52 lines
1.5 KiB
ReStructuredText
52 lines
1.5 KiB
ReStructuredText
|
Discrete and Maximum-Overlap Wavelet Transforms
|
||
|
|
================================================
|
||
|
|
|
||
|
|
The module :code:`timeseries_utils::wavelet` provides Haar and Daubechies
|
||
|
|
wavelet transforms with periodic boundary handling.
|
||
|
|
|
||
|
|
Filter banks
|
||
|
|
------------
|
||
|
|
|
||
|
|
For an orthogonal scaling filter :math:`\{h_k\}_{k=0}^{L-1}` the quadrature
|
||
|
|
mirror filter (QMF) is
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
g_k = (-1)^k\, h_{L - 1 - k},
|
||
|
|
|
||
|
|
so that :math:`\sum_k h_k = \sqrt{2}` and :math:`\sum_k g_k = 0`.
|
||
|
|
|
||
|
|
DWT (one level, periodic)
|
||
|
|
-------------------------
|
||
|
|
|
||
|
|
For an input vector :math:`x \in \mathbb{R}^N` with :math:`N` even,
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
a_n = \sum_{k=0}^{L-1} h_k\, x_{(2n + k)\bmod N},
|
||
|
|
\qquad
|
||
|
|
d_n = \sum_{k=0}^{L-1} g_k\, x_{(2n + k)\bmod N},
|
||
|
|
\qquad n = 0, \dots, N/2 - 1.
|
||
|
|
|
||
|
|
Successive levels apply the same filter to the previous approximation
|
||
|
|
:math:`a^{(j)}`.
|
||
|
|
|
||
|
|
MODWT (Maximum Overlap)
|
||
|
|
-----------------------
|
||
|
|
|
||
|
|
The MODWT does not downsample: at level :math:`j`, the filter is dilated
|
||
|
|
by inserting :math:`2^{j-1} - 1` zeros between successive taps and applied
|
||
|
|
in a periodic convolution. The result is shift-invariant.
|
||
|
|
|
||
|
|
API
|
||
|
|
---
|
||
|
|
|
||
|
|
.. code-block:: rust
|
||
|
|
|
||
|
|
pub enum WaveletFamily { Haar, Daubechies(u8) }
|
||
|
|
pub fn scaling_filter(family: WaveletFamily) -> Result<Vec<f64>>;
|
||
|
|
pub fn qmf(h: &[f64]) -> Vec<f64>;
|
||
|
|
pub fn dwt_step(x: &[f64], h: &[f64], g: &[f64]) -> (Vec<f64>, Vec<f64>);
|
||
|
|
pub fn dwt(x: &[f64], family: WaveletFamily, levels: usize) -> Result<Vec<Vec<f64>>>;
|
||
|
|
pub fn modwt_step(x: &[f64], h: &[f64], g: &[f64], level: usize) -> (Vec<f64>, Vec<f64>);
|