Python pywt.Wavelet() Examples

def _wavelet_coefs(data, wavelet_name='db4'):
    """Compute Discrete Wavelet Transform coefficients.

    Parameters
    ----------
    data : ndarray, shape (n_channels, n_times)

    wavelet_name : str (default: db4)
         Wavelet name (to be used with ``pywt.Wavelet``). The full list of
         Wavelet names are given by: ``[name for family in pywt.families() for
         name in pywt.wavelist(family)]``.

    Returns
    -------
    coefs : list of ndarray
         Coefficients of a DWT (Discrete Wavelet Transform). ``coefs[0]`` is
         the array of approximation coefficient and ``coefs[1:]`` is the list
         of detail coefficients.
    """
    wavelet = pywt.Wavelet(wavelet_name)
    levdec = min(pywt.dwt_max_level(data.shape[-1], wavelet.dec_len), 6)
    coefs = pywt.wavedec(data, wavelet=wavelet, level=levdec)
    return coefs 

def test_swt_dtypes():
    wavelet = pywt.Wavelet('haar')
    for dt_in, dt_out in zip(dtypes_in, dtypes_out):
        errmsg = "wrong dtype returned for {0} input".format(dt_in)

        # swt
        x = np.ones(8, dtype=dt_in)
        (cA2, cD2), (cA1, cD1) = pywt.swt(x, wavelet, level=2)
        assert_(cA2.dtype == cD2.dtype == cA1.dtype == cD1.dtype == dt_out,
                "swt: " + errmsg)

        # swt2
        with warnings.catch_warnings():
            warnings.simplefilter('ignore', FutureWarning)
            x = np.ones((8, 8), dtype=dt_in)
            cA, (cH, cV, cD) = pywt.swt2(x, wavelet, level=1)[0]
            assert_(cA.dtype == cH.dtype == cV.dtype == cD.dtype == dt_out,
                    "swt2: " + errmsg) 

def test_waverecn_coeff_reshape_odd():
    # verify round trip is correct:
    #   wavedecn - >coeffs_to_array-> array_to_coeffs -> waverecn
    rng = np.random.RandomState(1234)
    x1 = rng.randn(35, 33)
    for mode in pywt.Modes.modes:
        for wave in ['haar', ]:
            w = pywt.Wavelet(wave)
            maxlevel = pywt.dwt_max_level(np.min(x1.shape), w.dec_len)
            if maxlevel == 0:
                continue
            coeffs = pywt.wavedecn(x1, w, mode=mode)
            coeff_arr, coeff_slices = pywt.coeffs_to_array(coeffs)
            coeffs2 = pywt.array_to_coeffs(coeff_arr, coeff_slices)
            x1r = pywt.waverecn(coeffs2, w, mode=mode)
            # truncate reconstructed values to original shape
            x1r = x1r[[slice(s) for s in x1.shape]]
            assert_allclose(x1, x1r, rtol=1e-4, atol=1e-4) 

def test_wavedecn_coeff_reshape_even():
    # verify round trip is correct:
    #   wavedecn - >coeffs_to_array-> array_to_coeffs -> waverecn
    # This is done for wavedec{1, 2, n}
    rng = np.random.RandomState(1234)
    params = {'wavedec': {'d': 1, 'dec': pywt.wavedec, 'rec': pywt.waverec},
              'wavedec2': {'d': 2, 'dec': pywt.wavedec2, 'rec': pywt.waverec2},
              'wavedecn': {'d': 3, 'dec': pywt.wavedecn, 'rec': pywt.waverecn}}
    N = 28
    for f in params:
        x1 = rng.randn(*([N] * params[f]['d']))
        for mode in pywt.Modes.modes:
            for wave in wavelist:
                w = pywt.Wavelet(wave)
                maxlevel = pywt.dwt_max_level(np.min(x1.shape), w.dec_len)
                if maxlevel == 0:
                    continue

                coeffs = params[f]['dec'](x1, w, mode=mode)
                coeff_arr, coeff_slices = pywt.coeffs_to_array(coeffs)
                coeffs2 = pywt.array_to_coeffs(coeff_arr, coeff_slices,
                                               output_format=f)
                x1r = params[f]['rec'](coeffs2, w, mode=mode)

                assert_allclose(x1, x1r, rtol=1e-4, atol=1e-4) 

def test_3D_reconstruct_complex():
    # All dimensions even length so `take` does not need to be specified
    data = np.array([
        [[0, 4, 1, 5, 1, 4],
         [0, 5, 26, 3, 2, 1],
         [5, 8, 2, 33, 4, 9],
         [2, 5, 19, 4, 19, 1]],
        [[1, 5, 1, 2, 3, 4],
         [7, 12, 6, 52, 7, 8],
         [2, 12, 3, 52, 6, 8],
         [5, 2, 6, 78, 12, 2]]])
    data = data + 1j

    wavelet = pywt.Wavelet('haar')
    d = pywt.dwtn(data, wavelet)
    # idwtn creates even-length shapes (2x dwtn size)
    original_shape = [slice(None, s) for s in data.shape]
    assert_allclose(data, pywt.idwtn(d, wavelet)[original_shape],
                    rtol=1e-13, atol=1e-13) 

def test_stride():
    wavelet = pywt.Wavelet('haar')

    for dtype in ('float32', 'float64'):
        data = np.array([[0, 4, 1, 5, 1, 4],
                         [0, 5, 6, 3, 2, 1],
                         [2, 5, 19, 4, 19, 1]],
                        dtype=dtype)

        for mode in pywt.Modes.modes:
            expected = pywt.dwtn(data, wavelet)
            strided = np.ones((3, 12), dtype=data.dtype)
            strided[::-1, ::2] = data
            strided_dwtn = pywt.dwtn(strided[::-1, ::2], wavelet)
            for key in expected.keys():
                assert_allclose(strided_dwtn[key], expected[key]) 

def test_idwtn_missing():
    # Test to confirm missing data behave as zeroes
    data = np.array([
        [0, 4, 1, 5, 1, 4],
        [0, 5, 6, 3, 2, 1],
        [2, 5, 19, 4, 19, 1]])

    wavelet = pywt.Wavelet('haar')

    coefs = pywt.dwtn(data, wavelet)

    # No point removing zero, or all
    for num_missing in range(1, len(coefs)):
        for missing in combinations(coefs.keys(), num_missing):
            missing_coefs = coefs.copy()
            for key in missing:
                del missing_coefs[key]
            LL = missing_coefs.get('aa', None)
            HL = missing_coefs.get('da', None)
            LH = missing_coefs.get('ad', None)
            HH = missing_coefs.get('dd', None)

            assert_allclose(pywt.idwt2((LL, (HL, LH, HH)), wavelet),
                            pywt.idwtn(missing_coefs, 'haar'), atol=1e-15) 

def test_error_on_invalid_keys():
    data = np.array([
        [0, 4, 1, 5, 1, 4],
        [0, 5, 6, 3, 2, 1],
        [2, 5, 19, 4, 19, 1]])

    wavelet = pywt.Wavelet('haar')

    LL, (HL, LH, HH) = pywt.dwt2(data, wavelet)

    # unexpected key
    d = {'aa': LL, 'da': HL, 'ad': LH, 'dd': HH, 'ff': LH}
    assert_raises(ValueError, pywt.idwtn, d, wavelet)

    # mismatched key lengths
    d = {'a': LL, 'da': HL, 'ad': LH, 'dd': HH}
    assert_raises(ValueError, pywt.idwtn, d, wavelet) 

def test_custom_wavelet():
    haar_custom1 = pywt.Wavelet('Custom Haar Wavelet',
                                filter_bank=_CustomHaarFilterBank())
    haar_custom1.orthogonal = True
    haar_custom1.biorthogonal = True

    val = np.sqrt(2) / 2
    filter_bank = ([val]*2, [-val, val], [val]*2, [val, -val])
    haar_custom2 = pywt.Wavelet('Custom Haar Wavelet',
                                filter_bank=filter_bank)

    # check expected default wavelet properties
    assert_(~haar_custom2.orthogonal)
    assert_(~haar_custom2.biorthogonal)
    assert_(haar_custom2.symmetry == 'unknown')
    assert_(haar_custom2.family_name == '')
    assert_(haar_custom2.short_family_name == '')
    assert_(haar_custom2.vanishing_moments_phi == 0)
    assert_(haar_custom2.vanishing_moments_psi == 0)

    # Some properties can be set by the user
    haar_custom2.orthogonal = True
    haar_custom2.biorthogonal = True 

def check_coefficients(wavelet):
    epsilon = 5e-11
    level = 10
    w = pywt.Wavelet(wavelet)
    # Lowpass filter coefficients sum to sqrt2
    res = np.sum(w.dec_lo)-np.sqrt(2)
    msg = ('[RMS_REC > EPSILON] for Wavelet: %s, rms=%.3g' % (wavelet, res))
    assert_(res < epsilon, msg=msg)
    # sum even coef = sum odd coef = 1 / sqrt(2)
    res = np.sum(w.dec_lo[::2])-1./np.sqrt(2)
    msg = ('[RMS_REC > EPSILON] for Wavelet: %s, rms=%.3g' % (wavelet, res))
    assert_(res < epsilon, msg=msg)

    res = np.sum(w.dec_lo[1::2])-1./np.sqrt(2)
    msg = ('[RMS_REC > EPSILON] for Wavelet: %s, rms=%.3g' % (wavelet, res))
    assert_(res < epsilon, msg=msg)
    # Highpass filter coefficients sum to zero
    res = np.sum(w.dec_hi)
    msg = ('[RMS_REC > EPSILON] for Wavelet: %s, rms=%.3g' % (wavelet, res))
    assert_(res < epsilon, msg=msg) 

def test_3D_reconstruct():
    data = np.array([
        [[0, 4, 1, 5, 1, 4],
         [0, 5, 26, 3, 2, 1],
         [5, 8, 2, 33, 4, 9],
         [2, 5, 19, 4, 19, 1]],
        [[1, 5, 1, 2, 3, 4],
         [7, 12, 6, 52, 7, 8],
         [2, 12, 3, 52, 6, 8],
         [5, 2, 6, 78, 12, 2]]])

    wavelet = pywt.Wavelet('haar')
    for mode in pywt.Modes.modes:
        d = pywt.dwtn(data, wavelet, mode=mode)
        assert_allclose(data, pywt.idwtn(d, wavelet, mode=mode),
                        rtol=1e-13, atol=1e-13) 

def test_wavelet_properties():
    w = pywt.Wavelet('db3')

    # Name
    assert_(w.name == 'db3')
    assert_(w.short_family_name == 'db')
    assert_(w.family_name, 'Daubechies')

    # String representation
    fields = ('Family name', 'Short name', 'Filters length', 'Orthogonal',
              'Biorthogonal', 'Symmetry')
    for field in fields:
        assert_(field in str(w))

    # Filter coefficients
    dec_lo = [0.03522629188210, -0.08544127388224, -0.13501102001039,
              0.45987750211933, 0.80689150931334, 0.33267055295096]
    dec_hi = [-0.33267055295096, 0.80689150931334, -0.45987750211933,
              -0.13501102001039, 0.08544127388224, 0.03522629188210]
    rec_lo = [0.33267055295096, 0.80689150931334, 0.45987750211933,
              -0.13501102001039, -0.08544127388224, 0.03522629188210]
    rec_hi = [0.03522629188210, 0.08544127388224, -0.13501102001039,
              -0.45987750211933, 0.80689150931334, -0.33267055295096]
    assert_allclose(w.dec_lo, dec_lo)
    assert_allclose(w.dec_hi, dec_hi)
    assert_allclose(w.rec_lo, rec_lo)
    assert_allclose(w.rec_hi, rec_hi)

    assert_(len(w.filter_bank) == 4)

    # Orthogonality
    assert_(w.orthogonal)
    assert_(w.biorthogonal)

    # Symmetry
    assert_(w.symmetry)

    # Vanishing moments
    assert_(w.vanishing_moments_phi == 0)
    assert_(w.vanishing_moments_psi == 3) 

def test_swt2_axes():
    atol = 1e-14
    current_wavelet = pywt.Wavelet('db2')
    input_length_power = int(np.ceil(np.log2(max(
        current_wavelet.dec_len,
        current_wavelet.rec_len))))
    input_length = 2**(input_length_power)
    X = np.arange(input_length**2).reshape(input_length, input_length)
    with warnings.catch_warnings():
        warnings.simplefilter('ignore', FutureWarning)
        (cA1, (cH1, cV1, cD1)) = pywt.swt2(X, current_wavelet, level=1)[0]
        # opposite order
        (cA2, (cH2, cV2, cD2)) = pywt.swt2(X, current_wavelet, level=1,
                                           axes=(1, 0))[0]
        assert_allclose(cA1, cA2, atol=atol)
        assert_allclose(cH1, cV2, atol=atol)
        assert_allclose(cV1, cH2, atol=atol)
        assert_allclose(cD1, cD2, atol=atol)

        # duplicate axes not allowed
        assert_raises(ValueError, pywt.swt2, X, current_wavelet, 1,
                      axes=(0, 0))
        # too few axes
        assert_raises(ValueError, pywt.swt2, X, current_wavelet, 1, axes=(0, )) 

def compute_wavelet_feature_vector(image, wavelet='db6'):
    image_np = np.array(image)
    rgb = [image_np[:, :, i] for i in (0, 1, 2)]

    if isinstance(wavelet, basestring):
        wavelet = pywt.Wavelet(wavelet)

    feature_vector = []
    for c in rgb:
        level = pywt.dwt_max_level(min(c.shape[0], c.shape[1]), wavelet.dec_len)
        levels = pywt.wavedec2(c, wavelet, mode='sym', level=level)
        for coeffs in levels:
            if not isinstance(coeffs, tuple):
                coeffs = (coeffs,)
            for w in coeffs:
                w_flat = w.flatten()
                feature_vector += [float(np.mean(w_flat)), float(np.std(w_flat))]

    return feature_vector 

def test_accuracy_pymatbridge():
    rstate = np.random.RandomState(1234)
    # max RMSE (was 1.0e-10, is reduced to 5.0e-5 due to different coefficents)
    epsilon = 5.0e-5
    epsilon_pywt_coeffs = 1.0e-10
    mlab.start()
    try:
        for wavelet in wavelets:
            w = pywt.Wavelet(wavelet)
            mlab.set_variable('wavelet', wavelet)
            for N in _get_data_sizes(w):
                data = rstate.randn(N)
                mlab.set_variable('data', data)
                for pmode, mmode in modes:
                    ma, md = _compute_matlab_result(data, wavelet, mmode)
                    yield _check_accuracy, data, w, pmode, ma, md, wavelet, epsilon
                    ma, md = _load_matlab_result_pywt_coeffs(data, wavelet, mmode)
                    yield _check_accuracy, data, w, pmode, ma, md, wavelet, epsilon_pywt_coeffs

    finally:
        mlab.stop() 

def _compute_matlab_result(data, wavelet, mmode):
    """ Compute the result using MATLAB.

    This function assumes that the Matlab variables `wavelet` and `data` have
    already been set externally.
    """
    if np.any((wavelet == np.array(['coif6', 'coif7', 'coif8', 'coif9', 'coif10', 'coif11', 'coif12', 'coif13', 'coif14', 'coif15', 'coif16', 'coif17'])),axis=0):
        w = pywt.Wavelet(wavelet)
        mlab.set_variable('Lo_D', w.dec_lo)
        mlab.set_variable('Hi_D', w.dec_hi)
        mlab_code = ("[ma, md] = dwt(data, Lo_D, Hi_D, 'mode', '%s');" % mmode)
    else:
        mlab_code = "[ma, md] = dwt(data, wavelet, 'mode', '%s');" % mmode
    res = mlab.run_code(mlab_code)
    if not res['success']:
        raise RuntimeError("Matlab failed to execute the provided code. "
                           "Check that the wavelet toolbox is installed.")
    # need np.asarray because sometimes the output is a single float64
    ma = np.asarray(mlab.get_variable('ma'))
    md = np.asarray(mlab.get_variable('md'))
    return ma, md 

def test_dwt_wavelet_kwd():
    x = np.array([3, 7, 1, 1, -2, 5, 4, 6])
    w = pywt.Wavelet('sym3')
    cA, cD = pywt.dwt(x, wavelet=w, mode='constant')
    cA_expect = [4.38354585, 3.80302657, 7.31813271, -0.58565539, 4.09727044,
                 7.81994027]
    cD_expect = [-1.33068221, -2.78795192, -3.16825651, -0.67715519,
                 -0.09722957, -0.07045258]
    assert_allclose(cA, cA_expect)
    assert_allclose(cD, cD_expect) 

def test_intwave_orthogonal():
    w = pywt.Wavelet('db1')
    int_psi, x = pywt.integrate_wavelet(w, precision=12)
    ix = x < 0.5
    # For x < 0.5, the integral is equal to x
    assert_allclose(int_psi[ix], x[ix])
    # For x > 0.5, the integral is equal to (1 - x)
    # Ignore last point here, there x > 1 and something goes wrong
    assert_allclose(int_psi[~ix][:-1], 1 - x[~ix][:-1], atol=1e-10) 

def test_scal2frq_scale():
    scale = 2
    w = pywt.Wavelet('db1')
    expected = 1. / scale
    result = pywt.scale2frequency(w, scale, precision=12)
    assert_almost_equal(result, expected, decimal=3) 

def test_dwt_coeff_len():
    x = np.array([3, 7, 1, 1, -2, 5, 4, 6])
    w = pywt.Wavelet('sym3')
    ln_modes = [pywt.dwt_coeff_len(len(x), w.dec_len, mode) for mode in
                pywt.Modes.modes]

    expected_result = [6, ] * len(pywt.Modes.modes)
    expected_result[pywt.Modes.modes.index('periodization')] = 4

    assert_allclose(ln_modes, expected_result)
    ln_modes = [pywt.dwt_coeff_len(len(x), w, mode) for mode in
                pywt.Modes.modes]
    assert_allclose(ln_modes, expected_result) 

def test_dwdtn_idwtn_allwavelets():
    rstate = np.random.RandomState(1234)
    r = rstate.randn(16, 16)
    # test 2D case only for all wavelet types
    wavelist = pywt.wavelist()
    if 'dmey' in wavelist:
        wavelist.remove('dmey')
    for wavelet in wavelist:
        if isinstance(pywt.DiscreteContinuousWavelet(wavelet), pywt.Wavelet):
            for mode in pywt.Modes.modes:
                coeffs = pywt.dwtn(r, wavelet, mode=mode)
                assert_allclose(pywt.idwtn(coeffs, wavelet, mode=mode),
                                r, rtol=1e-7, atol=1e-7) 

def recon(self,z1):
        """
        Wavelet reconstruction:  coefficients -> image
        """
        coeffs = pywt.array_to_coeffs(z1, self.coeff_slices, \
            output_format='wavedec2')
        z0 = pywt.waverec2(coeffs, wavelet=self.wavelet, mode=self.mode)
        return z0 

def test_idwtn_idwt2():
    data = np.array([
        [0, 4, 1, 5, 1, 4],
        [0, 5, 6, 3, 2, 1],
        [2, 5, 19, 4, 19, 1]])

    wavelet = pywt.Wavelet('haar')

    LL, (HL, LH, HH) = pywt.dwt2(data, wavelet)
    d = {'aa': LL, 'da': HL, 'ad': LH, 'dd': HH}

    for mode in pywt.Modes.modes:
        assert_allclose(pywt.idwt2((LL, (HL, LH, HH)), wavelet, mode=mode),
                        pywt.idwtn(d, wavelet, mode=mode),
                        rtol=1e-14, atol=1e-14) 

def test_idwtn_idwt2_complex():
    data = np.array([
        [0, 4, 1, 5, 1, 4],
        [0, 5, 6, 3, 2, 1],
        [2, 5, 19, 4, 19, 1]])
    data = data + 1j
    wavelet = pywt.Wavelet('haar')

    LL, (HL, LH, HH) = pywt.dwt2(data, wavelet)
    d = {'aa': LL, 'da': HL, 'ad': LH, 'dd': HH}

    for mode in pywt.Modes.modes:
        assert_allclose(pywt.idwt2((LL, (HL, LH, HH)), wavelet, mode=mode),
                        pywt.idwtn(d, wavelet, mode=mode),
                        rtol=1e-14, atol=1e-14) 

def test_dwt2_idwt2_dtypes():
    wavelet = pywt.Wavelet('haar')
    for dt_in, dt_out in zip(dtypes_in, dtypes_out):
        x = np.ones((4, 4), dtype=dt_in)
        errmsg = "wrong dtype returned for {0} input".format(dt_in)

        cA, (cH, cV, cD) = pywt.dwt2(x, wavelet)
        assert_(cA.dtype == cH.dtype == cV.dtype == cD.dtype,
                "dwt2: " + errmsg)

        x_roundtrip = pywt.idwt2((cA, (cH, cV, cD)), wavelet)
        assert_(x_roundtrip.dtype == dt_out, "idwt2: " + errmsg) 

def test_dwtn_idwtn_dtypes():
    wavelet = pywt.Wavelet('haar')
    for dt_in, dt_out in zip(dtypes_in, dtypes_out):
        x = np.ones((4, 4), dtype=dt_in)
        errmsg = "wrong dtype returned for {0} input".format(dt_in)

        coeffs = pywt.dwtn(x, wavelet)
        for k, v in coeffs.items():
            assert_(v.dtype == dt_out, "dwtn: " + errmsg)

        x_roundtrip = pywt.idwtn(coeffs, wavelet)
        assert_(x_roundtrip.dtype == dt_out, "idwtn: " + errmsg) 

def test_dwt2_dimension_error():
    data = np.ones(16)
    wavelet = pywt.Wavelet('haar')

    # wrong number of input dimensions
    assert_raises(ValueError, pywt.dwt2, data, wavelet)

    # too many axes
    data2 = np.ones((8, 8))
    assert_raises(ValueError, pywt.dwt2, data2, wavelet, axes=(0, 1, 1)) 

def dt_max_level(data, first_stage, wavelet, axis=-1):
    """
    Returns the maximum decomposition level possible from the dual-tree complex wavelet transform.

    Parameters
    ----------
    data : ndarray
        Input data. Can be of any dimension.
    first_stage : str
        Wavelet to use for the first stage. See :func:`skued.available_first_stage_filters` for a list of suitable arguments
    wavelet : str
        Wavelet to use in stages > 1. Must be appropriate for the dual-tree complex wavelet transform.
        See :func:`skued.available_dt_filters` for possible values.
    axis : int, optional
        Axis over which to compute the transform. Default is -1
        
    Returns
    -------
    max_level : int
        Maximum decomposition level.
    """
    real_wavelet, imag_wavelet = dualtree_wavelet(wavelet)
    return dwt_max_level(
        data_len=data.shape[axis],
        filter_len=max([real_wavelet.dec_len, imag_wavelet.dec_len]),
    ) 

def _single_tree_analysis_1d(data, first_stage, wavelet, level, mode, axis):
    """
    Single tree of the forward dual-tree complex wavelet transform.
    
    Parameters
    ----------
    data : ndarray, ndim 1
    first_stage : Wavelet object
    wavelet : 2-tuple of Wavelet object
    level : int
    mode : str
    axis : int

    Returns
    -------
    [cA_n, cD_n, cD_n-1, ..., cD2, cD1] : list
        Ordered list of coefficients arrays
        where `n` denotes the level of decomposition. The first element
        (`cA_n`) of the result is approximation coefficients array and the
        following elements (`cD_n` - `cD_1`) are details coefficients arrays.
    """
    approx, first_detail = dwt(data=data, wavelet=first_stage, mode=mode, axis=axis)
    # Use of a deque vs. a list is because deque.appendleft is O(1)
    coeffs_list = deque([first_detail])
    for i, wav in zip(range(level - 1), cycle(wavelet)):
        approx, detail = dwt(data=approx, wavelet=wav, mode=mode, axis=axis)
        coeffs_list.appendleft(detail)

    # Format list ot be compatible to PyWavelet's format. See pywt.wavedec source.
    coeffs_list.appendleft(approx)
    return coeffs_list 

def _single_tree_synthesis_1d(coeffs, first_stage, wavelet, mode, axis):
    """
    Single tree of the inverse dual-tree complex wavelet transform.

    Parameters
    ----------
    coeffs : list
    first_stage : Wavelet object
    wavelet : 2-tuple of Wavelet objects
    mode : str
    axis : int

    Returns
    -------
    reconstructed : ndarray, ndim 1
    """
    # Determine the level except first stage:
    # The order of wavelets depends on whether
    # the level is even or odd.
    level = len(coeffs) - 1
    approx, detail_coeffs, first_stage_detail = coeffs[0], coeffs[1:-1], coeffs[-1]

    # Set the right alternating order between real and imag wavelet
    if level % 2 == 1:
        wavelet = reversed(wavelet)

    # In the case of level = 1, coeffs[1:-1] is an empty list. Then, the following
    # loop is not run since zip() iterates as long as the shortest iterable.
    for detail, wav in zip(detail_coeffs, cycle(wavelet)):
        approx = _normalize_size_axis(approx, detail, axis=axis)
        approx = idwt(cA=approx, cD=detail, wavelet=wav, mode=mode, axis=axis)

    approx = _normalize_size_axis(approx, first_stage_detail, axis=axis)
    return idwt(
        cA=approx, cD=first_stage_detail, wavelet=first_stage, mode=mode, axis=axis
    )