d and l are zero-indexed.

class dtcwt.numpy.Transform3d(biort='near_sym_a', qshift='qshift_a', ext_mode=4)

An implementation of the 3D DT-CWT via NumPy. biort and qshift are the wavelets which parameterise the transform. Valid values are documented in dtcwt.coeffs.biort() and dtcwt.coeffs.qshift().

forward(X, nlevels=3, include_scale=False, discard_level_1=False)

Perform a n-level DTCWT-3D decompostion on a 3D matrix X.

Parameters: X – 3D real array-like object nlevels – Number of levels of wavelet decomposition biort – Level 1 wavelets to use. See dtcwt.coeffs.biort(). qshift – Level >= 2 wavelets to use. See dtcwt.coeffs.qshift(). discard_level_1 – True if level 1 high-pass bands are to be discarded. a dtcwt.Pyramid instance

Each element of the Pyramid highpasses tuple is a 4D complex array with the 4th dimension having size 28. The 3D slice [l][:,:,:,d] corresponds to the complex higpass coefficients for direction d at level l where d and l are both 0-indexed.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

There are two values for ext_mode, either 4 or 8. If ext_mode = 4, check whether 1st level is divisible by 2 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coefs can be divided by 4. If any dimension size is not a multiple of 4, append extra coefs by repeating the edges. If ext_mode = 8, check whether 1st level is divisible by 4 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coeffs can be divided by 8. If any dimension size is not a multiple of 8, append extra coeffs by repeating the edges twice.

If discard_level_1 is True the highpass coefficients at level 1 will not be discarded. (And, in fact, will never be calculated.) This turns the transform from being 8:1 redundant to being 1:1 redundant at the cost of no-longer allowing perfect reconstruction. If this option is selected then the first element of the highpasses tuple will be None. Note that dtcwt.Transform3d.inverse() will accept the first element being None and will treat it as being zero.

inverse(pyramid)

Perform an n-level dual-tree complex wavelet (DTCWT) 3D reconstruction.

Parameters: pyramid – The dtcwt.Pyramid-like instance representing the transformed signal. biort – Level 1 wavelets to use. See biort(). qshift – Level >= 2 wavelets to use. See qshift(). ext_mode – Extension mode. See below. Reconstructed real image matrix.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

There are two values for ext_mode, either 4 or 8. If ext_mode = 4, check whether 1st level is divisible by 2 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coefs can be divided by 4. If any dimension size is not a multiple of 4, append extra coefs by repeating the edges. If ext_mode = 8, check whether 1st level is divisible by 4 (if not we raise a ValueError). Also check whether from 2nd level onwards, the coeffs can be divided by 8. If any dimension size is not a multiple of 8, append extra coeffs by repeating the edges twice.

dtcwt.numpy.lowlevel.colfilter(X, h)

Filter the columns of image X using filter vector h, without decimation. If len(h) is odd, each output sample is aligned with each input sample and Y is the same size as X. If len(h) is even, each output sample is aligned with the mid point of each pair of input samples, and Y.shape = X.shape + [1 0].

Parameters: X – an image whose columns are to be filtered h – the filter coefficients. the filtered image.
dtcwt.numpy.lowlevel.colifilt(X, ha, hb)

Filter the columns of image X using the two filters ha and hb = reverse(ha). ha operates on the odd samples of X and hb on the even samples. Both filters should be even length, and h should be approx linear phase with a quarter sample advance from its mid pt (i.e :math:|h(m/2)| > |h(m/2 + 1)|).

                  ext       left edge                      right edge       ext
Level 2:        !               |               !               |               !
+q filt on x      b       b       a       a       a       a       b       b
-q filt on o          a       a       b       b       b       b       a       a
Level 1:        !               |               !               |               !
odd filt on .    b   b   b   b   a   a   a   a   a   a   a   a   b   b   b   b
odd filt on .      a   a   a   a   b   b   b   b   b   b   b   b   a   a   a   a


The output is interpolated by two from the input sample rate and the results from the two filters, Ya and Yb, are interleaved to give Y. Symmetric extension with repeated end samples is used on the composite X columns before each filter is applied.

dtcwt.numpy.lowlevel.coldfilt(X, ha, hb)

Filter the columns of image X using the two filters ha and hb = reverse(ha). ha operates on the odd samples of X and hb on the even samples. Both filters should be even length, and h should be approx linear phase with a quarter sample advance from its mid pt (i.e. $$|h(m/2)| > |h(m/2 + 1)|$$).

                  ext        top edge                     bottom edge       ext
Level 1:        !               |               !               |               !
odd filt on .    b   b   b   b   a   a   a   a   a   a   a   a   b   b   b   b
odd filt on .      a   a   a   a   b   b   b   b   b   b   b   b   a   a   a   a
Level 2:        !               |               !               |               !
+q filt on x      b       b       a       a       a       a       b       b
-q filt on o          a       a       b       b       b       b       a       a


The output is decimated by two from the input sample rate and the results from the two filters, Ya and Yb, are interleaved to give Y. Symmetric extension with repeated end samples is used on the composite X columns before each filter is applied.

Raises ValueError if the number of rows in X is not a multiple of 4, the length of ha does not match hb or the lengths of ha or hb are non-even.

### OpenCL¶

Provide low-level OpenCL accelerated operations. This backend requires that PyOpenCL be installed.

class dtcwt.opencl.Pyramid(lowpass, highpasses, scales=None)

An interface-compatible version of dtcwt.Pyramid where the initialiser arguments are assumed to by pyopencl.array.Array instances.

The attributes defined in dtcwt.Pyramid are implemented via properties. The original OpenCL arrays may be accessed via the cl_... attributes.

Note

The copy from device to host is performed once and then memoized. This makes repeated access to the host-side attributes efficient but will mean that any changes to the device-side arrays will not be reflected in the host-side attributes after their first access. You should not be modifying the arrays once you return an instance of this class anyway but if you do, beware!

cl_lowpass

The CL array containing the lowpass image.

cl_highpasses

A tuple of CL arrays containing the subband images.

cl_scales

(optional) Either None or a tuple of lowpass images for each scale.

class dtcwt.opencl.Transform2d(biort='near_sym_a', qshift='qshift_a', queue=None)

An implementation of the 2D DT-CWT via OpenCL. biort and qshift are the wavelets which parameterise the transform.

If queue is non-None it is an instance of pyopencl.CommandQueue which is used to compile and execute the OpenCL kernels which implement the transform. If it is None, the first available compute device is used.

If biort or qshift are strings, they are used as an argument to the dtcwt.coeffs.biort() or dtcwt.coeffs.qshift() functions. Otherwise, they are interpreted as tuples of vectors giving filter coefficients. In the biort case, this should be (h0o, g0o, h1o, g1o). In the qshift case, this should be (h0a, h0b, g0a, g0b, h1a, h1b, g1a, g1b).

Note

At the moment only the forward transform is accelerated. The inverse transform uses the NumPy backend.

forward(X, nlevels=3, include_scale=False)

Perform a n-level DTCWT-2D decompostion on a 2D matrix X.

Parameters: X – 2D real array nlevels – Number of levels of wavelet decomposition A dtcwt.Pyramid compatible object representing the transform-domain signal

Note

X may be a pyopencl.array.Array instance which has already been copied to the device. In which case, it must be 2D. (I.e. a vector will not be auto-promoted.)

inverse(pyramid, gain_mask=None)

Perform an n-level dual-tree complex wavelet (DTCWT) 2D reconstruction.

Parameters: pyramid – A dtcwt.Pyramid-like class holding the transform domain representation to invert. gain_mask – Gain to be applied to each subband. A numpy-array compatible instance with the reconstruction.

The (d, l)-th element of gain_mask is gain for subband with direction d at level l. If gain_mask[d,l] == 0, no computation is performed for band (d,l). Default gain_mask is all ones. Note that both d and l are zero-indexed.

dtcwt.opencl.lowlevel.axis_convolve(X, h, axis=0, queue=None, output=None)

Filter along an of X using filter vector h. If h has odd length, each output sample is aligned with each input sample and Y is the same size as X. If h has even length, each output sample is aligned with the mid point of each pair of input samples, and the output matrix’s shape is increased by one along the convolution axis.

After convolution, the pyopencl.array.Array instance holding the device-side output is returned. This may be accessed on the host via to_array().

The axis of convolution is specified by axis. The default direction of convolution is column-wise.

If queue is non-None, it should be a pyopencl.CommandQueue instance which is used to perform the computation. If None, a default global queue is used.

If output is non-None, it should be a pyopencl.array.Array instance which the result is written into. If None, an output array is created.

dtcwt.opencl.lowlevel.coldfilt(X, ha, hb, queue=None)

Filter the columns of image X using the two filters ha and hb = reverse(ha). ha operates on the odd samples of X and hb on the even samples. Both filters should be even length, and h should be approx linear phase with a quarter sample advance from its mid pt (i.e. $$|h(m/2)| > |h(m/2 + 1)|$$).

                  ext        top edge                     bottom edge       ext
Level 1:        !               |               !               |               !
odd filt on .    b   b   b   b   a   a   a   a   a   a   a   a   b   b   b   b
odd filt on .      a   a   a   a   b   b   b   b   b   b   b   b   a   a   a   a
Level 2:        !               |               !               |               !
+q filt on x      b       b       a       a       a       a       b       b
-q filt on o          a       a       b       b       b       b       a       a


The output is decimated by two from the input sample rate and the results from the two filters, Ya and Yb, are interleaved to give Y. Symmetric extension with repeated end samples is used on the composite X columns before each filter is applied.

Raises ValueError if the number of rows in X is not a multiple of 4, the length of ha does not match hb or the lengths of ha or hb are non-even.

dtcwt.opencl.lowlevel.colfilter(X, h)

Filter the columns of image X using filter vector h, without decimation. If len(h) is odd, each output sample is aligned with each input sample and Y is the same size as X. If len(h) is even, each output sample is aligned with the mid point of each pair of input samples, and Y.shape = X.shape + [1 0].

The filtering will be accelerated via OpenCL.

Parameters: X – an image whose columns are to be filtered h – the filter coefficients. the filtered image.
dtcwt.opencl.lowlevel.colifilt(X, ha, hb, queue=None)

Filter the columns of image X using the two filters ha and hb = reverse(ha). ha operates on the odd samples of X and hb on the even samples. Both filters should be even length, and h should be approx linear phase with a quarter sample advance from its mid pt (i.e :math:|h(m/2)| > |h(m/2 + 1)|).

                  ext       left edge                      right edge       ext
Level 2:        !               |               !               |               !
+q filt on x      b       b       a       a       a       a       b       b
-q filt on o          a       a       b       b       b       b       a       a
Level 1:        !               |               !               |               !
odd filt on .    b   b   b   b   a   a   a   a   a   a   a   a   b   b   b   b
odd filt on .      a   a   a   a   b   b   b   b   b   b   b   b   a   a   a   a


The output is interpolated by two from the input sample rate and the results from the two filters, Ya and Yb, are interleaved to give Y. Symmetric extension with repeated end samples is used on the composite X columns before each filter is applied.

dtcwt.opencl.lowlevel.get_default_queue(*args, **kwargs)

Return the default queue used for computation if one is not specified.

This function is memoized and so only one queue is created after multiple invocations.