PK 0OmCȼ67 67 dtcwt-0.7/backends.html
The dtcwt library currently provides two backends for computing the wavelet transform: a NumPy based implementation and an OpenCL implementation which uses the PyOpenCL bindings for Python.
The top-level transform routines, such as dtcwt.dtwavexfm2(), will automatically use the NumPy backend. If you are not primarily focussed on speed, this is the correct choice since the NumPy backend has the fullest feature support, is the best tested and behaves correctly given single- and double-precision input.
If you care about speed and need only single-precision calculations, the OpenCL backend can provide significant speed-up. On the author’s system, the 2D transform sees around a times 10 speed improvement.
The NumPy and OpenCL backends live in the dtcwt.backend.backend_numpy and dtcwt.backend.backend_opencl modules respectively. Both provide the same base API as defined in dtcwt.backend.base.
Access to the 2D transform is via a Transform2d instance. For example, to compute the 2D DT-CWT of the 2D real array in X:
>>> from dtcwt.backend.backend_numpy import Transform2d
>>> trans = Transform2d() # You may optionally specify which wavelets to use here
>>> Y = trans.forward(X, nlevels=4) # Perform a 4-level transform of X
>>> imshow(Y.lowpass) # Show coarsest scale low-pass image
>>> imshow(Y.subbands[-1][:,:,0]) # Show first coarsest scale subband
In this case Y is an instance of a class which behaves like dtcwt.backend.base.TransformDomainSignal. Backends are free to return whatever result they like as long as the result can be used like this base class. (For example, the OpenCL backend returns a dtcwt.backend.backend_opencl.TransformDomainSignal instance which keeps the device-side results available.)
The 3d_dtcwt_directionality.py script in the examples directory shows how one may demonstrate the directional sensitivity of the 3D DT-CWT complex subband coefficients. It computes empirically the maximally sensitive directions for each subband and plots them in an interactive figure using matplotlib. A screenshot is reproduced below:
There are some points to note about this diagram. Each subband is labeled sich that ‘1’ refers to the first subband, ‘5’ the fifth and so forth. On this diagram the subbands are all four apart reflecting the fact that, for example, subbands 2, 3 and 4 are positioned in the other four quadrants of the upper hemisphere reflecting the position of subband 1. There are seven visible subband directions in the +ve quadrant of the hemisphere and hence there are 28 directions in total over all four quadrants.
The source for the script is shown below:
#!/bin/python
"""
An example of the directional selectivity of 3D DT-CWT coefficients.
This example creates a 3D array holding an image of a sphere and performs the
3D DT-CWT transform on it. The locations of maxima (and their images about the
mid-point of the image) are determined for each complex coefficient at level 2.
These maxima points are then shown on a single plot to demonstrate the
directions in which the 3D DT-CWT transform is selective.
"""
# Import the libraries we need
from matplotlib.pyplot import *
import numpy as np
from mpl_toolkits.mplot3d import Axes3D
from mpl_toolkits.mplot3d.art3d import Poly3DCollection
from dtcwt import dtwavexfm3, dtwaveifm3, biort, qshift
# Specify details about sphere and grid size
GRID_SIZE = 128
SPHERE_RAD = int(0.45 * GRID_SIZE) + 0.5
# Compute an image of the sphere
grid = np.arange(-(GRID_SIZE>>1), GRID_SIZE>>1)
X, Y, Z = np.meshgrid(grid, grid, grid)
r = np.sqrt(X*X + Y*Y + Z*Z)
sphere = (0.5 + np.clip(SPHERE_RAD-r, -0.5, 0.5)).astype(np.float32)
# Specify number of levels and wavelet family to use
nlevels = 2
b = biort('near_sym_a')
q = qshift('qshift_a')
# Form the DT-CWT of the sphere. We use discard_level_1 since we're
# uninterested in the inverse transform and this saves us some memory.
Yl, Yh = dtwavexfm3(sphere, nlevels, b, q, discard_level_1=True)
# Plot maxima
figure(figsize=(8,8))
ax = gcf().add_subplot(1,1,1, projection='3d')
ax.set_aspect('equal')
ax.view_init(35, 75)
# Plot unit sphere +ve octant
thetas = np.linspace(0, np.pi/2, 10)
phis = np.linspace(0, np.pi/2, 10)
def sphere_to_xyz(r, theta, phi):
st, ct = np.sin(theta), np.cos(theta)
sp, cp = np.sin(phi), np.cos(phi)
return r * np.asarray((st*cp, st*sp, ct))
tris = []
rad = 0.99 # so that points plotted latter are not z-clipped
for t1, t2 in zip(thetas[:-1], thetas[1:]):
for p1, p2 in zip(phis[:-1], phis[1:]):
tris.append([
sphere_to_xyz(rad, t1, p1),
sphere_to_xyz(rad, t1, p2),
sphere_to_xyz(rad, t2, p2),
sphere_to_xyz(rad, t2, p1),
])
sphere = Poly3DCollection(tris, facecolor='w', edgecolor=(0.6,0.6,0.6))
ax.add_collection3d(sphere)
locs = []
scale = 1.1
for idx in xrange(Yh[-1].shape[3]):
Z = Yh[-1][:,:,:,idx]
C = np.abs(Z)
max_loc = np.asarray(np.unravel_index(np.argmax(C), C.shape)) - np.asarray(C.shape)*0.5
max_loc /= np.sqrt(np.sum(max_loc * max_loc))
# Only record directions in the +ve octant (or those from the -ve quadrant
# which can be flipped).
if np.all(np.sign(max_loc) == 1):
locs.append(max_loc)
ax.text(max_loc[0] * scale, max_loc[1] * scale, max_loc[2] * scale, str(idx+1))
elif np.all(np.sign(max_loc) == -1):
locs.append(-max_loc)
ax.text(-max_loc[0] * scale, -max_loc[1] * scale, -max_loc[2] * scale, str(idx+1))
# Plot all directions as a scatter plot
locs = np.asarray(locs)
ax.scatter(locs[:,0], locs[:,1], locs[:,2], c=np.arange(locs.shape[0]))
w = 1.1
ax.auto_scale_xyz([0, w], [0, w], [0, w])
legend()
title('3D DT-CWT subband directions for +ve hemisphere quadrant')
tight_layout()
show()
# vim:sw=4:sts=4:et
This section will guide you through using the dtcwt library. Once installed, you are most likely to use one of these functions:
See API Reference for full details on how to call these functions. We shall present some simple usage below.
The easiest way to install dtcwt is via easy_install or pip:
$ pip install dtcwt
If you want to check out the latest in-development version, look at the project’s GitHub page. Once checked out, installation is based on setuptools and follows the usual conventions for a Python project:
$ python setup.py install
(Although the develop command may be more useful if you intend to perform any significant modification to the library.) A test suite is provided so that you may verify the code works on your system:
$ python setup.py nosetests
This will also write test-coverage information to the cover/ directory.
There is a pre-built version of this documentation available online and you can build your own copy via the Sphinx documentation system:
$ python setup.py build_sphinx
Compiled documentation may be found in build/docs/html/.
This example generates two 1D random walks and demonstrates reconstructing them using the forward and inverse 1D transforms. Note that dtcwt.dtwavexfm() and dtcwt.dtwaveifm() will transform columns of an input array independently:
import numpy as np
from matplotlib.pyplot import *
# Generate a 300x2 array of a random walk
vecs = np.cumsum(np.random.rand(300,2) - 0.5, 0)
# Show input
figure(1)
plot(vecs)
title('Input')
import dtcwt
# 1D transform
Yl, Yh = dtcwt.dtwavexfm(vecs)
# Inverse
vecs_recon = dtcwt.dtwaveifm(Yl, Yh)
# Show output
figure(2)
plot(vecs_recon)
title('Output')
# Show error
figure(3)
plot(vecs_recon - vecs)
title('Reconstruction error')
print('Maximum reconstruction error: {0}'.format(np.max(np.abs(vecs - vecs_recon))))
show()
Using the pylab environment (part of matplotlib) we can perform a simple example where we transform the standard ‘Lena’ image and show the level 2 wavelet coefficients:
# Load the Lena image from the Internet into a StringIO object
from StringIO import StringIO
from urllib2 import urlopen
LENA_URL = 'http://www.ece.rice.edu/~wakin/images/lena512.pgm'
lena_file = StringIO(urlopen(LENA_URL).read())
# Parse the lena file and rescale to be in the range (0,1]
from scipy.misc import imread
lena = imread(lena_file) / 255.0
from matplotlib.pyplot import *
import numpy as np
# Show lena on the left
figure(1)
imshow(lena, cmap=cm.gray, clim=(0,1))
import dtcwt
# Compute two levels of dtcwt with the defaul wavelet family
Yh, Yl = dtcwt.dtwavexfm2(lena, 2)
# Show the absolute images for each direction in level 2.
# Note that the 2nd level has index 1 since the 1st has index 0.
figure(2)
for slice_idx in xrange(Yl[1].shape[2]):
subplot(2, 3, slice_idx)
imshow(np.abs(Yl[1][:,:,slice_idx]), cmap=cm.spectral, clim=(0, 1))
# Show the phase images for each direction in level 2.
figure(3)
for slice_idx in xrange(Yl[1].shape[2]):
subplot(2, 3, slice_idx)
imshow(np.angle(Yl[1][:,:,slice_idx]), cmap=cm.hsv, clim=(-np.pi, np.pi))
show()
If the library is correctly installed and you also have matplotlib installed, you should see these three figures:
In the examples below I assume you’ve imported pyplot and numpy and, of course, the dtcwt library itself:
import numpy as np
from matplotlib.pyplot import *
from dtcwt import *
We can demonstrate the 3D transform by generating a 64x64x64 array which contains the image of a sphere:
GRID_SIZE = 64
SPHERE_RAD = int(0.45 * GRID_SIZE) + 0.5
grid = np.arange(-(GRID_SIZE>>1), GRID_SIZE>>1)
X, Y, Z = np.meshgrid(grid, grid, grid)
r = np.sqrt(X*X + Y*Y + Z*Z)
sphere = 0.5 + 0.5 * np.clip(SPHERE_RAD-r, -1, 1)
If we look at the central slice of this image, it looks like a circle:
imshow(sphere[:,:,GRID_SIZE>>1], interpolation='none', cmap=cm.gray)
Performing the 3 level DT-CWT with the defaul wavelet selection is easy:
Yl, Yh = dtwavexfm3(sphere, 3)
The function returns the lowest level low pass image and a tuple of complex subband coefficients:
>>> print(Yl.shape)
(16, 16, 16)
>>> for subbands in Yh:
... print(subbands.shape)
(32, 32, 32, 28)
(16, 16, 16, 28)
(8, 8, 8, 28)
Performing the inverse transform should result in perfect reconstruction:
>>> Z = dtwaveifm3(Yl, Yh)
>>> print(np.abs(Z - ellipsoid).max()) # Should be < 1e-12
8.881784197e-15
If you plot the locations of the large complex coefficients, you can see the directional sensitivity of the transform:
from mpl_toolkits.mplot3d import Axes3D
figure(figsize=(16,16))
nplts = Yh[-1].shape[3]
nrows = np.ceil(np.sqrt(nplts))
ncols = np.ceil(nplts / nrows)
W = np.max(Yh[-1].shape[:3])
for idx in xrange(Yh[-1].shape[3]):
C = np.abs(Yh[-1][:,:,:,idx])
ax = gcf().add_subplot(nrows, ncols, idx+1, projection='3d')
ax.set_aspect('equal')
good = C > 0.2*C.max()
x,y,z = np.nonzero(good)
ax.scatter(x, y, z, c=C[good].ravel())
ax.auto_scale_xyz((0,W), (0,W), (0,W))
tight_layout()
For a further directional sensitivity example, see Showing 3D Directional Sensitivity.
These functions provide API-level compatibility with MATLAB.
Note
The functionality of dtwavexfm2b and dtwaveifm2b have 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.
Perform a n-level DTCWT decompostion on a 1D column vector X (or on the columns of a matrix X).
Parameters: | |
---|---|
Returns Yl: | The real lowpass image from the final level |
Returns Yh: | A tuple containing the (N, M, 6) shape complex highpass subimages for each level. |
Returns Yscale: | 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 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).
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')
Perform an n-level dual-tree complex wavelet (DTCWT) 1D reconstruction.
Parameters: | |
---|---|
Returns Z: | 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 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).
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')
Perform a n-level DTCWT-2D decompostion on a 2D matrix X.
Parameters: | |
---|---|
Returns Yl: | The real lowpass image from the final level |
Returns Yh: | A tuple containing the complex highpass subimages for each level. |
Returns Yscale: | 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 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).
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')
Perform an n-level dual-tree complex wavelet (DTCWT) 2D reconstruction.
Parameters: | |
---|---|
Returns Z: | 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 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).
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')
Perform a n-level DTCWT-2D decompostion on a 2D matrix X.
Parameters: | |
---|---|
Returns Yl: | The real lowpass image from the final level |
Returns Yh: | A tuple containing the complex highpass subimages for each level. |
Returns Yscale: | 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 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).
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')
Perform an n-level dual-tree complex wavelet (DTCWT) 2D reconstruction.
Parameters: | |
---|---|
Returns Z: | 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 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).
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')
Perform a n-level DTCWT-3D decompostion on a 3D matrix X.
Parameters: | |
---|---|
Returns Yl: | The real lowpass image from the final level |
Returns Yh: | 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 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).
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')
Perform an n-level dual-tree complex wavelet (DTCWT) 3D reconstruction.
Parameters: | |
---|---|
Returns Z: | Reconstructed real image matrix. |
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).
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')
Load level 1 wavelet by name.
Parameters: | name – a string specifying the wavelet family name |
---|---|
Returns: | 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: |
|
---|
Load level >=2 wavelet by name,
Parameters: | name – a string specifying the wavelet family name |
---|---|
Returns: | 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: |
|
---|
A representation of the reconstructed signal from the inverse transform. A backend is free to implement their own version of this class providing it corresponds to the interface documented.
A NumPy-compatible array containing the reconstructed signal.
An implementation of a 2D DT-CWT transformation. Backends must provide a transform class which provides an interface compatible with this base class.
Parameters: |
|
---|
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).
In some cases the tuples may have more elements. This is used to represent the Rotational symmetry modified wavelet transform.
Perform a n-level DTCWT-2D decompostion on a 2D matrix X.
Parameters: |
|
---|---|
Returns: | A dtcwt.backend.TransformDomainSignal compatible object representing the transform-domain signal |
Perform an n-level dual-tree complex wavelet (DTCWT) 2D reconstruction.
Parameters: |
|
---|---|
Returns: | A dtcwt.backend.ReconstructedSignal 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.
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.
A NumPy-compatible array containing the coarsest scale lowpass signal.
A tuple where each element is the complex subband coefficients for corresponding scales finest to coarsest.
(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.
A backend which uses NumPy to perform the filtering. This backend should always be available.
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.
A NumPy-compatible array containing the coarsest scale lowpass signal.
A tuple where each element is the complex subband coefficients for corresponding scales finest to coarsest.
(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.
An implementation of the 2D DT-CWT via NumPy. biort and qshift are the wavelets which parameterise the transform. Valid values are documented in dtcwt.dtwavexfm2().
Perform a n-level DTCWT-2D decompostion on a 2D matrix X.
Parameters: |
|
---|---|
Returns: | A dtcwt.backend.TransformDomainSignal compatible object representing the transform-domain signal |
Perform an n-level dual-tree complex wavelet (DTCWT) 2D reconstruction.
Parameters: |
|
---|---|
Returns: | A dtcwt.backend.ReconstructedSignal 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.
Provide low-level OpenCL accelerated operations. This backend requires that PyOpenCL be installed.
An interface-compatible version of dtcwt.backend.TransformDomainSignal where the initialiser arguments are assumed to by pyopencl.array.Array instances.
The attributes defined in dtcwt.backend.TransformDomainSignal 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!
The CL array containing the lowpass image.
A tuple of CL arrays containing the subband images.
(optional) Either None or a tuple of lowpass images for each scale.
An implementation of the 2D DT-CWT via OpenCL. biort and qshift are the wavelets which parameterise the transform. Valid values are documented in dtcwt.dtwavexfm2().
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.
Note
At the moment only the forward transform is accelerated. The inverse transform uses the NumPy backend.
Perform a n-level DTCWT-2D decompostion on a 2D matrix X.
Parameters: |
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Returns: | A dtcwt.backend.TransformDomainSignal 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.)
Perform an n-level dual-tree complex wavelet (DTCWT) 2D reconstruction.
Parameters: |
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Returns: | A dtcwt.backend.ReconstructedSignal 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.
Parameters: |
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Returns: | (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_subbands 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 |
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fauqueur | Geometric mean of absolute values[1] | alpha, beta |
bendale | Minimum absolute value[2] | none |
kingsbury | Cross-product of orthogonal subbands | 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.
Rescaling and re-sampling high- and low-pass subbands.
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: |
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Raises ValueError: | |
if xs and ys have differing shapes |
As sample() except that the highpass image is first phase shifted to be centred on approximately DC.
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.
As rescale() except that the highpass image is first phase shifted to be centred on approximately DC.
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: |
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If method is None, the default sampling method 'lanczos' is used. The following sampling methods are supported:
Name | Description |
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nearest | Nearest-neighbour sampling |
bilinear | Bilinear sampling |
lanczos | Lanczos sampling with window radius of 3 |
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.
A normal user should not need to call these functions but they are documented here just in case you do.
Useful utilities for testing the 2-D DTCWT with synthetic images
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().
Return v as a column vector with shape (N,1).
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.
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.
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.
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.
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.
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: |
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Returns Y: | the filtered image. |
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.
In addition to the basic 1, 2 and 3 dimensional DT-CWT, this library also supports a selection of variant transforms.
For some applications, one may prefer the subband responses to be more rotationally similar.
In the original 2-D DTCWT, the 45 and 135 degree subbands have passbands whose centre frequencies are somewhat further from the origin than those of the other four subbands. This results from the combination of two highpass 1-D wavelet filters to produce 2-D wavelets. The remaining subbands combine highpass and lowpass 1-D filters, and hence their centre frequencies are a factor of approximately sqrt(1.8) closer to the origin of the frequency plane.
The dtwavexfm2b() function employs an alternative bandpass 1-D filter in place of the highpass filter for the appropriate subbands. The image below illustrates the relevant differences in impulse and frequency responses[1].
Usage is very similar to the standard 2-D transform function, but the only supported parameters are ‘near_sym_b_bp’, ‘qshift_b_bp’. These arguments are optional, but it is best practice to include them so that your intentions are clear (and because it is easier for others to spot than the difference between 2() and 2b().
Yl, Yh = dtcwt.dtwavexfm2b(image, tfmlevel, 'near_sym_b_bp', 'qshift_b_bp')
While the Hilbert transform property of the DTCWT is preserved, perfect reconstruction is lost. However, in applications such as machine vision, where all subsequent operations on the image take place in the transform domain, this is of relatively minor importance.
For full details, refer to:
[1] N. G. Kingsbury. Rotation-invariant local feature matching with complex wavelets. In Proc. European Conference on Signal Processing (EUSIPCO), pages 901–904, 2006. 2, 18, 21
Working on the Lena image, the standard 2-D DTCWT achieves perfect reconstruction:
# Perform the standard 2-D DTCWT
Yl, Yh = dtcwt.dtwavexfm2(image, tfmlevel, 'near_sym_b', 'qshift_b')
# Perform the inverse transform
Z = dtcwt.dtwaveifm2(Yl, Yh, biort='near_sym_b', qshift='qshift_b')
# Show the error
imshow(Z-image, cmap=cm.gray)
The error signal appears to be just noise, which we can attribute to floating-point precision.
Using the modified wavelets yields the following result:
# Perform the symmetry-modified 2-D DTCWT
Yl, Yh = dtcwt.dtwavexfm2b(image, tfmlevel, 'near_sym_b_bp', 'qshift_b_bp')
# Perform the inverse transform
Z = dtcwt.dtwaveifm2b(Yl, Yh, biort='near_sym_b_bp', qshift='qshift_b_bp')
# Show the error
imshow(Z-image, cmap=cm.gray)
As we would expect, the error is more significant, but only near 45 and 135 degree edge features.
This library provides support for computing 1D, 2D and 3D dual-tree complex wavelet transforms and their inverse in Python. The interface is simple and easy to use. As a quick example, a 1D DT-CWT can be performed from the Python console in a single line:
>>> import dtcwt
>>> Yl, Yh = dtcwt.dtwavexfm([1,2,3,4], nlevels=3) # 3 levels, default wavelets
The interface is intentionally similar to the existing MATLAB dual-tree complex wavelet transform toolbox provided by Prof. Nick Kingsbury. This library is intended to ease the porting of algorithms written using the original MATLAB toolbox to Python.
The library also includes an OpenCL-based implementation of the wavelet transform which can accelerate computation even on CPU-only devices.
The features of the dtcwt library are:
The original toolbox is copyrighted and there are some restrictions on use which are outlined in the file ORIGINAL_README.txt. Aside from portions directly derived from the original MATLAB toolbox, any additions in this library and this documentation are licensed under the 2-clause BSD licence as documented in the file COPYING.txt.