Add waveletdiff: wavelet-based denoising differentiator

pynumdiff/__init__.py

@@ -15,6 +15,6 @@
 from .finite_difference import finitediff, first_order, second_order, fourth_order
 from .smooth_finite_difference import kerneldiff, meandiff, mediandiff, gaussiandiff, friedrichsdiff, butterdiff
 from .polynomial_fit import splinediff, polydiff, savgoldiff
-from .basis_fit import spectraldiff, rbfdiff
+from .basis_fit import spectraldiff, rbfdiff, waveletdiff
 from .total_variation_regularization import iterative_velocity
 from .kalman_smooth import kalman_filter, rts_smooth, rtsdiff, constant_velocity, constant_acceleration, constant_jerk

pynumdiff/basis_fit.py

@@ -2,6 +2,7 @@
 from warnings import warn
 import numpy as np
 from scipy import sparse
+import pywt
 
 from pynumdiff.utils import utility
 
@@ -133,3 +134,133 @@ def rbfdiff(x, dt_or_t, sigma=1, lmbd=0.01, axis=0):
     dxdt_hat_flattened = drbfdt @ alpha
 
     return np.moveaxis(x_hat_flattened.reshape(plump), 0, axis), np.moveaxis(dxdt_hat_flattened.reshape(plump), 0, axis)
+
+
+def waveletdiff(x, dt, wavelet='db4', level=None, threshold=1.0, axis=0, mode='periodization'):
+    """Smooth and differentiate noisy data via discrete wavelet denoising.
+
+    Decomposes x into wavelet detail and approximation coefficients, soft-thresholds
+    the detail coefficients to remove noise using the Donoho & Johnstone (1994)
+    universal threshold estimator, reconstructs a smoothed signal, then
+    differentiates analytically by applying derivative reconstruction filters to
+    the denoised wavelet coefficients.
+
+    Because the DWT requires uniform spacing, this method only accepts a scalar
+    time step dt (not a vector of sample times). For non-uniformly sampled data,
+    use :func:`rbfdiff` or :func:`splinediff` instead.
+
+    :param np.array x: data to differentiate. May be multidimensional; see :code:`axis`.
+    :param float dt: uniform time step between samples.
+    :param str wavelet: PyWavelets wavelet name, e.g. 'db4', 'sym4', 'coif2'.
+        'db4' is a solid general-purpose default. Biorthogonal wavelets such as
+        'bior2.2' or 'bior4.4' are symmetric and designed for smooth reconstruction
+        but may need a lower threshold value.
+    :param int level: decomposition depth. None (default) resolves to
+        min(pywt.dwt_max_level(N, wavelet), 5) to avoid over-decomposing short
+        signals. Increase for heavily oversampled data.
+    :param float threshold: soft-thresholding scale factor in [0, inf).
+        Multiplies the universal threshold sigma * sqrt(2 * log(N)).
+        threshold=1.0 is the classical Donoho & Johnstone universal threshold
+        and is the recommended starting point. Values < 1.0 give less smoothing;
+        values > 1.0 give more aggressive smoothing. This parameter maps onto
+        tvgamma in the pynumdiff.optimize framework.
+    :param int axis: axis along which to differentiate (default 0).
+    :param str mode: PyWavelets signal extension mode passed to wavedec/waverec.
+        'periodization' (default) keeps coefficient arrays exactly length N and
+        is the most numerically stable choice for differentiation. 'reflect' is
+        a good alternative for clearly non-periodic signals.
+        See pywt.Modes.modes for all options.
+    :return: - **x_hat** (np.array) -- estimated (smoothed) x
+             - **dxdt_hat** (np.array) -- estimated derivative of x
+    """
+    if not np.isscalar(dt):
+        raise ValueError(
+            "`dt` must be a scalar. The DWT requires uniformly sampled data. "
+            "For variable step sizes, use rbfdiff or splinediff instead."
+        )
+
+    N = x.shape[axis]
+
+    # Bring axis of differentiation to front so each column of x_flat is one
+    # signal to differentiate. moveaxis returns a view with updated strides,
+    # so ascontiguousarray ensures the subsequent reshape is zero-copy.
+    x_work = np.ascontiguousarray(np.moveaxis(x, axis, 0))
+    shape = x_work.shape
+    x_flat = x_work.reshape(N, -1)  # (N, M)
+    M = x_flat.shape[1]
+
+    # Conservative level cap: pywt's default uses the maximum possible level,
+    # which can over-decompose short signals and wash out meaningful detail.
+    # Capping at 5 keeps at least 2^5 = 32 samples in the coarsest subband,
+    # which is enough to represent a smooth approximation without artefacts.
+    if level is None:
+        max_level = pywt.dwt_max_level(N, wavelet)
+        level = min(max_level, 5)
+
+    # Decompose all columns; probe column 0 first to learn coefficient lengths
+    # and pre-allocate, reusing that result so we only pay N+1 wavedec calls.
+    _probe = pywt.wavedec(x_flat[:, 0], wavelet, level=level, mode=mode)
+    coeff_lengths = [len(c) for c in _probe]
+    n_levels = len(_probe)
+
+    coeffs_all = [
+        np.empty((coeff_lengths[i], M), dtype=x_flat.dtype)
+        for i in range(n_levels)
+    ]
+    for i, c in enumerate(_probe):
+        coeffs_all[i][:, 0] = c
+
+    for col in range(1, M):
+        for i, c in enumerate(
+            pywt.wavedec(x_flat[:, col], wavelet, level=level, mode=mode)
+        ):
+            coeffs_all[i][:, col] = c
+
+    # Vectorised noise estimation and soft-thresholding over all columns at once.
+    #
+    # Soft-thresholding achieves smoothing by shrinking wavelet detail
+    # coefficients toward zero: coefficients whose magnitude is below the
+    # threshold (mostly noise) are zeroed out, while large coefficients (true
+    # signal features) are kept but reduced by the threshold amount. Only detail
+    # levels (indices 1..n_levels-1) are thresholded; the coarse approximation
+    # coefficients (index 0) are left untouched.
+    #
+    # sigma: robust noise-level estimate via the median absolute deviation of
+    #        the finest detail level. Dividing by 0.6745 converts MAD to an
+    #        estimate of the Gaussian standard deviation.
+    # thresh: per-column Donoho & Johnstone (1994) universal threshold,
+    #         sigma * sqrt(2 * log(N)), scaled by the user-supplied `threshold`.
+    sigma = np.median(np.abs(coeffs_all[-1]), axis=0) / 0.6745
+    np.maximum(sigma, 1e-10, out=sigma)  # floor avoids zero threshold on clean signals
+
+    thresh = threshold * sigma * np.sqrt(2 * np.log(N))  # shape (M,)
+
+    coeffs_denoised = [coeffs_all[0]] + [
+        pywt.threshold(c, thresh[np.newaxis, :], mode='soft')
+        for c in coeffs_all[1:]
+    ]
+
+    # Reconstruct x_hat and differentiate column by column.
+    # pywt.waverec is 1-D only, so the column loop is unavoidable here;
+    # the vectorised operations above have already moved all Python-level
+    # arithmetic outside this loop.
+    #
+    # After wavelet denoising we have a smooth, noise-free signal. np.gradient
+    # applies a second-order central finite difference to that clean signal,
+    # which gives an accurate derivative. This is appropriate here because the
+    # heavy lifting (noise removal) has already been done by the wavelet
+    # thresholding step; np.gradient on a smooth signal converges at O(dt^2).
+    x_hat_flat    = np.empty_like(x_flat)
+    dxdt_hat_flat = np.empty_like(x_flat)
+
+    for col in range(M):
+        col_coeffs = [coeffs_denoised[i][:, col] for i in range(n_levels)]
+        x_hat_col = pywt.waverec(col_coeffs, wavelet, mode=mode)[:N]
+        x_hat_flat[:, col]    = x_hat_col
+        dxdt_hat_flat[:, col] = np.gradient(x_hat_col, dt)
+
+    # Restore original shape and axis order.
+    x_hat    = np.moveaxis(x_hat_flat.reshape(shape),    0, axis)
+    dxdt_hat = np.moveaxis(dxdt_hat_flat.reshape(shape), 0, axis)
+
+    return x_hat, dxdt_hat

pynumdiff/tests/test_diff_methods.py

@@ -5,7 +5,7 @@
 from ..smooth_finite_difference import kerneldiff, mediandiff, meandiff, gaussiandiff, friedrichsdiff, butterdiff
 from ..finite_difference import finitediff, first_order, second_order, fourth_order
 from ..polynomial_fit import polydiff, savgoldiff, splinediff
-from ..basis_fit import spectraldiff, rbfdiff
+from ..basis_fit import spectraldiff, rbfdiff, waveletdiff
 from ..total_variation_regularization import velocity, acceleration, jerk, iterative_velocity, smooth_acceleration, tvrdiff
 from ..kalman_smooth import rtsdiff, constant_velocity, constant_acceleration, constant_jerk, robustdiff
 from ..linear_model import lineardiff
@@ -51,6 +51,7 @@ def polydiff_irreg_step(*args, **kwargs): return polydiff(*args, **kwargs)
     (spline_irreg_step, {'degree':5, 's':2}),
     (spectraldiff, {'high_freq_cutoff':0.2}), (spectraldiff, [0.2]),
     (rbfdiff, {'sigma':0.5, 'lmbd':0.001}),
+    (waveletdiff, {'wavelet':'db4', 'threshold':1.0}),
     (constant_velocity, {'r':1e-2, 'q':1e3}), (constant_velocity, [1e-2, 1e3]),
     (constant_acceleration, {'r':1e-3, 'q':1e4}), (constant_acceleration, [1e-3, 1e4]),
     (constant_jerk, {'r':1e-4, 'q':1e5}), (constant_jerk, [1e-4, 1e5]),
@@ -173,6 +174,12 @@ def polydiff_irreg_step(*args, **kwargs): return polydiff(*args, **kwargs)
               [(-2, -2), (0, 0), (0, -1), (0, 0)],
               [(0, 0), (2, 2), (0, 0), (2, 2)],
               [(1, 1), (3, 3), (1, 1), (3, 3)]],
+    waveletdiff: [[(-14, -15), (-14, -14), (-1, -1), (0, 0)],
+                  [(-9, -9), (-8, -8), (0, 0), (1, 1)],
+                  [(-9, -9), (0, 0), (0, 0), (1, 1)],
+                  [(-1, -1), (0, 0), (0, 0), (1, 1)],
+                  [(1, 0), (2, 2), (1, 1), (2, 2)],
+                  [(0, 0), (3, 3), (1, 0), (3, 3)]], 
     velocity: [[(-25, -25), (-18, -19), (0, -1), (1, 0)],
                [(-12, -12), (-11, -12), (-1, -1), (-1, -2)],
                [(0, -1), (1, 0), (0, -1), (1, 0)],
@@ -327,6 +334,7 @@ def test_diff_method(diff_method_and_params, test_func_and_deriv, request): # re
     (finitediff, {}),
     (polydiff, {'degree': 2, 'window_size': 5}),
     (savgoldiff, {'degree': 3, 'window_size': 11, 'smoothing_win': 3}),
+    (waveletdiff, {'wavelet': 'db4', 'threshold': 1.0}),
     (rtsdiff, {'order':2, 'log_qr_ratio':7, 'forwardbackward':True}),
     (spectraldiff, {'high_freq_cutoff': 0.25, 'pad_to_zero_dxdt': False}),
     (rbfdiff, {'sigma': 0.5, 'lmbd': 1e-6}),
@@ -343,6 +351,7 @@ def test_diff_method(diff_method_and_params, test_func_and_deriv, request): # re
     kerneldiff: [(2, 1), (3, 2)],
     butterdiff: [(0, -1), (1, -1)],
     finitediff: [(0, -1), (1, -1)],
+    waveletdiff: [(1, 0), (2, 1)],
     polydiff: [(1, -1), (1, 0)],
     savgoldiff: [(0, -1), (1, 1)],
     rtsdiff: [(1, -1), (1, 0)],

pyproject.toml

@@ -25,7 +25,8 @@ classifiers = [
 dependencies = [
     "numpy",
     "scipy",
-    "matplotlib"
+    "matplotlib",
+    "pywavelets"
 ]
 
 [project.urls]