Proor of concept on how to fix issues PyWavelets#531, PyWavelets#535 …

…and PyWavelets#570

pywt/_cwt.py

@@ -1,8 +1,9 @@
 from math import floor, ceil
+from scipy import interpolate
 
 from ._extensions._pywt import (DiscreteContinuousWavelet, ContinuousWavelet,
                                 Wavelet, _check_dtype)
-from ._functions import integrate_wavelet, scale2frequency
+from ._functions import evaluate_wavelet, scale2frequency
 
 
 __all__ = ["cwt"]
@@ -123,13 +124,16 @@ def cwt(data, scales, wavelet, sampling_period=1., method='conv', axis=-1):
     dt_out = dt_cplx if wavelet.complex_cwt else dt
     out = np.empty((np.size(scales),) + data.shape, dtype=dt_out)
     precision = 10
-    int_psi, x = integrate_wavelet(wavelet, precision=precision)
-    int_psi = np.conj(int_psi) if wavelet.complex_cwt else int_psi
+    psi, x = evaluate_wavelet(wavelet, precision=precision)
+    psi = np.conj(psi) if wavelet.complex_cwt else psi
 
-    # convert int_psi, x to the same precision as the data
-    dt_psi = dt_cplx if int_psi.dtype.kind == 'c' else dt
-    int_psi = np.asarray(int_psi, dtype=dt_psi)
+    # convert psi, x to the same precision as the data
+    dt_psi = dt_cplx if psi.dtype.kind == 'c' else dt
+    psi = np.asarray(psi, dtype=dt_psi)
     x = np.asarray(x, dtype=data.real.dtype)
+    # FIXME: The original wavelet function could be used here, but
+    # interpolation is computationally more efficient.
+    wavefun = interpolate.interp1d(x, psi, kind='cubic', assume_sorted=True)
 
     if method == 'fft':
         size_scale0 = -1
@@ -146,41 +150,44 @@ def cwt(data, scales, wavelet, sampling_period=1., method='conv', axis=-1):
         data = data.reshape((-1, data.shape[-1]))
 
     for i, scale in enumerate(scales):
-        step = x[1] - x[0]
-        j = np.arange(scale * (x[-1] - x[0]) + 1) / (scale * step)
-        j = j.astype(int)  # floor
-        if j[-1] >= int_psi.size:
-            j = np.extract(j < int_psi.size, j)
-        int_psi_scale = int_psi[j][::-1]
+        # FIXME: Boundary points might be discarded erroneously
+        if np.sign(x[0])*np.sign(x[-1])<0:
+            # Wavelet is sampled at 0.0 if the range includes it
+            xsl = np.arange(0.0, x[0], -1.0/scale)
+            xsr = np.arange(0.0, x[-1], 1.0/scale)
+            xs = np.concatenate((xsl[:0:-1], xsr))
+        else:
+            xs = np.arange(x[0], x[-1], 1.0/scale)
+        psi_scale = wavefun(xs)[::-1]
 
         if method == 'conv':
             if data.ndim == 1:
-                conv = np.convolve(data, int_psi_scale)
+                conv = np.convolve(data, psi_scale)
             else:
                 # batch convolution via loop
                 conv_shape = list(data.shape)
-                conv_shape[-1] += int_psi_scale.size - 1
+                conv_shape[-1] += psi_scale.size - 1
                 conv_shape = tuple(conv_shape)
                 conv = np.empty(conv_shape, dtype=dt_out)
                 for n in range(data.shape[0]):
-                    conv[n, :] = np.convolve(data[n], int_psi_scale)
+                    conv[n, :] = np.convolve(data[n], psi_scale)
         else:
             # The padding is selected for:
             # - optimal FFT complexity
             # - to be larger than the two signals length to avoid circular
             #   convolution
             size_scale = next_fast_len(
-                data.shape[-1] + int_psi_scale.size - 1
+                data.shape[-1] + psi_scale.size - 1
             )
             if size_scale != size_scale0:
                 # Must recompute fft_data when the padding size changes.
                 fft_data = fftmodule.fft(data, size_scale, axis=-1)
             size_scale0 = size_scale
-            fft_wav = fftmodule.fft(int_psi_scale, size_scale, axis=-1)
+            fft_wav = fftmodule.fft(psi_scale, size_scale, axis=-1)
             conv = fftmodule.ifft(fft_wav * fft_data, axis=-1)
-            conv = conv[..., :data.shape[-1] + int_psi_scale.size - 1]
+            conv = conv[..., :data.shape[-1] + psi_scale.size - 1]
 
-        coef = - np.sqrt(scale) * np.diff(conv, axis=-1)
+        coef = conv / np.sqrt(scale)
         if out.dtype.kind != 'c':
             coef = coef.real
         # transform axis is always -1 due to the data reshape above

pywt/_functions.py

@@ -17,8 +17,8 @@
 from ._extensions._pywt import DiscreteContinuousWavelet, Wavelet, ContinuousWavelet
 
 
-__all__ = ["integrate_wavelet", "central_frequency", "scale2frequency", "qmf",
-           "orthogonal_filter_bank",
+__all__ = ["integrate_wavelet", "evaluate_wavelet", "central_frequency", 
+           "scale2frequency", "qmf", "orthogonal_filter_bank",
            "intwave", "centrfrq", "scal2frq", "orthfilt"]
 
 
@@ -119,6 +119,57 @@ def integrate_wavelet(wavelet, precision=8):
         return _integrate(psi_d, step), _integrate(psi_r, step), x
 
 
+def evaluate_wavelet(wavelet, precision=8):
+    """
+    Evaluate `psi` wavelet function between lower and upper bound.
+
+    Parameters
+    ----------
+    wavelet : Wavelet instance or str
+        Wavelet to evaluate.  If a string, should be the name of a wavelet.
+    precision : int, optional
+        Number of wavelet function points computed with Wavelet's 
+        wavefun(level=precision) method (default: 8).
+
+    Returns
+    -------
+    [psi, x] :
+        for orthogonal wavelets
+    [psi_d, psi_r, x] :
+        for other wavelets
+
+
+    Examples
+    --------
+    >>> from pywt import Wavelet, evaluate_wavelet
+    >>> wavelet1 = Wavelet('db2')
+    >>> [psi, x] = evaluate_wavelet(wavelet1, precision=5)
+    >>> wavelet2 = Wavelet('bior1.3')
+    >>> [psi_d, psi_r, x] = evaluate_wavelet(wavelet2, precision=5)
+
+    """
+
+    if type(wavelet) in (tuple, list):
+        psi, x = np.asarray(wavelet[0]), np.asarray(wavelet[1])
+        return psi, x
+    elif not isinstance(wavelet, (Wavelet, ContinuousWavelet)):
+        wavelet = DiscreteContinuousWavelet(wavelet)
+
+    functions_approximations = wavelet.wavefun(precision)
+
+    if len(functions_approximations) == 2:      # continuous wavelet
+        psi, x = functions_approximations
+        return psi, x
+
+    elif len(functions_approximations) == 3:    # orthogonal wavelet
+        phi, psi, x = functions_approximations
+        return psi, x
+
+    else:                                       # biorthogonal wavelet
+        phi_d, psi_d, phi_r, psi_r, x = functions_approximations
+        return psi_d, psi_r, x
+
+
 def central_frequency(wavelet, precision=8):
     """
     Computes the central frequency of the `psi` wavelet function.