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fig6a_irreg_time_samp.py
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fig6a_irreg_time_samp.py
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from __future__ import division
import datetime
import time
import os
import numpy as np
from scipy import linalg
from matplotlib import rcParams
import matplotlib.pyplot as plt
# import bokeh.plotting as b_plt
# from bokeh.io import vplot, hplot, output_file, show
# from bokeh.models.tools import WheelZoomTool
from alg_tools_1d import dirac_recon_time, periodicSinc, distance
# for latex rendering
os.environ['PATH'] = os.environ['PATH'] + ':/usr/texbin' + \
':/opt/local/bin' + ':/Library/TeX/texbin/'
rcParams['text.usetex'] = True
rcParams['text.latex.unicode'] = True
if __name__ == '__main__':
# various experiment settings
save_fig = True # save figure or not
fig_format = r'png' # file type used to save the figure, e.g., pdf, png, etc.
stop_cri = 'max_iter' # stopping criteria: 1) mse; or 2) max_iter
web_fig = False # generate html file for the figures
K = 5 # number of Diracs
M = K
tau = 1 # period of the Dirac stream
# number of time domain samples
L = (2 * M + 1)
Tmax = tau / L # the average sampling step size (had we used a uniform sampling setup)
# generate the random sampling time instances
t_samp = np.arange(0, L, dtype=float) * Tmax
t_samp += np.sign(np.random.randn(L)) * np.random.rand(L) * Tmax / 2.
# round t_samp to [0, tau)
t_samp -= np.floor(t_samp / tau) * tau
# generate parameters for the periodic stream of Diracs
B = (2. * M + 1.) / tau # bandwidth of the sampling filter
# amplitudes of the Diracs
ak = np.sign(np.random.randn(K)) * (1 + (np.random.rand(K) - 0.5) / 1.)
# locations of the Diracs
if K == 1:
tk = np.random.rand()
else:
a = 4. / L
uk = np.random.exponential(scale=1. / K, size=(K - 1, 1))
tk = np.cumsum(a + (1. - K * a) * (1 - 0.1 * np.random.rand()) / uk.sum() * uk)
tk = np.sort(np.hstack((np.random.rand() * tk[0] / 2., tk)) + (1 - tk[-1]) / 2.) * tau
# save Dirac parameter
time_stamp = datetime.datetime.now().strftime("%d-%m_%H_%M")
file_name = './data/Dirac_Data_' + time_stamp + '.npz'
np.savez(file_name, tk=tk, ak=ak, K=K, time_stamp=time_stamp)
# load saved data
time_stamp = r'20-12_02_22'
stored_param = np.load(r'./data/Dirac_Data_' + time_stamp + r'.npz')
tk = stored_param['tk']
ak = stored_param['ak']
print(r'time stamp: ' + time_stamp +
'\n=======================================\n')
# compute the noiseless Fourier series coefficients
tk_grid, m_grid_gt = np.meshgrid(tk, np.arange(-np.floor(B * tau / 2.), 1 + np.floor(B * tau / 2.)))
x_hat_noiseless = 1. / tau * np.dot(np.exp(-2j * np.pi / tau * m_grid_gt * tk_grid), ak)
m_grid, t_samp_grid = np.meshgrid(np.arange(-np.floor(B * tau / 2.), 1 + np.floor(B * tau / 2.)), t_samp)
# build the linear transformation matrix that links x_hat with the samples
G = 1. / B * np.exp(2j * np.pi / tau * m_grid * t_samp_grid)
y_ell_noiseless = np.real(np.dot(G, x_hat_noiseless))
# add noise
P = float('inf')
noise = np.random.randn(L)
noise = noise / linalg.norm(noise) * linalg.norm(y_ell_noiseless) * 10 ** (-P / 20.)
y_ell = y_ell_noiseless + noise
# noise energy, in the noiseless case 1e-10 is considered as 0
noise_level = np.max([1e-10, linalg.norm(noise)])
max_ini = 100 # maximum number of random initialisations
# FRI reconstruction
xhat_recon, min_error, c_opt, ini = dirac_recon_time(G, y_ell, K, noise_level, max_ini, stop_cri)
print(r'Noise level: {0:.2e}'.format(noise_level))
print(r'Minimum approximation error |a - Gb|_2: {0:.2e}'.format(min_error))
# reconstruct Diracs' locations tk
z = np.roots(c_opt)
z = z / np.abs(z)
tk_recon = np.real(tau * 1j / (2 * np.pi) * np.log(z))
tk_recon = np.sort(tk_recon - np.floor(tk_recon / tau) * tau)
# reconstruct amplitudes ak
Phi_recon = periodicSinc(np.pi * B * (np.reshape(t_samp, (-1, 1), order='F') -
np.reshape(tk_recon, (1, -1), order='F')),
B * tau)
ak_recon = np.real(linalg.lstsq(Phi_recon, y_ell)[0])
# location estimation error
t_error = distance(tk_recon, tk)[0]
# plot reconstruction
plt.close()
fig = plt.figure(num=1, figsize=(5.5, 2.5), dpi=90)
# sub-figure 1
ax1 = plt.axes([0.125, 0.59, 0.85, 0.31])
markerline211_1, stemlines211_1, baseline211_1 = \
ax1.stem(tk, ak, label='Original Diracs')
plt.setp(stemlines211_1, linewidth=1.5, color=[0, 0.447, 0.741])
plt.setp(markerline211_1, marker='^', linewidth=1.5, markersize=8,
markerfacecolor=[0, 0.447, 0.741], mec=[0, 0.447, 0.741])
plt.setp(baseline211_1, linewidth=0)
markerline211_2, stemlines211_2, baseline211_2 = \
plt.stem(tk_recon, ak_recon, label='Estimated Diracs')
plt.setp(stemlines211_2, linewidth=1.5, color=[0.850, 0.325, 0.098])
plt.setp(markerline211_2, marker='*', linewidth=1.5, markersize=10,
markerfacecolor=[0.850, 0.325, 0.098], mec=[0.850, 0.325, 0.098])
plt.setp(baseline211_2, linewidth=0)
plt.axhline(0, color='k')
plt.xlim([0, tau])
plt.ylim([1.17 * np.min(np.concatenate((ak, ak_recon, np.array(0)[np.newaxis]))),
1.17 * np.max(np.concatenate((ak, ak_recon, np.array(0)[np.newaxis])))])
# plt.xlabel(r'$t$', fontsize=12)
plt.ylabel('amplitudes', fontsize=12)
ax1.yaxis.set_label_coords(-0.095, 0.5)
plt.legend(numpoints=1, loc=0, fontsize=9, framealpha=0.3,
handletextpad=.2, columnspacing=0.6, labelspacing=0.05, ncol=2)
t_error_pow = np.int(np.floor(np.log10(t_error)))
if np.isinf(P):
plt.title(r'$K={0}$, $L={1}$, '
r'$\mbox{{SNR}}=\mbox{{inf }}$dB, '
r'$t_{{\mbox{{\footnotesize err}}}}={2:.2f}\times10^{other}$'.format(repr(K), repr(L),
t_error / 10 ** t_error_pow,
other='{' + str(
t_error_pow) + '}'),
fontsize=12)
else:
plt.title(r'$K={0}$, $L={1}$, '
r'$\mbox{{SNR}}={2}$dB, '
r'$t_{{\mbox{{\footnotesize err}}}}={3:.2f}\times10^{other}$'.format(repr(K), repr(L), repr(P),
t_error / 10 ** t_error_pow,
other='{' + str(
t_error_pow) + '}'),
fontsize=12)
# sub-figure 2
t_plt = np.linspace(0, tau, num=np.max([10 * L, 1000]))
m_plt_grid, t_plt_grid = np.meshgrid(np.arange(-np.floor(B * tau / 2.),
1 + np.floor(B * tau / 2.)),
t_plt)
G_plt = 1. / B * np.exp(2j * np.pi / tau * m_plt_grid * t_plt_grid)
y_plt = np.real(np.dot(G_plt, x_hat_noiseless)) # for plotting purposes only
ax2 = plt.axes([0.125, 0.18, 0.85, 0.31])
line212_1 = ax2.plot(t_plt, y_plt, label='Ground Truth')
plt.setp(line212_1, linestyle='-', color=[0, 0.447, 0.741], linewidth=1)
line212_2 = ax2.plot(t_samp, y_ell, label='Samples')
plt.setp(line212_2, marker='.', linestyle='None', markersize=5, color=[0.850, 0.325, 0.098])
plt.ylim([1.05 * np.min(np.concatenate((y_plt, y_ell))),
1.05 * np.max(np.concatenate((y_plt, y_ell)))])
plt.ylabel(r'$x(t) * \mathrm{{sinc}}(B t)$', fontsize=12)
plt.xlabel(r'$t$', fontsize=12)
ax2.xaxis.set_label_coords(0.5, -0.21)
ax2.yaxis.set_label_coords(-0.095, 0.5)
plt.legend(numpoints=1, loc=0, fontsize=9, framealpha=0.3,
handletextpad=.2, columnspacing=0.6, labelspacing=0.05, ncol=2)
if save_fig:
file_name = (r'./result/TSP_eg1_K_{0}_L_{1}_noise_{2}dB' +
time_stamp + r'.' + fig_format).format(repr(K), repr(L), repr(P))
plt.savefig(file_name, format=fig_format, dpi=300, transparent=True)
plt.show()
# for web rendering
# if web_fig:
# output_file((r'./html/TSP_eg1_K_{0}_L_{1}_noise_{2}dB' +
# time_stamp + '.html').format(repr(K), repr(L), repr(P)))
# TOOLS = 'pan,box_zoom,box_select,reset'
# p_hdl1 = b_plt.figure(title='K={0}, L={1}, SNR={2:.1f}dB, error={3:.2e}'.format(repr(K), repr(L), P, t_error),
# tools=TOOLS,
# x_axis_label='time', y_axis_label='amplitudes',
# plot_width=550, plot_height=220,
# x_range=(0, tau),
# y_range=(1.17 * np.min(np.concatenate((ak, ak_recon,
# np.array(0)[np.newaxis]))),
# 1.17 * np.max(np.concatenate((ak, ak_recon,
# np.array(0)[np.newaxis]))))
# )
# p_hdl1.title.text_font_size = '12pt'
# p_hdl1.add_tools(WheelZoomTool(dimensions=["width"]))
# p_hdl1.triangle(x=tk, y=ak,
# color='#0072BD',
# fill_color='#0072BD',
# line_width=1.5, size=8,
# legend='Original Diracs')
# p_hdl1.multi_line(xs=np.vstack((tk, tk)).T.tolist(),
# ys=np.vstack((np.zeros(ak.shape), ak)).T.tolist(),
# color='#0072BD',
# line_width=1.5,
# line_color='#0072BD')
# p_hdl1.diamond(x=tk_recon, y=ak_recon,
# color='#D95319',
# line_width=1.5, size=10,
# legend='Estimated Diracs')
# p_hdl1.multi_line(xs=np.vstack((tk_recon, tk_recon)).T.tolist(),
# ys=np.vstack((np.zeros(ak_recon.shape), ak_recon)).T.tolist(),
# color='#D95319',
# line_width=1.5,
# line_color='#D95319')
# p_hdl1.legend.location = 'bottom_right'
# p_hdl1.legend.border_line_alpha = 0.6
# p_hdl1.xaxis.axis_label_text_font_size = "11pt"
# p_hdl1.yaxis.axis_label_text_font_size = "11pt"
# p_hdl1.legend.legend_spacing = 1
# p_hdl1.legend.legend_padding = 5
# p_hdl1.legend.label_text_font_size = "9pt"
#
# # subplot 2
# TOOLS2 = 'pan,box_zoom,wheel_zoom,box_select,reset'
# p_hdl2 = b_plt.figure(tools=TOOLS2, x_axis_label='time', y_axis_label='lowpssed signal',
# plot_width=550, plot_height=220,
# x_range=p_hdl1.x_range,
# y_range=(1.05 * np.min(np.concatenate((y_plt, y_ell))),
# 1.05 * np.max(np.concatenate((y_plt, y_ell))))
# )
#
# p_hdl2.line(x=t_plt, y=y_plt,
# color='#0072BD',
# line_color='#0072BD',
# line_width=1.5,
# legend='Ground Truth')
# p_hdl2.circle(x=t_samp, y=y_ell,
# color='#D95319',
# fill_color='#D95319',
# line_width=1.5, size=2,
# legend='Samples')
#
# p_hdl2.xaxis.axis_label_text_font_size = "11pt"
# p_hdl2.yaxis.axis_label_text_font_size = "11pt"
# p_hdl2.legend.location = 'bottom_right'
# p_hdl2.legend.border_line_alpha = 0.6
# p_hdl2.legend.legend_spacing = 1
# p_hdl2.legend.legend_padding = 5
# p_hdl2.legend.label_text_font_size = "9pt"
#
# p_hdl = b_plt.gridplot([[p_hdl1], [p_hdl2]], toolbar_location='above')
# show(p_hdl)