# -*- coding: utf-8 -*-
import numpy as np
from . import Filter # prevent circular import in Python < 3.5
[docs]class Held(Filter):
r"""Design 2 filters with the Held construction (tight frame).
This function create a parseval filterbank of :math:`2` filters.
The low-pass filter is defined by the function
.. math:: f_{l}=\begin{cases} 1 & \mbox{if }x\leq a\\
\sin\left(2\pi\mu\left(\frac{x}{8a}\right)\right) & \mbox{if }a<x\leq2a\\
0 & \mbox{if }x>2a \end{cases}
with
.. math:: \mu(x) = -1+24x-144*x^2+256*x^3
The high pass filter is adaptated to obtain a tight frame.
Parameters
----------
G : graph
a : float
See equation above for this parameter
The spectrum is scaled between 0 and 2 (default = 2/3)
Examples
--------
Filter bank's representation in Fourier and time (ring graph) domains.
>>> import matplotlib.pyplot as plt
>>> G = graphs.Ring(N=20)
>>> G.estimate_lmax()
>>> G.set_coordinates('line1D')
>>> g = filters.Held(G)
>>> s = g.localize(G.N // 2)
>>> fig, axes = plt.subplots(1, 2)
>>> g.plot(ax=axes[0])
>>> G.plot_signal(s, ax=axes[1])
"""
def __init__(self, G, a=2./3, **kwargs):
g = [lambda x: held(x * (2./G.lmax), a)]
g.append(lambda x: np.real(np.sqrt(1 - (held(x * (2./G.lmax), a))
** 2)))
def held(val, a):
y = np.empty(np.shape(val))
l1 = a
l2 = 2 * a
mu = lambda x: -1. + 24.*x - 144.*x**2 + 256*x**3
r1ind = (val >= 0) * (val < l1)
r2ind = (val >= l1) * (val < l2)
r3ind = (val >= l2)
y[r1ind] = 1
y[r2ind] = np.sin(2*np.pi*mu(val[r2ind]/(8.*a)))
y[r3ind] = 0
return y
super(Held, self).__init__(G, g, **kwargs)