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fig11d_radio_ast_eg2.py
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fig11d_radio_ast_eg2.py
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from __future__ import division
import datetime
import os
import numpy as np
from scipy import linalg, stats
import sympy
import matplotlib
if os.environ.get('DISPLAY') is None:
matplotlib.use('Agg')
else:
matplotlib.use('Qt5Agg')
from matplotlib import rcParams
import matplotlib.pyplot as plt
from alg_tools_1d import distance
from alg_tools_2d import mtx_space2freq, mtx_freq2space, fista, \
recon_2d_dirac, plot_2d_dirac_loc, plot_2d_dirac_spec, detect_peaks
# 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__':
num_realisation = 1000
save_fig = True # save figure or not
fig_format = r'png' # file type used to save the figure, e.g., pdf, png, etc.
# number of Dirac
K = 5
K_est = 5 # estimated number of Diracs
# M * tau1 and N * tau2 are odd numbers
M = 12 # period of the spectrum along x-axis: M * tau1 must be an ODD number
N = 12 # period of the spectrum along y-axis: N * tau2 must be an ODD number
tau1 = 1
tau2 = 1
# amplitudes of the Dirac
alpha_k = np.random.lognormal(mean=np.log(2), sigma=0.5, size=(K,))
# locations of Diracs
a1 = 1 / M
a2 = 1 / N
uk_x = np.random.exponential(1. / (K - 1), (1, K - 1))
# multiplied by 0.9 is to prevent the Dirac from being located too close to the boundary
xk = 0.9 * np.cumsum(a1 + (1 - (K - 1) * a1) * (1. - 0.1 * np.random.rand(1, 1)) / np.sum(uk_x) * uk_x)
offset = 0.06 * np.sqrt(tau1 ** 2 + tau2 ** 2)
angle = 2 * np.pi * np.random.rand()
xk = np.append(xk, xk[np.int(K / 2)] - offset * np.cos(angle))
xk -= 0.45 * tau1
uk_y = np.random.exponential(1. / (K - 1), (1, K - 1))
yk = 0.9 * np.cumsum(a2 + (1 - (K - 1) * a2) * (1. - 0.1 * np.random.rand(1, 1)) / np.sum(uk_y) * uk_y)
yk -= 0.45 * tau2
yk = yk[np.random.permutation(K - 1)]
yk = np.append(yk, yk[np.int(K / 2)] - offset * np.sin(angle))
# irregular frequency domain measurements
L = 8500
'''
# generate random Fourier domain sampling locations, which have denser distributions in low frequencies
# cross-correlation is symmetric, so only need to specify half of the frequencies
Lhalf = np.int(np.ceil(L / 2.))
rand_num1 = (np.random.rand(Lhalf) + np.random.rand(Lhalf) + np.random.rand(Lhalf)) / 3.
rand_num2 = (np.random.rand(Lhalf) + np.random.rand(Lhalf) + np.random.rand(Lhalf)) / 3.
omega_ell_x_half = np.pi * (rand_num1 * (2 * M) - M)
omega_ell_y_half = np.pi * (rand_num2 * (2 * N) - N)
omega_ell_x = np.concatenate((omega_ell_x_half, -omega_ell_x_half))
omega_ell_y = np.concatenate((omega_ell_y_half, -omega_ell_y_half))
'''
# load saved data
time_stamp = r'4-2_02_28'
stored_param = np.load(r'./data/Dirac_Data_' + time_stamp + r'.npz')
xk = stored_param['xk']
yk = stored_param['yk']
alpha_k = stored_param['alpha_k']
K = stored_param['K'].tolist()
omega_ell_x = stored_param['omega_ell_x']
omega_ell_y = stored_param['omega_ell_y']
print(r'time stamp: ' + time_stamp +
'\n=======================================\n')
L = omega_ell_x.size
Lhalf = np.int(np.ceil(L / 2.))
xk_grid, omega_grid_x = np.meshgrid(xk, omega_ell_x)
yk_grid, omega_grid_y = np.meshgrid(yk, omega_ell_y)
# Fourier measurements at frequencies omega_ell
Ihat_omega_ell = np.dot(np.exp(- 1j * omega_grid_x * xk_grid
- 1j * omega_grid_y * yk_grid), alpha_k)
# add noise
P = 5 # SNR in [dB]
file_name_summary = r'./result/radio_ast_eg3_batch_res' + repr(num_realisation) + r'.npz'
if os.path.isfile(file_name_summary):
result_all = np.load(file_name_summary)
fri_recon_all = result_all['fri_recon_all']
ell1_recon_all = result_all['ell1_recon_all']
else:
# initialisation
fri_recon_all = np.zeros((num_realisation, 3 * K + 1))
ell1_recon_all = np.zeros((num_realisation, 2 * K + 1))
for realisations in range(num_realisation):
# the added noise is Hermitian symmetric because the noise is added
# to EM waves at each antenna. The Fourier transform is obtained via
# cross-correlation. Hence, it will also be Hermitian symmetric.
noise_half = np.random.randn(Lhalf) + 1j * np.random.randn(Lhalf)
noise = np.concatenate((noise_half, np.conj(noise_half)))
noise = noise / linalg.norm(noise) * linalg.norm(Ihat_omega_ell) * 10 ** (-P / 20.)
Ihat_omega_ell_noisy = Ihat_omega_ell + noise
# reconstruction algorithm to get denoised Fourier measurements on a uniform grid
max_ini = 25
stop_cri = 'max_iter' # stopping criteria: 1) mse; or 2) max_iter
noise_level = np.max([1e-10, linalg.norm(noise)])
taus = np.array([tau1, tau2])
omega_ell = np.column_stack((omega_ell_x, omega_ell_y))
xk_recon, yk_recon, alpha_k_recon = \
recon_2d_dirac(Ihat_omega_ell_noisy, K_est, tau1, tau2,
sympy.Rational(15, 12), sympy.Rational(15, 12),
omega_ell, M, N, noise_level,
max_ini, stop_cri, num_rotation=12)
# calculate reconstruction error
r_est_error, index = distance(xk + 1j * yk, xk_recon + 1j * yk_recon)
xk = xk[index[:, 0]]
yk = yk[index[:, 0]]
alpha_k = alpha_k[index[:, 0]]
xk_recon = xk_recon[index[:, 1]]
yk_recon = yk_recon[index[:, 1]]
alpha_k_recon = np.real(alpha_k_recon[index[:, 1]])
# order the results for the ease of comparison
ind_order = np.argsort(xk)
fri_recon_all[realisations, :] = np.hstack((np.reshape(xk_recon[ind_order],
(1, K), order='F'),
np.reshape(yk_recon[ind_order],
(1, K), order='F'),
np.reshape(alpha_k_recon[ind_order],
(1, K), order='F'),
np.reshape(r_est_error, (1, 1), order='F')))
print('Position estimation error: {0:.2e}\n').format(r_est_error)
# --------------------------------------------------------------------------
# plot results
plt.close('all')
file_name_loc = (r'./result/TSP_eg4_K_{0}_L_{1}_' +
r'noise_{2}dB_locations.' +
fig_format).format(repr(K), repr(L), repr(P))
plot_2d_dirac_loc(xk_recon, yk_recon, alpha_k_recon, xk, yk, alpha_k, K, L, P, tau1, tau2,
save_figure=True, fig_format=fig_format, file_name=file_name_loc)
# --------------------------------------------------------------------------
file_name_spec = (r'./result/TSP_eg4_K_{0}_L_{1}_' +
r'noise_{2}dB_spectrum').format(repr(K), repr(L), repr(P))
plot_2d_dirac_spec(xk_recon, yk_recon, alpha_k_recon, Ihat_omega_ell_noisy, Ihat_omega_ell,
omega_ell_x, omega_ell_y, M, N, P, L,
save_figure=True, fig_format=fig_format,
file_name=file_name_spec)
# ==========================================================================
# the ell1 minimisation result with FISTA
num_pixel_x = 515
num_pixel_y = 515
A = lambda img: mtx_space2freq(img, omega_ell_x, omega_ell_y,
num_pixel_x, num_pixel_y, tau1, tau2)
At = lambda img_hat: np.real(mtx_freq2space(img_hat, omega_ell_x, omega_ell_y,
num_pixel_x, num_pixel_y, tau1, tau2))
img_recon_ell1, reg_weight = fista(Ihat_omega_ell_noisy, A, At,
4e-3, linalg.norm(noise) ** 2,
max_iter=200, max_iter_reg=200)
# --------------------------------------------------------------------------
# detect local maximum points
peak_locs = detect_peaks(img_recon_ell1 * (img_recon_ell1 > 0))[2]
xk_recon_ell1 = tau1 * peak_locs[1, :] / num_pixel_x - 0.5 * tau1
yk_recon_ell1 = tau2 * peak_locs[0, :] / num_pixel_y - 0.5 * tau2
if peak_locs.shape[1] == 0:
r_est_error_ell1, index_ell1 = distance(xk + 1j * yk,
np.zeros(K, dtype=complex))
else:
r_est_error_ell1, index_ell1 = distance(xk + 1j * yk,
xk_recon_ell1 + 1j * yk_recon_ell1)
xk = xk[index_ell1[:, 0]]
yk = yk[index_ell1[:, 0]]
alpha_k = alpha_k[index_ell1[:, 0]]
xk_recon_ell1 = xk_recon_ell1[index_ell1[:, 1]]
yk_recon_ell1 = yk_recon_ell1[index_ell1[:, 1]]
# order the results for the ease of comparison
ind_order_ell1 = np.argsort(xk)
ell1_recon_all[realisations, :] = np.hstack((np.reshape(xk_recon_ell1[ind_order_ell1],
(1, K), order='F'),
np.reshape(yk_recon_ell1[ind_order_ell1],
(1, K), order='F'),
np.reshape(r_est_error_ell1,
(1, 1), order='F')))
print('Number of detected sources: {0}\n'.format(repr(peak_locs.shape[1])))
# --------------------------------------------------------------------------
# plot ell-1 reconstruction
plt.figure(figsize=(3, 3), dpi=90)
ax2 = plt.axes([0.2, 0.067, 0.75, 0.75])
plt_ell1recon = ax2.imshow(np.real(img_recon_ell1) * (img_recon_ell1 > 0), origin='lower',
cmap='Spectral_r')
plt.xticks(np.linspace(num_pixel_x / 12., num_pixel_x - num_pixel_x / 12., num=5),
(r'$-0.4$', r'$-0.2$', r'$0.0$', r'$0.2$', r'$0.4$'))
plt.yticks(np.linspace(num_pixel_y / 12., num_pixel_y - num_pixel_y / 12., num=5),
(r'$-0.4$', r'$-0.2$', r'$0.0$', r'$0.2$', r'$0.4$'))
ax2c = plt.colorbar(plt_ell1recon, use_gridspec=False,
anchor=(-0.15, 0.5), shrink=0.8, spacing='proportional')
ax2c.ax.tick_params(labelsize=8.5)
plt.xlabel(r'horizontal position $x$', fontsize=12)
plt.ylabel(r'vertical position $y$', fontsize=12)
ax2.xaxis.set_label_coords(0.5, -0.11)
ax2.yaxis.set_label_coords(-0.19, 0.5)
r_est_error_ell1_pow = np.int(np.floor(np.log10(r_est_error_ell1)))
plt.title(r'Average $\mathbf{{r}}_{{\mbox{{\footnotesize error}}}}'
r'={0:.2f}\times10^{other}$'.format(r_est_error_ell1 /
10 ** r_est_error_ell1_pow,
other='{' + str(r_est_error_ell1_pow) + '}'),
fontsize=11)
file_name_ell1_recon = (r'./result/TSP_eg4_K_{0}_L_{1}_' +
r'noise_{2}dB_ell1_recon_{3}by{4}.' +
fig_format).format(repr(K), repr(L), repr(P),
repr(num_pixel_y), repr(num_pixel_x))
plt.savefig(file_name_ell1_recon, format=fig_format, dpi=300, transparent=True)
plt.show()
# save batch run results
np.savez(file_name_summary, fri_recon_all=fri_recon_all,
ell1_recon_all=ell1_recon_all)
# plot the aggregated results
plt.figure(figsize=(3, 3), dpi=90)
ax3 = plt.axes([0.2, 0.15, 0.75, 0.75])
endpoint = fri_recon_all[:, 0].nonzero()[0][-1]
for k in range(K):
data_xy = np.vstack((np.reshape(fri_recon_all[:endpoint + 1, k],
(1, -1), order='F'),
np.reshape(fri_recon_all[:endpoint + 1, k + K],
(1, -1), order='F')
))
z_loop = stats.gaussian_kde(data_xy, 'silverman')(data_xy)
idx_loop = z_loop.argsort()
cax = ax3.scatter(fri_recon_all[:endpoint + 1, k][idx_loop],
fri_recon_all[:endpoint + 1, k + K][idx_loop],
s=0.25, edgecolor='none', c=z_loop[idx_loop], cmap='Spectral_r')
if k == 0:
ax3.hold(True)
cbar = plt.colorbar(cax, shrink=0.8, anchor=(-0.1, 0.5), location='right',
ticks=[[z_loop.min(), (z_loop.min() + z_loop.max()) / 2.,
z_loop.max()]])
cbar.ax.set_yticklabels(['low', 'medium', 'high'], fontsize=8, rotation=270)
cbar.ax.xaxis.set_tick_params(pad=0.3)
plt.axis('scaled')
plt.xlim([-0.5 * tau1, 0.5 * tau1])
plt.ylim([-0.5 * tau2, 0.5 * tau2])
plt.xlabel(r'horizontal position $x$', fontsize=11)
plt.ylabel(r'vertical position $y$', fontsize=11)
ax3.xaxis.set_label_coords(0.5, -0.11)
ax3.yaxis.set_label_coords(-0.19, 0.5)
plt.title(r'Probability Density', fontsize=11)
if save_fig:
file_name_aggregated = (r'./result/TSP_intro_K_{0}_L_{1}_' +
r'noise_{2}dB_aggregated.' +
fig_format).format(repr(K), repr(L), repr(P))
plt.savefig(file_name_aggregated, format=fig_format, dpi=300, transparent=True)
plt.show()