ucurv.meyerwavelet

Functions

meyer_wavelet(N[, engine])

Generate forward and inverse 1D Meyer wavelet filters of length N.

meyerfwd1d(img, dim[, engine])

Perform a 1-D Meyer wavelet forward transform along a specified axis.

meyerfwdmd(img[, engine])

N-dimensional forward Meyer wavelet transform.

meyerinv1d(h1, h2, dim[, engine])

Inverse 1-D Meyer wavelet transform along a chosen axis.

meyerinvmd(band[, engine])

Inverse N-dimensional Meyer wavelet transform.
ucurv.meyerwavelet.meyer_wavelet(N, engine: str = 'auto')[source]

Generate forward and inverse 1D Meyer wavelet filters of length N.

Parameters:

N (int) – Number of samples (length of the wavelet filters).

Returns:

  • f1 (ndarray) – The forward Meyer wavelet filter, computed as the square root of the FFT-shifted Meyer window on a grid of length N.

  • f2 (ndarray) – The inverse Meyer wavelet filter, computed as the square root of the unshifted Meyer window on the same grid.

Notes

This function constructs a spatial grid: x = linspace(0, 2π, N, endpoint=False) - π/2, and parameters: prm = π * [-1/3, 1/3, 2/3, 4/3]. It then calls fun_meyer(x, prm) to generate the Meyer window, applies FFT-shift for the forward filter, and takes the square roots.

ucurv.meyerwavelet.meyerfwd1d(img, dim, engine: str = 'auto')[source]

Perform a 1-D Meyer wavelet forward transform along a specified axis.

This applies Meyer filters in the frequency domain, then splits and downsamples into low and high-frequency sub-bands.

Parameters:

  • img (ndarray) – Incput array of arbitrary shape. The transform is applied along one axis; all other dimensions are preserved.

  • dim (int) – Axis (0 ≤ dim < img.ndim) along which to compute the 1-D Meyer transform.

Returns:

  • h1 (ndarray) – The low-frequency sub-band. Same shape as img except along dim, where the length is N/2 (even samples).

  • h2 (ndarray) – The high frequency sub band (odd samples); shape identical to h1.

Notes

Internally this function:

  1. Swaps axis dim with the last axis.

  2. Computes length N Meyer filters f1 and f2.

  3. FFTs the incput along that axis.

  4. Multiplies by f1/f2 and IFFTs back.

  5. Takes the real part and downsamples by 2: h1 = [..., ::2] (even), h2 = [..., 1::2] (odd).

  6. Swaps axes back to restore the original layout.

Examples

>>> import numpy as ncp
>>> x = ncp.random.randn(100, 200)
>>> low, high = meyerfwd1d(x, dim=1)
>>> low.shape, high.shape
((100, 100), (100, 100))
ucurv.meyerwavelet.meyerfwdmd(img, engine: str = 'auto')[source]

N-dimensional forward Meyer wavelet transform.

The routine applies the 1-D forward Meyer filters successively along each axis of the incput array, splitting the data into low- and high-frequency sub-bands at every step.

Parameters:

img (ndarray) – Incput array of arbitrary dimensionality. The transform is applied along axis 0, then axis 1, and so on up to img.ndim - 1.

Returns:

cband – List of length 2**N containing the sub-band arrays, where N = img.ndim. Each decomposition level doubles the number of bands by splitting every current band into its low- and high-frequency components.

Return type:

list of ndarray

Notes

  • Axis 0 - the first call to meyerfwd1d() splits img into [h1, h2].

  • Axis i ( i ≥ 1 ) - every existing band is split again along axis i, so after processing axis i the total number of bands is 2**(i + 1).

Examples

>>> import numpy as ncp
>>> x = ncp.random.randn(8, 8, 8)   # 3-D array (N = 3)
>>> subbands = meyerfwdmd(x)
>>> len(subbands)
8
ucurv.meyerwavelet.meyerinv1d(h1, h2, dim, engine: str = 'auto')[source]

Inverse 1-D Meyer wavelet transform along a chosen axis.

The routine reconstructs the original signal by interleaving the low- and high-frequency sub-bands, applying the Meyer synthesis filters in the frequency domain, and summing the results.

Parameters:

  • h1 (ndarray) – Low-frequency sub-band. Same shape as the original incput except along dim, where its length is N/2 (even samples).

  • h2 (ndarray) – High-frequency sub-band; shape identical to h1 (odd samples).

  • dim (int) – Axis along which the forward transform was taken (0 <= dim < h1.ndim).

Returns:

imrecon – Reconstructed array with the same shape and dtype as the original incput to meyerfwd1d().

Return type:

ndarray

Notes

Internally the function proceeds as follows:

  1. Swap axis dim with the last axis.

  2. Build two length-N arrays g1 and g2 by placing h1 in the even indices ([..., ::2]) and h2 in the odd indices ([..., 1::2]).

  3. Compute Meyer synthesis filters f1 and f2 of length N.

  4. FFT g1 and g2 along the last axis, multiply by f1/f2, and sum in the frequency domain.

  5. Apply the inverse FFT, take the real part, and scale by 2.

  6. Swap axes back to restore the original layout.

Examples

>>> import numpy as ncp
>>> x = ncp.random.randn(64, 128)
>>> low, high = meyerfwd1d(x, dim=1)
>>> x_rec = meyerinv1d(low, high, dim=1)
>>> ncp.allclose(x, x_rec, atol=1e-6)
True
ucurv.meyerwavelet.meyerinvmd(band, engine: str = 'auto')[source]

Inverse N-dimensional Meyer wavelet transform.

Reconstruct the original array from the 2**N sub-bands returned by meyerfwdmd() by successively merging low- and high-frequency components along each axis.

Parameters:

band (list[numpy.ndarray]) – Sequence of 2**N sub-band arrays produced by meyerfwdmd(). The order must match exactly the order returned by that function for the same incput dimensions.

Returns:

img_recon – Array with the same shape and dtype as the data that was passed to meyerfwdmd().

Return type:

numpy.ndarray

Notes

The reconstruction proceeds in reverse axis order:

  1. Start with axis N - 1; pair each low/high band and merge them with meyerinv1d() along that axis, halving the number of bands.

  2. Repeat the pairing-and-merge step for the next smaller axis.

  3. Continue until only one band remains—the fully reconstructed signal.

Examples

>>> import numpy as ncp
>>> x = ncp.random.randn(8, 8, 8)
>>> subbands = meyerfwdmd(x)
>>> x_rec = meyerinvmd(subbands)
>>> ncp.allclose(x, x_rec, atol=1e-6)
True