def compute_spectra(data, dt, scaling='amp', nseg=1):
Nr, Nt, Ns = data.shape
if scaling == 'amp':
N = nextPow2(Nt)
freqs = np.fft.rfftfreq(N, dt)
A = np.abs(np.fft.rfft(data, axis=1, n=N))
elif scaling == 'pow':
N = nextPow2(Nt)
freqs = np.fft.rfftfreq(N, dt)
A = np.abs(np.fft.rfft(data, axis=1, n=N))**2
elif scaling == 'psd':
if nseg > 1:
N = nextPow2(Nt // nseg)
else:
N = None
freqs, A = welch(data,
fs=1.0 / dt,
nperseg=N,
scaling='density',
axis=1)
# Average the spectra over sources and receivers
A = np.sum(A, axis=(0, 2)) / (Nr * Ns)
return freqs, A
def fft_and_window(X, dt, double_length):
# Transform X into the frequency domain and apply window around nonzero
# frequency components
if double_length:
N = nextPow2(2 * X.shape[1])
else:
N = nextPow2(X.shape[1])
X = np.fft.rfft(X, n=N, axis=1)
if plotParams['fmax'] is None:
freqs = np.fft.rfftfreq(N, dt)
plotParams['fmax'] = np.max(freqs)
pickle.dump(plotParams, open('plotParams.pkl', 'wb'),
pickle.HIGHEST_PROTOCOL)
# Apply the frequency window
fmin = plotParams['fmin']
fmax = plotParams['fmax']
fu = plotParams['fu'] # frequency units (e.g., Hz)
if fu != '':
print('Applying frequency window: [%0.2f %s, %0.2f %s]' %
(fmin, fu, fmax, fu))
else:
print('Applying frequency window: [%0.2f, %0.2f]' % (fmin, fmax))
df = 1.0 / (N * dt)
startIndex = int(round(fmin / df))
stopIndex = int(round(fmax / df))
finterval = np.arange(startIndex, stopIndex, 1)
X = X[:, finterval, :]
return X
def compute_spectrum(data, dt, power=False):
Nr, Nt, Ns = data.shape
N = nextPow2(Nt)
freqs = np.fft.rfftfreq(N, dt)
amplitudes = np.abs(np.fft.rfft(data, axis=1, n=N))
if power:
amplitudes = amplitudes**2 / N
A = np.sum(amplitudes, axis=(0, 2)) / (Nr * Ns)
return freqs, A
def solver(medium, s, Uh, V, alpha, domain):
#==============================================================================
# Load the receiver coordinates and recording times from the data directory
datadir = np.load('datadir.npz')
times = np.load(str(datadir['recordingTimes']))
receiverPoints = np.load(str(datadir['receivers']))
# Compute length of time step.
# This parameter is used for FFT shifting and time windowing
dt = times[1] - times[0]
# Load the windowing parameters for the receiver and time axes of
# the 3D data array
if Path('window.npz').exists():
windowDict = np.load('window.npz')
# Time window parameters (with units of time)
tstart = windowDict['tstart']
tstop = windowDict['tstop']
# Convert time window parameters to corresponding array indices
tstart = int(round(tstart / dt))
tstop = int(round(tstop / dt))
tstep = windowDict['tstep']
# Receiver window parameters
rstart = windowDict['rstart']
rstop = windowDict['rstop']
rstep = windowDict['rstep']
else:
# Set default window parameters if user did
# not specify window parameters.
# Time window parameters (integers corresponding to indices in an array)
tstart = 0
tstop = len(times)
tstep = 1
# Receiver window parameters
rstart = 0
rstop = receiverPoints.shape[0]
rstep = 1
# Slice the recording times according to the time window parameters
# to create a time window array
tinterval = np.arange(tstart, tstop, tstep)
times = times[tinterval]
T = times[-1] - times[0]
times = np.linspace(-T, T, 2 * len(times) - 1)
# Slice the receiverPoints array according to the receiver window parameters
rinterval = np.arange(rstart, rstop, rstep)
receiverPoints = receiverPoints[rinterval, :]
Nr = receiverPoints.shape[0]
Nt = len(times) # number of samples in time window
# Get information about the pulse function used to
# generate the interrogating wave (These parameters are
# used to help Vezda decide if it needs to recompute the
# test functions in the case the user changes these parameters.)
velocity = pulseFun.velocity # only used if medium == constant
peakFreq = pulseFun.peakFreq # peak frequency
peakTime = pulseFun.peakTime # time at which the pulse amplitude is maximum
# Used for getting time and frequency units
if Path('plotParams.pkl').exists():
plotParams = pickle.load(open('plotParams.pkl', 'rb'))
else:
plotParams = default_params()
# Get machine precision
eps = np.finfo(float).eps # about 2e-16 (used in division
# so we never divide by zero)
#==============================================================================
# Load the user-specified space-time sampling grid
try:
samplingGrid = np.load('samplingGrid.npz')
except FileNotFoundError:
samplingGrid = None
if samplingGrid is None:
sys.exit(
textwrap.dedent('''
A space-time sampling grid needs to be set up before running the
\'vzsolve\' command. Enter:
vzgrid --help
from the command-line for more information on how to set up a
sampling grid.
'''))
if 'z' not in samplingGrid:
# Apply linear sampling method to three-dimensional space-time
x = samplingGrid['x']
y = samplingGrid['y']
tau = samplingGrid['tau']
z = None
# Get number of sampling points in space and time
Nx = len(x)
Ny = len(y)
X, Y = np.meshgrid(x, y, indexing='ij')
# Initialize the Histogram for storing images at each sampling point in time.
# Initialize the Image (time-integrated Histogram with respect to L2 norm)
Image = np.zeros(X.shape)
if medium == 'constant':
# Vezda will compute free-space test functions over the space-time
# sampling grid via function calls to 'FundamentalSolutions.py'. This is
# much more efficient than applying a forward and inverse FFT pair to
# shift the test functions in time corresponding to different sampling
# points in time. FFT pairs are only used when medium == variable.
pulse = lambda t: pulseFun.pulse(t)
# Previously computed test functions and parameters from pulseFun module
# are stored in 'VZTestFuncs.npz'. If the current space-time sampling grid
# and pulseFun module parameters are consistent with those stored in
# 'VZTestFuncs.npz', then Vezda will load the previously computed test
# functions. Otherwise, Vezda will recompute the test functions. This reduces
# computational cost by only computing test functions when necessary.
if Path('VZTestFuncs.npz').exists():
print(
'\nDetected that free-space test functions have already been computed...'
)
print(
'Checking consistency with current space-time sampling grid and pulse function...'
)
TFDict = np.load('VZTestFuncs.npz')
if samplingIsCurrent(TFDict, receiverPoints, times, velocity,
tau, x, y, z, peakFreq, peakTime):
print('Moving forward to imaging algorithm...')
TFarray = TFDict['TFarray']
else:
if tau[0] != 0:
tu = plotParams['tu']
if tu != '':
print(
'Recomputing test functions for focusing time %0.2f %s...'
% (tau[0], tu))
else:
print(
'Recomputing test functions for focusing time %0.2f...'
% (tau[0]))
TFarray, samplingPoints = sampleSpace(
receiverPoints, times - tau[0], velocity, x, y, z,
pulse)
else:
print('Recomputing test functions...')
TFarray, samplingPoints = sampleSpace(
receiverPoints, times, velocity, x, y, z, pulse)
np.savez('VZTestFuncs.npz',
TFarray=TFarray,
time=times,
receivers=receiverPoints,
peakFreq=peakFreq,
peakTime=peakTime,
velocity=velocity,
x=x,
y=y,
tau=tau,
samplingPoints=samplingPoints)
else:
print(
'\nComputing free-space test functions for the current space-time sampling grid...'
)
if tau[0] != 0:
tu = plotParams['tu']
if tu != '':
print(
'Computing test functions for focusing time %0.2f %s...'
% (tau[0], tu))
else:
print(
'Computing test functions for focusing time %0.2f...'
% (tau[0]))
TFarray, samplingPoints = sampleSpace(
receiverPoints, times - tau[0], velocity, x, y, z,
pulse)
else:
TFarray, samplingPoints = sampleSpace(
receiverPoints, times, velocity, x, y, z, pulse)
np.savez('VZTestFuncs.npz',
TFarray=TFarray,
time=times,
receivers=receiverPoints,
peakFreq=peakFreq,
peakTime=peakTime,
velocity=velocity,
x=x,
y=y,
tau=tau,
samplingPoints=samplingPoints)
#==============================================================================
if domain == 'freq':
# Transform test functions into the frequency domain and bandpass for efficient solution
# to near-field equation
print('Transforming test functions to the frequency domain...')
N = nextPow2(Nt)
TFarray = np.fft.rfft(TFarray, n=N, axis=1)
if plotParams['fmax'] is None:
freqs = np.fft.rfftfreq(N, tstep * dt)
plotParams['fmax'] = np.max(freqs)
pickle.dump(plotParams, open('plotParams.pkl', 'wb'),
pickle.HIGHEST_PROTOCOL)
# Apply the frequency window
fmin = plotParams['fmin']
fmax = plotParams['fmax']
fu = plotParams['fu'] # frequency units (e.g., Hz)
if fu != '':
print('Applying bandpass filter: [%0.2f %s, %0.2f %s]' %
(fmin, fu, fmax, fu))
else:
print('Applying bandpass filter: [%0.2f, %0.2f]' %
(fmin, fmax))
df = 1.0 / (N * tstep * dt)
startIndex = int(round(fmin / df))
stopIndex = int(round(fmax / df))
finterval = np.arange(startIndex, stopIndex, 1)
TFarray = TFarray[:, finterval, :]
N = TFarray.shape[1]
#==============================================================================
# Solve the near-field equation for each sampling point
print('Localizing the source...')
# Compute the Tikhonov-regularized solution to the near-field equation N * phi = tf.
# 'tf' is a test function
# 'alpha' is the regularization parameter
# 'phi_alpha' is the regularized solution given 'alpha'
k = 0 # counter for spatial sampling points
for ix in trange(Nx, desc='Solving system'):
for iy in range(Ny):
tf = np.reshape(TFarray[:, :, k], (N * Nr, 1))
phi_alpha = Tikhonov(Uh, s, V, tf, alpha)
Image[ix, iy] = 1.0 / (norm(phi_alpha) + eps)
k += 1
Imin = np.min(Image)
Imax = np.max(Image)
Image = (Image - Imin) / (Imax - Imin + eps)
elif medium == 'variable':
if 'testFuncs' in datadir:
# Load the user-provided test functions
TFarray = np.load(str(datadir['testFuncs']))
# Apply the receiver/time windows, if any
TFarray = TFarray[rinterval, :, :]
TFarray = TFarray[:, tinterval, :]
#==============================================================================
if domain == 'freq':
# Transform test functions into the frequency domain and bandpass for efficient solution
# to near-field equation
print(
'Transforming test functions to the frequency domain...'
)
N = nextPow2(Nt)
TFarray = np.fft.rfft(TFarray, n=N, axis=1)
if plotParams['fmax'] is None:
freqs = np.fft.rfftfreq(N, tstep * dt)
plotParams['fmax'] = np.max(freqs)
pickle.dump(plotParams, open('plotParams.pkl', 'wb'),
pickle.HIGHEST_PROTOCOL)
# Apply the frequency window
fmin = plotParams['fmin']
fmax = plotParams['fmax']
fu = plotParams['fu'] # frequency units (e.g., Hz)
if fu != '':
print(
'Applying bandpass filter: [%0.2f %s, %0.2f %s]' %
(fmin, fu, fmax, fu))
else:
print('Applying bandpass filter: [%0.2f, %0.2f]' %
(fmin, fmax))
df = 1.0 / (N * tstep * dt)
startIndex = int(round(fmin / df))
stopIndex = int(round(fmax / df))
finterval = np.arange(startIndex, stopIndex, 1)
TFarray = TFarray[:, finterval, :]
N = TFarray.shape[1]
# Load the sampling points
samplingPoints = np.load(str(datadir['samplingPoints']))
else:
sys.exit(
textwrap.dedent('''
FileNotFoundError: Attempted to load file containing test
functions, but no such file exists. If a file exists containing
the test functions, run:
'vzdata --path=<path/to/data/>'
and specify the file containing the test functions when prompted.
Otherwise, specify 'no' when asked if a file containing the test
functions exists.
'''))
userResponded = False
print(
textwrap.dedent('''
In what order was the sampling grid spanned to compute the test functions?
Enter 'xy' if for each x, loop over y. (Default)
Enter 'yx' if for each y, loop over x.
Enter 'q/quit' to abort the calculation.
'''))
while userResponded == False:
order = input('Order: ')
if order == '' or order == 'xy':
print('Proceeding with order \'xy\'...')
print('Localizing the source...')
# Compute the Tikhonov-regularized solution to the near-field equation N * phi = tf.
# 'tf' is a test function
# 'alpha' is the regularization parameter
# 'phi_alpha' is the regularized solution given 'alpha'
k = 0 # counter for spatial sampling points
for ix in trange(Nx, desc='Solving system'):
for iy in range(Ny):
tf = np.reshape(TFarray[:, :, k], (N * Nr, 1))
phi_alpha = Tikhonov(Uh, s, V, tf, alpha)
Image[ix, iy] = 1.0 / (norm(phi_alpha) + eps)
k += 1
Imin = np.min(Image)
Imax = np.max(Image)
Image = (Image - Imin) / (Imax - Imin + eps)
userResponded = True
break
elif order == 'yx':
print('Proceeding with order \'yx\'...')
print('Localizing the source...')
# Compute the Tikhonov-regularized solution to the near-field equation N * phi = tf.
# 'tf' is a test function
# 'alpha' is the regularization parameter
# 'phi_alpha' is the regularized solution given 'alpha'
k = 0 # counter for spatial sampling points
for iy in trange(Ny, desc='Solving system'):
for ix in range(Nx):
tf = np.reshape(TFarray[:, :, k], (N * Nr, 1))
phi_alpha = Tikhonov(Uh, s, V, tf, alpha)
Image[ix, iy] = 1.0 / (norm(phi_alpha) + eps)
k += 1
Imin = np.min(Image)
Imax = np.max(Image)
Image = (Image - Imin) / (Imax - Imin + eps)
userResponded = True
break
elif order == 'q' or order == 'quit':
sys.exit('Aborting calculation.')
else:
print(
textwrap.dedent('''
Invalid response. Please enter one of the following:
Enter 'xy' if for each x, loop over y. (Default)
Enter 'yx' if for each y, loop over x.
Enter 'q/quit' to abort the calculation.
'''))
if domain == 'freq':
np.savez('imageNFE.npz', Image=Image, alpha=alpha, X=X, Y=Y)
else:
np.savez('imageNFE.npz',
Image=Image,
alpha=alpha,
X=X,
Y=Y,
tau=tau)
#==============================================================================
else:
# Apply linear sampling method to four-dimensional space-time
x = samplingGrid['x']
y = samplingGrid['y']
z = samplingGrid['z']
tau = samplingGrid['tau']
# Get number of sampling points in space and time
Nx = len(x)
Ny = len(y)
Nz = len(z)
X, Y, Z = np.meshgrid(x, y, z, indexing='ij')
# Initialize the Histogram for storing images at each sampling point in time.
# Initialize the Image (time-integrated Histogram with respect to L2 norm)
Image = np.zeros(X.shape)
if medium == 'constant':
# Vezda will compute free-space test functions over the space-time
# sampling grid via function calls to 'FundamentalSolutions.py'. This is
# much more efficient than applying a forward and inverse FFT pair to
# shift the test functions in time corresponding to different sampling
# points in time. FFT pairs are only used when medium == variable.
pulse = lambda t: pulseFun.pulse(t)
# Previously computed test functions and parameters from pulseFun module
# are stored in 'VZTestFuncs.npz'. If the current space-time sampling grid
# and pulseFun module parameters are consistent with those stored in
# 'VZTestFuncs.npz', then Vezda will load the previously computed test
# functions. Otherwise, Vezda will recompute the test functions. This reduces
# computational cost by only computing test functions when necessary.
if Path('VZTestFuncs.npz').exists():
print(
'\nDetected that free-space test functions have already been computed...'
)
print(
'Checking consistency with current space-time sampling grid and pulse function...'
)
TFDict = np.load('VZTestFuncs.npz')
if samplingIsCurrent(TFDict, receiverPoints, times, velocity,
tau, x, y, z, peakFreq, peakTime):
print('Moving forward to imaging algorithm...')
TFarray = TFDict['TFarray']
else:
if tau[0] != 0:
tu = plotParams['tu']
if tu != '':
print(
'Recomputing test functions for focusing time %0.2f %s...'
% (tau[0], tu))
else:
print(
'Recomputing test functions for focusing time %0.2f...'
% (tau[0]))
TFarray, samplingPoints = sampleSpace(
receiverPoints, times - tau[0], velocity, x, y, z,
pulse)
else:
print('Recomputing test functions...')
TFarray, samplingPoints = sampleSpace(
receiverPoints, times, velocity, x, y, z, pulse)
np.savez('VZTestFuncs.npz',
TFarray=TFarray,
time=times,
receivers=receiverPoints,
peakFreq=peakFreq,
peakTime=peakTime,
velocity=velocity,
x=x,
y=y,
z=z,
tau=tau,
samplingPoints=samplingPoints)
else:
print(
'\nComputing free-space test functions for the current space-time sampling grid...'
)
if tau[0] != 0:
tu = plotParams['tu']
if tu != '':
print(
'Computing test functions for focusing time %0.2f %s...'
% (tau[0], tu))
else:
print(
'Computing test functions for focusing time %0.2f...'
% (tau[0]))
TFarray, samplingPoints = sampleSpace(
receiverPoints, times - tau[0], velocity, x, y, z,
pulse)
else:
TFarray, samplingPoints = sampleSpace(
receiverPoints, times, velocity, x, y, z, pulse)
np.savez('VZTestFuncs.npz',
TFarray=TFarray,
time=times,
receivers=receiverPoints,
peakFreq=peakFreq,
peakTime=peakTime,
velocity=velocity,
x=x,
y=y,
z=z,
tau=tau,
samplingPoints=samplingPoints)
#==============================================================================
if domain == 'freq':
# Transform test functions into the frequency domain and bandpass for efficient solution
# to near-field equation
print('Transforming test functions to the frequency domain...')
N = nextPow2(Nt)
TFarray = np.fft.rfft(TFarray, n=N, axis=1)
if plotParams['fmax'] is None:
freqs = np.fft.rfftfreq(N, tstep * dt)
plotParams['fmax'] = np.max(freqs)
pickle.dump(plotParams, open('plotParams.pkl', 'wb'),
pickle.HIGHEST_PROTOCOL)
# Apply the frequency window
fmin = plotParams['fmin']
fmax = plotParams['fmax']
fu = plotParams['fu'] # frequency units (e.g., Hz)
if fu != '':
print('Applying bandpass filter: [%0.2f %s, %0.2f %s]' %
(fmin, fu, fmax, fu))
else:
print('Applying bandpass filter: [%0.2f, %0.2f]' %
(fmin, fmax))
df = 1.0 / (N * tstep * dt)
startIndex = int(round(fmin / df))
stopIndex = int(round(fmax / df))
finterval = np.arange(startIndex, stopIndex, 1)
TFarray = TFarray[:, finterval, :]
N = TFarray.shape[1]
#==============================================================================
# Solve the near-field equation for each sampling point
print('Localizing the source...')
# Compute the Tikhonov-regularized solution to the near-field equation N * phi = tf.
# 'tf' is a test function
# 'alpha' is the regularization parameter
# 'phi_alpha' is the regularized solution given 'alpha'
k = 0 # counter for spatial sampling points
for ix in trange(Nx, desc='Solving system'):
for iy in range(Ny):
for iz in range(Nz):
tf = np.reshape(TFarray[:, :, k], (N * Nr, 1))
phi_alpha = Tikhonov(Uh, s, V, tf, alpha)
Image[ix, iy, iz] = 1.0 / (norm(phi_alpha) + eps)
k += 1
Imin = np.min(Image)
Imax = np.max(Image)
Image = (Image - Imin) / (Imax - Imin + eps)
elif medium == 'variable':
if 'testFuncs' in datadir:
# Load the user-provided test functions
TFarray = np.load(str(datadir['testFuncs']))
# Apply the receiver/time windows, if any
TFarray = TFarray[rinterval, :, :]
TFarray = TFarray[:, tinterval, :]
#==============================================================================
if domain == 'freq':
# Transform test functions into the frequency domain and bandpass for efficient solution
# to near-field equation
print(
'Transforming test functions to the frequency domain...'
)
N = nextPow2(Nt)
TFarray = np.fft.rfft(TFarray, n=N, axis=1)
if plotParams['fmax'] is None:
freqs = np.fft.rfftfreq(N, tstep * dt)
plotParams['fmax'] = np.max(freqs)
pickle.dump(plotParams, open('plotParams.pkl', 'wb'),
pickle.HIGHEST_PROTOCOL)
# Apply the frequency window
fmin = plotParams['fmin']
fmax = plotParams['fmax']
fu = plotParams['fu'] # frequency units (e.g., Hz)
if fu != '':
print(
'Applying bandpass filter: [%0.2f %s, %0.2f %s]' %
(fmin, fu, fmax, fu))
else:
print('Applying bandpass filter: [%0.2f, %0.2f]' %
(fmin, fmax))
df = 1.0 / (N * tstep * dt)
startIndex = int(round(fmin / df))
stopIndex = int(round(fmax / df))
finterval = np.arange(startIndex, stopIndex, 1)
TFarray = TFarray[:, finterval, :]
N = TFarray.shape[1]
# Load the sampling points
samplingPoints = np.load(str(datadir['samplingPoints']))
else:
sys.exit(
textwrap.dedent('''
FileNotFoundError: Attempted to load file containing test
functions, but no such file exists. If a file exists containing
the test functions, run:
'vzdata --path=<path/to/data/>'
and specify the file containing the test functions when prompted.
Otherwise, specify 'no' when asked if a file containing the test
functions exists.
'''))
userResponded = False
print(
textwrap.dedent('''
In what order was the sampling grid spanned to compute the test functions?
Enter 'xyz' if for each x, for each y, loop over z. (Default)
Enter 'xzy' if for each x, for each z, loop over y.
Enter 'yxz' if for each y, for each x, loop over z.
Enter 'yzx' if for each y, for each z, loop over x.
Enter 'zxy' if for each z, for each x, loop over y.
Enter 'zyx' if for each z, for each y, loop over x
Enter 'q/quit' to abort the calculation.
'''))
while userResponded == False:
order = input('Order: ')
if order == '' or order == 'xyz':
print('Proceeding with order \'xyz\'...')
print('Localizing the source...')
# Compute the Tikhonov-regularized solution to the near-field equation N * phi = tf.
# 'tf' is a test function
# 'alpha' is the regularization parameter
# 'phi_alpha' is the regularized solution given 'alpha'
k = 0 # counter for spatial sampling points
for ix in trange(Nx, desc='Solving system'):
for iy in range(Ny):
for iz in range(Nz):
tf = np.reshape(TFarray[:, :, k], (N * Nr, 1))
phi_alpha = Tikhonov(Uh, s, V, tf, alpha)
Image[ix, iy,
iz] = 1.0 / (norm(phi_alpha) + eps)
k += 1
Imin = np.min(Image)
Imax = np.max(Image)
Image = (Image - Imin) / (Imax - Imin + eps)
userResponded = True
break
elif order == 'xzy':
print('Proceeding with order \'xzy\'...')
print('Localizing the source...')
# Compute the Tikhonov-regularized solution to the near-field equation N * phi = tf.
# 'tf' is a test function
# 'alpha' is the regularization parameter
# 'phi_alpha' is the regularized solution given 'alpha'
k = 0 # counter for spatial sampling points
for ix in trange(Nx, desc='Solving system'):
for iz in range(Nz):
for iy in range(Ny):
tf = np.reshape(TFarray[:, :, k], (N * Nr, 1))
phi_alpha = Tikhonov(Uh, s, V, tf, alpha)
Image[ix, iy,
iz] = 1.0 / (norm(phi_alpha) + eps)
k += 1
Imin = np.min(Image)
Imax = np.max(Image)
Image = (Image - Imin) / (Imax - Imin + eps)
userResponded = True
break
elif order == 'yxz':
print('Proceeding with order \'yxz\'...')
print('Localizing the source...')
# Compute the Tikhonov-regularized solution to the near-field equation N * phi = tf.
# 'tf' is a test function
# 'alpha' is the regularization parameter
# 'phi_alpha' is the regularized solution given 'alpha'
k = 0 # counter for spatial sampling points
for iy in trange(Ny, desc='Solving system'):
for ix in range(Nx):
for iz in range(Nz):
tf = np.reshape(TFarray[:, :, k], (N * Nr, 1))
phi_alpha = Tikhonov(Uh, s, V, tf, alpha)
Image[ix, iy,
iz] = 1.0 / (norm(phi_alpha) + eps)
k += 1
Imin = np.min(Image)
Imax = np.max(Image)
Image = (Image - Imin) / (Imax - Imin + eps)
userResponded = True
break
elif order == 'yzx':
print('Proceeding with order \'yzx\'...')
print('Localizing the source...')
# Compute the Tikhonov-regularized solution to the near-field equation N * phi = tf.
# 'tf' is a test function
# 'alpha' is the regularization parameter
# 'phi_alpha' is the regularized solution given 'alpha'
k = 0 # counter for spatial sampling points
for iy in trange(Ny, desc='Solving system'):
for iz in range(Nz):
for ix in range(Nx):
tf = np.reshape(TFarray[:, :, k], (N * Nr, 1))
phi_alpha = Tikhonov(Uh, s, V, tf, alpha)
Image[ix, iy,
iz] = 1.0 / (norm(phi_alpha) + eps)
k += 1
Imin = np.min(Image)
Imax = np.max(Image)
Image = (Image - Imin) / (Imax - Imin + eps)
userResponded = True
break
elif order == 'zxy':
print('Proceeding with order \'zxy\'...')
print('Localizing the source...')
# Compute the Tikhonov-regularized solution to the near-field equation N * phi = tf.
# 'tf' is a test function
# 'alpha' is the regularization parameter
# 'phi_alpha' is the regularized solution given 'alpha'
k = 0 # counter for spatial sampling points
for iz in trange(Nz, desc='Solving system'):
for ix in range(Nx):
for iy in range(Ny):
tf = np.reshape(TFarray[:, :, k], (N * Nr, 1))
phi_alpha = Tikhonov(Uh, s, V, tf, alpha)
Image[ix, iy,
iz] = 1.0 / (norm(phi_alpha) + eps)
k += 1
Imin = np.min(Image)
Imax = np.max(Image)
Image = (Image - Imin) / (Imax - Imin + eps)
userResponded = True
break
elif order == 'zyx':
print('Proceeding with order \'zyx\'...')
print('Localizing the source...')
# Compute the Tikhonov-regularized solution to the near-field equation N * phi = tf.
# 'tf' is a test function
# 'alpha' is the regularization parameter
# 'phi_alpha' is the regularized solution given 'alpha'
k = 0 # counter for spatial sampling points
for iz in trange(Nz, desc='Solving system'):
for iy in range(Ny):
for ix in range(Nx):
tf = np.reshape(TFarray[:, :, k], (N * Nr, 1))
phi_alpha = Tikhonov(Uh, s, V, tf, alpha)
Image[ix, iy,
iz] = 1.0 / (norm(phi_alpha) + eps)
k += 1
Imin = np.min(Image)
Imax = np.max(Image)
Image = (Image - Imin) / (Imax - Imin + eps)
userResponded = True
break
elif order == 'q' or order == 'quit':
sys.exit('Aborting calculation.')
else:
print(
textwrap.dedent('''
Invalid response. Please enter one of the following:
Enter 'xyz' if for each x, for each y, loop over z. (Default)
Enter 'xzy' if for each x, for each z, loop over y.
Enter 'yxz' if for each y, for each x, loop over z.
Enter 'yzx' if for each y, for each z, loop over x.
Enter 'zxy' if for each z, for each x, loop over y.
Enter 'zyx' if for each z, for each y, loop over x
Enter 'q/quit' to abort the calculation.
'''))
if domain == 'freq':
np.savez('imageNFE.npz', Image=Image, alpha=alpha, X=X, Y=Y, Z=Z)
else:
np.savez('imageNFE.npz',
Image=Image,
alpha=alpha,
X=X,
Y=Y,
Z=Z,
tau=tau)
def asConvolutionalOperator(kernel):
'''
This function takes a 3D data array as input and defines matrix-vector products
for both the forward and adjoint convolution operators. The convolutional operator
can either be the near-field operator (NFO) or the Lippmann-Schwinger operator (LSO).
Input: a three-dimensional data array of shape Nr x Nm x Ns
matvec: Definition of the forward matrix-vector product
rmatvec: Definition of the adjoint matrix-vector product
*** Note ***
All reshape comands in this function definition use Fortran column-based
indexing (order='F'). This is because SciPy uses the Fortran library
ARPACK as a backend to compute spectral decompositions.
x: an arbitrary input vector
Nm: number of time/frequency samples
Nr: number of receivers
Ns: number of sources
Matrix: discretized kernel of the convolutional operator. The input data
form the kernel of the operator.
shape(Matrix) = (Nm * Nr) x (Nm * Ns)
shape(x) = (Nm * Ns) x 1 for forward operator
shape(x) = (Nm * Nr) x 1 for adjoint operator
Output: the operator M such that y = Mx
'''
Nr, Nm, Ns = kernel.shape
if np.issubdtype(kernel.dtype, np.floating):
# input data are real (time domain)
# get the next power of 2 greater than or equal to 2*Nm
# for efficient circular convolution via FFT
N = nextPow2(2 * Nm)
# Fourier transform the data over the time axis=1
U = np.fft.rfft(kernel, n=N, axis=1)
def forwardOperator(x):
# definition of the forward convolutional operator
#reshape x into a matrix and FFT over time axis=0
x = x.reshape((2 * Nm - 1, Ns), order='F')
x = np.fft.rfft(x, n=N, axis=0)
# initialize the output array y for the range of Matrix
y = np.zeros((2 * Nm - 1, Nr), dtype=kernel.dtype)
for i in range(Nr):
# Compute the matrix-vector product for Matrix * x
circularConvolution = np.fft.irfft(U[i, :, :] * x, axis=0)
circularConvolution = circularConvolution[:(2 * Nm - 1), :]
y[:, i] = np.sum(circularConvolution,
axis=1) # sum over sources
y = y.reshape(((2 * Nm - 1) * Nr, 1), order='F')
return y
def adjointOperator(y):
# definition of the adjoint convolutional operator
#reshape y into a matrix and FFT over time axis=0
y = y.reshape((2 * Nm - 1, Nr), order='F')
y = np.fft.rfft(y, n=N, axis=0)
# initialize the output array x for the range of Matrix.T
x = np.zeros((2 * Nm - 1, Ns), dtype=kernel.dtype)
for j in range(Ns):
# Compute the matrix-vector product for Matrix.T * y
circularConvolutionT = np.fft.irfft(U[:, :, j].conj().T * y,
axis=0)
circularConvolutionT = circularConvolutionT[:(2 * Nm - 1), :]
x[:, j] = np.sum(circularConvolutionT,
axis=1) # sum over receivers
x = x.reshape(((2 * Nm - 1) * Ns, 1), order='F')
return x
return LinearOperator(shape=((2 * Nm - 1) * Nr, (2 * Nm - 1) * Ns),
matvec=forwardOperator,
rmatvec=adjointOperator,
dtype=kernel.dtype)
else:
# input data are complex (frequency domain)
def forwardOperator(x):
# definition of the forward convolutional operator
#reshape x into a matrix
x = x.reshape((Nm, Ns), order='F')
# initialize the output array y for the range of Matrix
y = np.zeros((Nm, Nr), dtype=kernel.dtype)
for i in range(Nr):
y[:, i] = np.sum(kernel[i, :, :] * x,
axis=1) # sum over sources
y = y.reshape((Nm * Nr, 1), order='F')
return y
def adjointOperator(y):
# definition of the adjoint convolutional operator
#reshape y into a matrix
y = y.reshape((Nm, Nr), order='F')
# initialize the output array x for the range of Matrix.H
x = np.zeros((Nm, Ns), dtype=kernel.dtype)
for j in range(Ns):
x[:, j] = np.sum(kernel[:, :, j].conj().T * y,
axis=1) # sum over receivers
x = x.reshape((Nm * Ns, 1), order='F')
return x
return LinearOperator(shape=(Nm * Nr, Nm * Ns),
matvec=forwardOperator,
rmatvec=adjointOperator,
dtype=kernel.dtype)
import textwrap
sys.path.append(os.getcwd())
import pulseFun
rinterval, tinterval, tstep, dt, sinterval = get_user_windows()
datadir = np.load('datadir.npz')
recordingTimes = np.load(str(datadir['recordingTimes']))
recordingTimes = recordingTimes[tinterval]
Nt = len(recordingTimes)
T = recordingTimes[-1] - recordingTimes[0]
convolutionTimes = np.linspace(-T, T, 2 * Nt - 1)
N = nextPow2(2 * Nt)
freqs = np.fft.rfftfreq(N, tstep * dt)
Nf = len(freqs)
if 'sources' in datadir:
sourcePoints = np.load(str(datadir['sources']))
sourcePoints = sourcePoints[sinterval, :]
else:
sys.exit(textwrap.dedent('''
'''))
Ns = sourcePoints.shape[0]
try:
samplingGrid = np.load('samplingGrid.npz')
except FileNotFoundError:
def cli():
parser = argparse.ArgumentParser()
parser.add_argument('--nfo', action='store_true',
help='''Plot the singular-value decomposition of the
near-field operator (NFO).''')
parser.add_argument('--lso', action='store_true',
help='''Plot the singular-value decomposition of the
Lippmann-Schwinger operator (LSO).''')
parser.add_argument('--format', '-f', type=str, default='pdf', choices=['png', 'pdf', 'ps', 'eps', 'svg'],
help='''Specify the image format of the saved file. Accepted formats are png, pdf,
ps, eps, and svg. Default format is set to pdf.''')
parser.add_argument('--mode', type=str, choices=['light', 'dark'], required=False,
help='''Specify whether to view plots in light mode for daytime viewing
or dark mode for nighttime viewing.
Mode must be either \'light\' or \'dark\'.''')
args = parser.parse_args()
# See if an SVD already exists. If so, attempt to load it...
if args.nfo and not args.lso:
operatorName = 'near-field operator'
filename = 'NFO_SVD.npz'
elif not args.nfo and args.lso:
operatorName = 'Lippmann-Schwinger operator'
filename = 'LSO_SVD.npz'
elif args.nfo and args.lso:
sys.exit(textwrap.dedent(
'''
UsageError: Please specify only one of the arguments \'--nfo\' or \'--lso\'.
'''))
else:
sys.exit(textwrap.dedent(
'''
For which operator would you like to plot a singular-value decomposition?
Enter:
vzsvd --nfo
for the near-field operator or
vzsvd --lso
for the Lippmann-Schwinger operator.
'''))
try:
U, s, Vh = load_svd(filename)
except IOError:
sys.exit(textwrap.dedent(
'''
A singular-value decomposition of the {s} does not exist.
'''.format(s=operatorName)))
#==============================================================================
# Read in data files
#==============================================================================
datadir = np.load('datadir.npz')
receiverPoints = np.load(str(datadir['receivers']))
recordingTimes = np.load(str(datadir['recordingTimes']))
# Apply user-specified windows
rinterval, tinterval, tstep, dt, sinterval = get_user_windows()
receiverPoints = receiverPoints[rinterval, :]
recordingTimes = recordingTimes[tinterval]
# Load appropriate source points and source window
if args.nfo: # Near-field operator
if 'sources' in datadir:
sourcePoints = np.load(str(datadir['sources']))
sourcePoints = sourcePoints[sinterval, :]
else:
sourcePoints = None
else:
# if args.lso (Lippmann-Schwinger operator)
# in the case of the Lippmann-Schwinger operator, 'sourcePoints'
# correspond to sampling points, which should always exist.
if 'testFuncs' in datadir:
sourcePoints = np.load(str(datadir['samplingPoints']))
elif Path('VZTestFuncs.npz').exists():
TFDict = np.load('VZTestFuncs.npz')
sourcePoints = TFDict['samplingPoints']
else:
sys.exit(textwrap.dedent(
'''
Error: A sampling grid must exist and test functions computed
before a singular-value decomposition of the Lippmann-Schwinger
operator can be computed or plotted.
'''))
# update sinterval for test functions
sinterval = np.arange(0, sourcePoints.shape[0], 1)
# increment receiver/source intervals to be consistent with
# one-based indexing (i.e., count from one instead of zero)
rinterval += 1
sinterval += 1
#==============================================================================
# Determine whether to plot SVD in time domain or frequency domain
#==============================================================================
if np.issubdtype(U.dtype, np.complexfloating):
domain = 'freq'
else:
domain = 'time'
# Load plot parameters
if Path('plotParams.pkl').exists():
plotParams = pickle.load(open('plotParams.pkl', 'rb'))
else:
plotParams = default_params()
Nr = receiverPoints.shape[0]
Nt = len(recordingTimes)
k = len(s)
if domain == 'freq':
# plot singular vectors in frequency domain
N = nextPow2(2 * Nt)
freqs = np.fft.rfftfreq(N, tstep * dt)
if plotParams['fmax'] is None:
plotParams['fmax'] = np.max(freqs)
# Apply the frequency window
fmin = plotParams['fmin']
fmax = plotParams['fmax']
df = 1.0 / (N * tstep * dt)
startIndex = int(round(fmin / df))
stopIndex = int(round(fmax / df))
finterval = np.arange(startIndex, stopIndex, 1)
freqs = freqs[finterval]
M = len(freqs)
Ns = int(Vh.shape[1] / M)
U = U.toarray().reshape((Nr, M, k))
V = Vh.getH().toarray().reshape((Ns, M, k))
else: # domain == 'time'
M = 2 * Nt - 1
Ns = int(Vh.shape[1] / M)
U = U.reshape((Nr, M, k))
V = Vh.T.reshape((Ns, M, k))
T = recordingTimes[-1] - recordingTimes[0]
times = np.linspace(-T, T, M)
if args.mode is not None:
plotParams['view_mode'] = args.mode
pickle.dump(plotParams, open('plotParams.pkl', 'wb'), pickle.HIGHEST_PROTOCOL)
remove_keymap_conflicts({'left', 'right', 'up', 'down', 'save'})
if domain == 'freq':
# plot the left singular vectors
fig_lvec, ax_lvec_r, ax_lvec_i = setFigure(num_axes=2, mode=plotParams['view_mode'])
ax_lvec_r.volume = U.real
ax_lvec_i.volume = U.imag
ax_lvec_r.index = 0
ax_lvec_i.index = 0
fig_lvec.suptitle('Left-Singular Vector', color=ax_lvec_r.titlecolor, fontsize=16)
fig_lvec.subplots_adjust(bottom=0.27, top=0.86)
leftTitle_r = vector_title('left', ax_lvec_r.index + 1, 'real')
leftTitle_i = vector_title('left', ax_lvec_i.index + 1, 'imag')
for ax, title in zip([ax_lvec_r, ax_lvec_i], [leftTitle_r, leftTitle_i]):
left_im = plotFreqVectors(ax, ax.volume[:, :, ax.index], freqs, rinterval,
receiverPoints, title, 'left', plotParams)
lp0 = ax_lvec_r.get_position().get_points().flatten()
lp1 = ax_lvec_i.get_position().get_points().flatten()
left_cax = fig_lvec.add_axes([lp0[0], 0.12, lp1[2]-lp0[0], 0.03])
lcbar = fig_lvec.colorbar(left_im, left_cax, orientation='horizontal')
lcbar.outline.set_edgecolor(ax_lvec_r.cbaredgecolor)
lcbar.ax.tick_params(axis='x', colors=ax_lvec_r.labelcolor)
lcbar.ax.yaxis.set_major_formatter(FormatStrFormatter('%.1f'))
lcbar.set_label('Amplitude',
labelpad=5, rotation=0, fontsize=12, color=ax_lvec_r.labelcolor)
fig_lvec.canvas.mpl_connect('key_press_event', lambda event: process_key_vectors(event, freqs, rinterval, sinterval,
receiverPoints, sourcePoints, plotParams,
'cmplx_left'))
# plot the right singular vectors
fig_rvec, ax_rvec_r, ax_rvec_i = setFigure(num_axes=2, mode=plotParams['view_mode'])
ax_rvec_r.volume = V.real
ax_rvec_i.volume = V.imag
ax_rvec_r.index = 0
ax_rvec_i.index = 0
fig_rvec.suptitle('Right-Singular Vector', color=ax_rvec_r.titlecolor, fontsize=16)
fig_rvec.subplots_adjust(bottom=0.27, top=0.86)
rightTitle_r = vector_title('right', ax_rvec_r.index + 1, 'real')
rightTitle_i = vector_title('right', ax_rvec_i.index + 1, 'imag')
for ax, title in zip([ax_rvec_r, ax_rvec_i], [rightTitle_r, rightTitle_i]):
right_im = plotFreqVectors(ax, ax.volume[:, :, ax.index], freqs, sinterval,
sourcePoints, title, 'right', plotParams)
rp0 = ax_rvec_r.get_position().get_points().flatten()
rp1 = ax_rvec_i.get_position().get_points().flatten()
right_cax = fig_rvec.add_axes([rp0[0], 0.12, rp1[2]-rp0[0], 0.03])
rcbar = fig_rvec.colorbar(right_im, right_cax, orientation='horizontal')
rcbar.outline.set_edgecolor(ax_rvec_r.cbaredgecolor)
rcbar.ax.tick_params(axis='x', colors=ax_rvec_r.labelcolor)
rcbar.ax.yaxis.set_major_formatter(FormatStrFormatter('%.1f'))
rcbar.set_label('Amplitude',
labelpad=5, rotation=0, fontsize=12, color=ax_lvec_r.labelcolor)
fig_rvec.canvas.mpl_connect('key_press_event', lambda event: process_key_vectors(event, freqs, rinterval, sinterval,
receiverPoints, sourcePoints, plotParams,
'cmplx_right'))
else:
# domain == 'time'
fig_vec, ax_lvec, ax_rvec = setFigure(num_axes=2, mode=plotParams['view_mode'])
ax_lvec.volume = U
ax_lvec.index = 0
leftTitle = vector_title('left', ax_lvec.index + 1)
plotWiggles(ax_lvec, ax_lvec.volume[:, :, ax_lvec.index], times, rinterval,
receiverPoints, leftTitle, 'left', plotParams)
ax_rvec.volume = V
ax_rvec.index = 0
rightTitle = vector_title('right', ax_rvec.index + 1)
plotWiggles(ax_rvec, ax_rvec.volume[:, :, ax_rvec.index], times, sinterval,
sourcePoints, rightTitle, 'right', plotParams)
fig_vec.tight_layout()
fig_vec.canvas.mpl_connect('key_press_event', lambda event: process_key_vectors(event, times, rinterval, sinterval,
receiverPoints, sourcePoints, plotParams))
#==============================================================================
# plot the singular values
# figure and axis for singular values
fig_vals, ax_vals = setFigure(num_axes=1, mode=plotParams['view_mode'])
n = np.arange(1, k + 1, 1)
kappa = s[0] / s[-1] # condition number = max(s) / min(s)
ax_vals.plot(n, s, '.', clip_on=False, markersize=9, label=r'Condition Number: %0.1e' %(kappa), color=ax_vals.pointcolor)
ax_vals.set_xlabel('n', color=ax_vals.labelcolor)
ax_vals.set_ylabel('$\sigma_n$', color=ax_vals.labelcolor)
legend = ax_vals.legend(title='Singular Values', loc='upper center', bbox_to_anchor=(0.5, 1.25),
markerscale=0, handlelength=0, handletextpad=0, fancybox=True, shadow=True,
fontsize='large')
legend.get_title().set_fontsize('large')
ax_vals.set_xlim([1, k])
ax_vals.set_ylim(bottom=0)
ax_vals.locator_params(axis='y', nticks=6)
ax_vals.ticklabel_format(style='sci', axis='y', scilimits=(0,0))
fig_vals.tight_layout()
fig_vals.savefig('singularValues.' + args.format, format=args.format, bbox_inches='tight', facecolor=fig_vals.get_facecolor())
plt.show()
def cli():
parser = argparse.ArgumentParser()
parser.add_argument(
'--nfo',
action='store_true',
help='''Compute or plot the singular-value decomposition of the
near-field operator (NFO).''')
parser.add_argument(
'--lso',
action='store_true',
help='''Compute or plot the singular-value decomposition of the
Lippmann-Schwinger operator (LSO).''')
parser.add_argument(
'--numVals',
'-k',
type=int,
help='''Specify the number of singular values/vectors to compute.
Must a positive integer between 1 and the order of the square
input matrix.''')
parser.add_argument(
'--domain',
'-d',
type=str,
choices=['time', 'freq'],
help='''Specify whether to compute the singular-value decomposition in
the time domain or frequency domain. Default is set to frequency domain
for faster, more accurate performance.''')
parser.add_argument(
'--plot',
'-p',
action='store_true',
help='''Plot the computed singular values and vectors.''')
parser.add_argument(
'--format',
'-f',
type=str,
default='pdf',
choices=['png', 'pdf', 'ps', 'eps', 'svg'],
help=
'''Specify the image format of the saved file. Accepted formats are png, pdf,
ps, eps, and svg. Default format is set to pdf.''')
parser.add_argument(
'--mode',
type=str,
choices=['light', 'dark'],
required=False,
help='''Specify whether to view plots in light mode for daytime viewing
or dark mode for nighttime viewing.
Mode must be either \'light\' or \'dark\'.''')
args = parser.parse_args()
if args.nfo and not args.lso:
operatorType = 'near-field operator'
inputType = 'data'
try:
SVD = np.load('NFO_SVD.npz')
s = SVD['s']
Uh = SVD['Uh']
V = SVD['V']
domain = SVD['domain']
except FileNotFoundError:
s, Uh, V, domain = None, None, None, 'freq'
elif not args.nfo and args.lso:
operatorType = 'Lippmann-Schwinger operator'
inputType = 'test functions'
try:
SVD = np.load('LSO_SVD.npz')
s = SVD['s']
Uh = SVD['Uh']
V = SVD['V']
domain = SVD['domain']
except FileNotFoundError:
s, Uh, V, domain = None, None, None, 'freq'
elif args.nfo and args.lso:
sys.exit(
textwrap.dedent('''
UsageError: Please specify only one of the arguments \'--nfo\' or \'--lso\'.
'''))
else:
sys.exit(
textwrap.dedent('''
For which operator would you like to compute or plot a singular-value decomposition?
Enter:
vzsvd --nfo
for the near-field operator or
vzsvd --lso
for the Lippmann-Schwinger operator.
'''))
#==============================================================================
# if an SVD already exists...
if any(v is not None for v in
[s, Uh, V]) and args.numVals is not None and args.plot is True:
if args.numVals >= 1 and args.numVals == len(s):
userResponded = False
print(
textwrap.dedent('''
A singular-value decomposition of the {s} for {n} values/vectors already exists.
What would you like to do?
Enter '1' to specify a new number of values/vectors to compute. (Default)
Enter '2' to recompute a singular-value decomposition for {n} values/vectors.
Enter 'q/quit' to exit.
'''.format(s=operatorType, n=args.numVals)))
while userResponded == False:
answer = input('Action: ')
if answer == '' or answer == '1':
k = int(
input(
'Please specify the number of singular values/vectors to compute: '
))
if isValid(k):
print('Proceeding with numVals = %s...' % (k))
userResponded = True
computeSVD = True
break
else:
break
elif answer == '2':
k = args.numVals
print(
'Recomputing SVD of the %s for %s singular values/vectors...'
% (operatorType, k))
userResponded = True
computeSVD = True
elif answer == 'q' or answer == 'quit':
sys.exit('Exiting program.\n')
else:
print(
'Invalid response. Please enter \'1\', \'2\', or \'q/quit\'.'
)
elif args.numVals >= 1 and args.numVals != len(s):
k = args.numVals
computeSVD = True
elif args.numVals < 1:
userResponded = False
print(
textwrap.dedent('''
ValueError: Argument '-k/--numVals' must be a positive integer
between 1 and the order of the square input matrix. The parameter will
be set to the default value of 6.
What would you like to do?
Enter '1' to specify a value of the parameter. (Default)
Enter '2' to proceed with the default value.
Enter 'q/quit' exit the program.
'''))
while userResponded == False:
answer = input('Action: ')
if answer == '' or answer == '1':
k = int(
input(
'Please specify the number of singular values/vectors to compute: '
))
if isValid(k):
print('Proceeding with numVals = %s...' % (k))
userResponded = True
computeSVD = True
break
else:
break
elif answer == '2':
k = 6
print('Proceeding with the default value numVals = %s...' %
(k))
computeSVD = True
userResponded = True
break
elif answer == 'q' or answer == 'quit':
sys.exit('Exiting program.\n')
else:
print(
'Invalid response. Please enter \'1\', \'2\', or \'q/quit\'.'
)
elif all(v is not None for v in
[s, Uh, V]) and args.numVals is None and args.plot is True:
computeSVD = False
elif all(v is not None for v in
[s, Uh, V]) and args.numVals is not None and args.plot is False:
if args.numVals >= 1 and args.numVals == len(s):
userResponded = False
print(
textwrap.dedent('''
A singular-value decomposition of the {s} for {n} values/vectors already exists.
What would you like to do?
Enter '1' to specify a new number of values/vectors to compute. (Default)
Enter '2' to recompute a singular-value decomposition for {n} values/vectors.
Enter 'q/quit' to exit.
'''.format(s=operatorType, n=args.numVals)))
while userResponded == False:
answer = input('Action: ')
if answer == '' or answer == '1':
k = int(
input(
'Please specify the number of singular values/vectors to compute: '
))
if isValid(k):
print('Proceeding with numVals = %s...' % (k))
userResponded = True
computeSVD = True
break
else:
break
elif answer == '2':
k = args.numVals
print(
'Recomputing SVD of the %s for %s singular values/vectors...'
% (operatorType, k))
userResponded = True
computeSVD = True
elif answer == 'q' or answer == 'quit':
sys.exit('Exiting program.\n')
else:
print(
'Invalid response. Please enter \'1\', \'2\', or \'q/quit\'.'
)
elif args.numVals >= 1 and args.numVals != len(s):
k = args.numVals
computeSVD = True
elif args.numVals < 1:
userResponded = False
print(
textwrap.dedent('''
ValueError: Argument '-k/--numVals' must be a positive integer
between 1 and the order of the square input matrix. The parameter will
be set to the default value of 6.
What would you like to do?
Enter '1' to specify a value of the parameter. (Default)
Enter '2' to proceed with the default value.
Enter 'q/quit' exit the program.
'''))
while userResponded == False:
answer = input('Action: ')
if answer == '' or answer == '1':
k = int(
input(
'Please specify the number of singular values/vectors to compute: '
))
if isValid(k):
print('Proceeding with numVals = %s...' % (k))
userResponded = True
computeSVD = True
break
else:
break
elif answer == '2':
k = 6
print('Proceeding with the default value numVals = %s...' %
(k))
computeSVD = True
userResponded = True
break
elif answer == 'q' or answer == 'quit':
sys.exit('Exiting program.\n')
else:
print(
'Invalid response. Please enter \'1\', \'2\', or \'q/quit\'.'
)
elif all(v is not None for v in
[s, Uh, V]) and args.numVals is None and args.plot is False:
sys.exit(
textwrap.dedent('''
No action specified. A singular-value decomposition of the %s
for %s values/vectors already exists. Please specify at least one of '-k/--numVals'
or '-p/--plot' arguments with 'vzsvd' command.
''' % (operatorType, len(s))))
#==============================================================================
# if an SVD does not already exist...
elif any(v is None for v in
[s, Uh, V]) and args.numVals is not None and args.plot is True:
if args.numVals >= 1:
computeSVD = True
k = args.numVals
elif args.numVals < 1:
userResponded = False
print(
textwrap.dedent('''
ValueError: Argument '-k/--numVals' must be a positive integer
between 1 and the order of the square input matrix. The parameter will
be set to the default value of 6.
What would you like to do?
Enter '1' to specify a value of the parameter. (Default)
Enter '2' to proceed with the default value.
Enter 'q/quit' exit the program.
'''))
while userResponded == False:
answer = input('Action: ')
if answer == '' or answer == '1':
k = int(
input(
'Please specify the number of singular values/vectors to compute: '
))
if isValid(k):
print('Proceeding with numVals = %s...' % (k))
userResponded = True
computeSVD = True
break
else:
break
elif answer == '2':
k = 6
print('Proceeding with the default value numVals = %s...' %
(k))
computeSVD = True
userResponded = True
break
elif answer == 'q' or answer == 'quit':
sys.exit('Exiting program.\n')
else:
print(
'Invalid response. Please enter \'1\', \'2\', or \'q/quit\'.'
)
elif any(v is None for v in
[s, Uh, V]) and args.numVals is None and args.plot is True:
userResponded = False
print(
textwrap.dedent('''
PlotError: A singular-value decomposition of the {s} does not exist. A plot will be
generated after a singular-value decomposition has been computed.
Enter '1' to specify a number of singular values/vectors to compute. (Default)
Enter 'q/quit' to exit.
'''.format(s=operatorType)))
while userResponded == False:
answer = input('Action: ')
if answer == '' or answer == '1':
k = int(
input(
'Please specify the number of singular values/vectors to compute: '
))
if isValid(k):
print('Proceeding with numVals = %s...' % (k))
userResponded = True
computeSVD = True
break
else:
break
elif answer == 'q' or answer == 'quit':
sys.exit('Exiting program.\n')
else:
print('Invalid response. Please enter \'1\', or \'q/quit\'.')
elif any(v is None for v in
[s, Uh, V]) and args.numVals is not None and args.plot is False:
if args.numVals >= 1:
k = args.numVals
computeSVD = True
elif args.numVals < 1:
userResponded = False
print(
textwrap.dedent('''
ValueError: Argument '-k/--numVals' must be a positive integer
between 1 and the order of the square input matrix. The parameter will
be set to the default value of 6.
What would you like to do?
Enter '1' to specify a value of the parameter. (Default)
Enter '2' to proceed with the default value.
Enter 'q/quit' exit the program.
'''))
while userResponded == False:
answer = input('Action: ')
if answer == '' or answer == '1':
k = int(
input(
'Please specify the number of singular values/vectors to compute: '
))
if isValid(k):
print('Proceeding with numVals = %s...' % (k))
userResponded = True
computeSVD = True
break
else:
break
elif answer == '2':
k = 6
print('Proceeding with the default value numVals = %s...' %
(k))
computeSVD = True
userResponded = True
break
elif answer == 'q' or answer == 'quit':
sys.exit('Exiting program.\n')
else:
print(
'Invalid response. Please enter \'1\', \'2\', or \'q/quit\'.'
)
elif any(v is None for v in
[s, Uh, V]) and args.numVals is None and args.plot is False:
sys.exit(
textwrap.dedent('''
Nothing to be done. A singular-value decomposition of the {s} does not exist.
Please specify at least one of '-k/--numVals' or '-p/--plot'
arguments with 'vzsvd' command.
'''.format(s=operatorType)))
#==============================================================================
# Read in data files
datadir = np.load('datadir.npz')
receiverPoints = np.load(str(datadir['receivers']))
recordingTimes = np.load(str(datadir['recordingTimes']))
dt = recordingTimes[1] - recordingTimes[0]
if Path('window.npz').exists():
windowDict = np.load('window.npz')
# Apply the receiver window
rstart = windowDict['rstart']
rstop = windowDict['rstop']
rstep = windowDict['rstep']
# Apply the time window
tstart = windowDict['tstart']
tstop = windowDict['tstop']
tstep = windowDict['tstep']
# Convert time window parameters to corresponding array indices
Tstart = int(round(tstart / dt))
Tstop = int(round(tstop / dt))
else:
rstart = 0
rstop = receiverPoints.shape[0]
rstep = 1
tstart = recordingTimes[0]
tstop = recordingTimes[-1]
Tstart = 0
Tstop = len(recordingTimes)
tstep = 1
# Apply the receiver window
rinterval = np.arange(rstart, rstop, rstep)
receiverPoints = receiverPoints[rinterval, :]
# Apply the time window
tinterval = np.arange(Tstart, Tstop, tstep)
recordingTimes = recordingTimes[tinterval]
# Used for getting time and frequency units
if Path('plotParams.pkl').exists():
plotParams = pickle.load(open('plotParams.pkl', 'rb'))
else:
plotParams = default_params()
if computeSVD:
# get time units for printing time windows or time shifts
tu = plotParams['tu']
if args.nfo:
if Path('noisyData.npz').exists():
userResponded = False
print(
textwrap.dedent('''
Detected that band-limited noise has been added to the data array.
Would you like to compute an SVD of the noisy data? ([y]/n)
Enter 'q/quit' exit the program.
'''))
while userResponded == False:
answer = input('Action: ')
if answer == '' or answer == 'y' or answer == 'yes':
print(
'Proceeding with singular-value decomposition using noisy data...'
)
# read in the noisy data array
X = np.load('noisyData.npz')['noisyData']
userResponded = True
elif answer == 'n' or answer == 'no':
print(
'Proceeding with singular-value decomposition using noise-free data...'
)
# read in the recorded data array
X = np.load(str(datadir['recordedData']))
userResponded = True
elif answer == 'q' or answer == 'quit':
sys.exit('Exiting program.\n')
else:
print(
'Invalid response. Please enter \'y/yes\', \'n\no\', or \'q/quit\'.'
)
else:
# read in the recorded data array
X = np.load(str(datadir['recordedData']))
if Path('window.npz').exists():
print('Detected user-specified window:\n')
# For display/printing purposes, count receivers with one-based
# indexing. This amounts to incrementing the rstart parameter by 1
print('window @ receivers : start =', rstart + 1)
print('window @ receivers : stop =', rstop)
print('window @ receivers : step =', rstep, '\n')
if tu != '':
print('window @ time : start = %0.2f %s' % (tstart, tu))
print('window @ time : stop = %0.2f %s' % (tstop, tu))
else:
print('window @ time : start =', tstart)
print('window @ time : stop =', tstop)
print('window @ time : step =', tstep, '\n')
# Apply the source window
slabel = windowDict['slabel']
sstart = windowDict['sstart']
sstop = windowDict['sstop']
sstep = windowDict['sstep']
sinterval = np.arange(sstart, sstop, sstep)
# For display/printing purposes, count recordings/sources with one-based
# indexing. This amounts to incrementing the sstart parameter by 1
print('window @ %s : start = %s' % (slabel, sstart + 1))
print('window @ %s : stop = %s' % (slabel, sstop))
print('window @ %s : step = %s\n' % (slabel, sstep))
print('Applying window to data volume...')
X = X[rinterval, :, :]
X = X[:, tinterval, :]
X = X[:, :, sinterval]
Nr, Nt, Ns = X.shape
# Apply tapered cosine (Tukey) window to time signals.
# This ensures the fast fourier transform (FFT) used in
# the definition of the matrix-vector product below is
# acting on a function that is continuous at its edges.
peakFreq = pulseFun.peakFreq
# Np : Number of samples in the dominant period T = 1 / peakFreq
Np = int(round(1 / (tstep * dt * peakFreq)))
# alpha is set to taper over 6 of the dominant period of the
# pulse function (3 periods from each end of the signal)
alpha = 6 * Np / Nt
print('Tapering time signals with Tukey window: %d' %
(int(round(alpha * 100))) + '%')
TukeyWindow = tukey(Nt, alpha)
X *= TukeyWindow[None, :, None]
else:
Nr, Nt, Ns = X.shape
elif args.lso:
if Path('samplingGrid.npz').exists():
samplingGrid = np.load('samplingGrid.npz')
x = samplingGrid['x']
y = samplingGrid['y']
tau = samplingGrid['tau']
if 'z' in samplingGrid:
z = samplingGrid['z']
else:
z = None
else:
sys.exit(
textwrap.dedent('''
A sampling grid needs to be set up before computing a
singular-value decomposition of the %s.
Enter:
vzgrid --help
from the command-line for more information on how to set up a
sampling grid.
''' % (operatorType)))
pulse = lambda t: pulseFun.pulse(t)
velocity = pulseFun.velocity
peakFreq = pulseFun.peakFreq
peakTime = pulseFun.peakTime
if Path('VZTestFuncs.npz').exists():
print(
'\nDetected that free-space test functions have already been computed...'
)
print(
'Checking consistency with current space-time sampling grid...'
)
TFDict = np.load('VZTestFuncs.npz')
if samplingIsCurrent(TFDict, receiverPoints, recordingTimes,
velocity, tau, x, y, z, peakFreq,
peakTime):
X = TFDict['TFarray']
sourcePoints = TFDict['samplingPoints']
print('Moving forward to SVD...')
else:
print('Recomputing test functions...')
# set up the convolution times based on length of recording time interval
T = recordingTimes[-1] - recordingTimes[0]
convolutionTimes = np.linspace(-T, T,
2 * len(recordingTimes) - 1)
if tau[0] != 0:
if tu != '':
print(
'Recomputing test functions for focusing time %0.2f %s...'
% (tau[0], tu))
else:
print(
'Recomputing test functions for focusing time %0.2f...'
% (tau[0]))
X, sourcePoints = sampleSpace(
receiverPoints, convolutionTimes - tau[0],
velocity, x, y, z, pulse)
else:
X, sourcePoints = sampleSpace(receiverPoints,
convolutionTimes,
velocity, x, y, z, pulse)
if z is None:
np.savez('VZTestFuncs.npz',
TFarray=X,
time=recordingTimes,
receivers=receiverPoints,
peakFreq=peakFreq,
peakTime=peakTime,
velocity=velocity,
x=x,
y=y,
tau=tau,
samplingPoints=sourcePoints)
else:
np.savez('VZTestFuncs.npz',
TFarray=X,
time=recordingTimes,
receivers=receiverPoints,
peakFreq=peakFreq,
peakTime=peakTime,
velocity=velocity,
x=x,
y=y,
z=z,
tau=tau,
samplingPoints=sourcePoints)
else:
print(
'\nComputing free-space test functions for the current space-time sampling grid...'
)
if tau[0] != 0:
if tu != '':
print(
'Computing test functions for focusing time %0.2f %s...'
% (tau[0], tu))
else:
print(
'Computing test functions for focusing time %0.2f...'
% (tau[0]))
X, sourcePoints = sampleSpace(receiverPoints,
recordingTimes - tau[0],
velocity, x, y, z, pulse)
else:
X, sourcePoints = sampleSpace(receiverPoints,
recordingTimes, velocity, x,
y, z, pulse)
if z is None:
np.savez('VZTestFuncs.npz',
TFarray=X,
time=recordingTimes,
receivers=receiverPoints,
peakFreq=peakFreq,
peakTime=peakTime,
velocity=velocity,
x=x,
y=y,
tau=tau,
samplingPoints=sourcePoints)
else:
np.savez('VZTestFuncs.npz',
TFarray=X,
time=recordingTimes,
receivers=receiverPoints,
peakFreq=peakFreq,
peakTime=peakTime,
velocity=velocity,
x=x,
y=y,
z=z,
tau=tau,
samplingPoints=sourcePoints)
Nr, Nt, Ns = X.shape
#==============================================================================
if args.domain is not None:
domain = args.domain
if domain == 'freq':
# Transform convolutional operator into frequency domain and bandpass for efficient SVD
print('Transforming %s to the frequency domain...' % (inputType))
N = nextPow2(2 * Nt)
X = np.fft.rfft(X, n=N, axis=1)
if plotParams['fmax'] is None:
freqs = np.fft.rfftfreq(N, tstep * dt)
plotParams['fmax'] = np.max(freqs)
# Apply the frequency window
fmin = plotParams['fmin']
fmax = plotParams['fmax']
fu = plotParams['fu'] # frequency units (e.g., Hz)
if fu != '':
print('Applying bandpass filter: [%0.2f %s, %0.2f %s]' %
(fmin, fu, fmax, fu))
else:
print('Applying bandpass filter: [%0.2f, %0.2f]' %
(fmin, fmax))
df = 1.0 / (N * tstep * dt)
startIndex = int(round(fmin / df))
stopIndex = int(round(fmax / df))
finterval = np.arange(startIndex, stopIndex, 1)
X = X[:, finterval, :]
#==============================================================================
# Compute the k largest singular values (which='LM') of the operator A
# Singular values are elements of the vector 's'
# Left singular vectors are columns of 'U'
# Right singular vectors are columns of 'V'
A = asConvolutionalOperator(X)
if k == 1:
print('Computing SVD of the %s for 1 singular value/vector...' %
(operatorType))
else:
print('Computing SVD of the %s for %s singular values/vectors...' %
(operatorType, k))
startTime = time.time()
U, s, Vh = svds(A, k, which='LM')
endTime = time.time()
print('Elapsed time:', humanReadable(endTime - startTime), '\n')
# sort the singular values and corresponding vectors in descending order
# (i.e., largest to smallest)
index = s.argsort()[::-1]
s = s[index]
Uh = U[:, index].conj().T
V = Vh[index, :].conj().T
# Write binary output with numpy
if args.nfo:
np.savez('NFO_SVD.npz', s=s, Uh=Uh, V=V, domain=domain)
elif args.lso:
np.savez('LSO_SVD.npz', s=s, Uh=Uh, V=V, domain=domain)
#==============================================================================
if args.plot and all(v is not None for v in [s, Uh, V]):
Nr = receiverPoints.shape[0]
Nt = len(recordingTimes)
try:
k
except NameError:
k = len(s)
if args.domain is not None and domain != args.domain:
if domain == 'freq':
s1 = 'time'
s2 = 'frequency'
else:
s1 = 'frequency'
s2 = 'time'
sys.exit(
textwrap.dedent('''
Error: Attempted to plot the singular-value decomposition in the %s
domain, but the decomposition was computed in the %s domain.
''' % (s1, s2)))
if domain == 'freq':
# plot singular vectors in frequency domain
N = nextPow2(2 * Nt)
freqs = np.fft.rfftfreq(N, tstep * dt)
if plotParams['fmax'] is None:
plotParams['fmax'] = np.max(freqs)
# Apply the frequency window
fmin = plotParams['fmin']
fmax = plotParams['fmax']
df = 1.0 / (N * tstep * dt)
startIndex = int(round(fmin / df))
stopIndex = int(round(fmax / df))
finterval = np.arange(startIndex, stopIndex, 1)
freqs = freqs[finterval]
fmax = freqs[-1]
M = len(freqs)
Ns = int(V.shape[0] / M)
U = np.reshape(Uh.conj().T, (Nr, M, k))
V = np.reshape(V, (Ns, M, k))
else: # domain == 'time'
M = 2 * Nt - 1
Ns = int(V.shape[0] / M)
U = np.reshape(Uh.T, (Nr, M, k))
V = np.reshape(V, (Ns, M, k))
T = recordingTimes[-1] - recordingTimes[0]
times = np.linspace(-T, T, M)
if args.nfo: # Near-field operator
try:
sinterval
except NameError:
if Path('window.npz').exists():
sstart = windowDict['sstart']
sstop = windowDict['sstop']
sstep = windowDict['sstep']
else:
sstart = 0
sstop = Ns
sstep = 1
sinterval = np.arange(sstart, sstop, sstep)
if 'sources' in datadir:
sourcePoints = np.load(str(datadir['sources']))
sourcePoints = sourcePoints[sinterval, :]
else:
sourcePoints = None
else:
# if args.lso (Lippmann-Schwinger operator)
# in the case of the Lippmann-Schwinger operator, 'sourcePoints'
# correspond to sampling points, which should always exist.
try:
sourcePoints
except NameError:
if Path('VZTestFuncs.npz').exists():
TFDict = np.load('VZTestFuncs.npz')
sourcePoints = TFDict['samplingPoints']
else:
sys.exit(
textwrap.dedent('''
Error: A sampling grid must exist and test functions computed
before a singular-value decomposition of the Lippmann-Schwinger
operator can be computed or plotted.
'''))
sstart = 0
sstop = sourcePoints.shape[0]
sstep = 1
sinterval = np.arange(sstart, sstop, sstep)
# increment source/recording interval and receiver interval to be consistent
# with one-based indexing (i.e., count from one instead of zero)
sinterval += 1
rinterval += 1
rstart += 1
sstart += 1
if args.mode is not None:
plotParams['view_mode'] = args.mode
pickle.dump(plotParams, open('plotParams.pkl', 'wb'),
pickle.HIGHEST_PROTOCOL)
remove_keymap_conflicts({'left', 'right', 'up', 'down', 'save'})
if domain == 'freq':
# plot the left singular vectors
fig_lvec, ax_lvec_r, ax_lvec_i = setFigure(
num_axes=2, mode=plotParams['view_mode'])
ax_lvec_r.volume = U.real
ax_lvec_i.volume = U.imag
ax_lvec_r.index = 0
ax_lvec_i.index = 0
fig_lvec.suptitle('Left-Singular Vector',
color=ax_lvec_r.titlecolor,
fontsize=16)
fig_lvec.subplots_adjust(bottom=0.27, top=0.86)
leftTitle_r = vector_title('left', ax_lvec_r.index + 1, 'real')
leftTitle_i = vector_title('left', ax_lvec_i.index + 1, 'imag')
for ax, title in zip([ax_lvec_r, ax_lvec_i],
[leftTitle_r, leftTitle_i]):
left_im = plotFreqVectors(ax, ax.volume[:, :, ax.index], freqs,
fmin, fmax, rstart, rinterval,
receiverPoints, title, 'left',
plotParams)
lp0 = ax_lvec_r.get_position().get_points().flatten()
lp1 = ax_lvec_i.get_position().get_points().flatten()
left_cax = fig_lvec.add_axes([lp0[0], 0.12, lp1[2] - lp0[0], 0.03])
lcbar = fig_lvec.colorbar(left_im,
left_cax,
orientation='horizontal')
lcbar.outline.set_edgecolor(ax_lvec_r.cbaredgecolor)
lcbar.ax.tick_params(axis='x', colors=ax_lvec_r.labelcolor)
lcbar.ax.yaxis.set_major_formatter(FormatStrFormatter('%.1f'))
lcbar.set_label('Amplitude',
labelpad=5,
rotation=0,
fontsize=12,
color=ax_lvec_r.labelcolor)
fig_lvec.canvas.mpl_connect(
'key_press_event', lambda event: process_key_vectors(
event, freqs, fmin, fmax, rstart, sstart, rinterval,
sinterval, receiverPoints, sourcePoints, plotParams,
'cmplx_left'))
# plot the right singular vectors
fig_rvec, ax_rvec_r, ax_rvec_i = setFigure(
num_axes=2, mode=plotParams['view_mode'])
ax_rvec_r.volume = V.real
ax_rvec_i.volume = V.imag
ax_rvec_r.index = 0
ax_rvec_i.index = 0
fig_rvec.suptitle('Right-Singular Vector',
color=ax_rvec_r.titlecolor,
fontsize=16)
fig_rvec.subplots_adjust(bottom=0.27, top=0.86)
rightTitle_r = vector_title('right', ax_rvec_r.index + 1, 'real')
rightTitle_i = vector_title('right', ax_rvec_i.index + 1, 'imag')
for ax, title in zip([ax_rvec_r, ax_rvec_i],
[rightTitle_r, rightTitle_i]):
right_im = plotFreqVectors(ax, ax.volume[:, :, ax.index],
freqs, fmin, fmax, sstart,
sinterval, sourcePoints, title,
'right', plotParams)
rp0 = ax_rvec_r.get_position().get_points().flatten()
rp1 = ax_rvec_i.get_position().get_points().flatten()
right_cax = fig_rvec.add_axes(
[rp0[0], 0.12, rp1[2] - rp0[0], 0.03])
rcbar = fig_rvec.colorbar(right_im,
right_cax,
orientation='horizontal')
rcbar.outline.set_edgecolor(ax_rvec_r.cbaredgecolor)
rcbar.ax.tick_params(axis='x', colors=ax_rvec_r.labelcolor)
rcbar.ax.yaxis.set_major_formatter(FormatStrFormatter('%.1f'))
rcbar.set_label('Amplitude',
labelpad=5,
rotation=0,
fontsize=12,
color=ax_lvec_r.labelcolor)
fig_rvec.canvas.mpl_connect(
'key_press_event', lambda event: process_key_vectors(
event, freqs, fmin, fmax, rstart, sstart, rinterval,
sinterval, receiverPoints, sourcePoints, plotParams,
'cmplx_right'))
else:
# domain == 'time'
fig_vec, ax_lvec, ax_rvec = setFigure(num_axes=2,
mode=plotParams['view_mode'])
ax_lvec.volume = U
ax_lvec.index = 0
leftTitle = vector_title('left', ax_lvec.index + 1)
plotWiggles(ax_lvec, ax_lvec.volume[:, :, ax_lvec.index], times,
-T, T, rstart, rinterval, receiverPoints, leftTitle,
'left', plotParams)
ax_rvec.volume = V
ax_rvec.index = 0
rightTitle = vector_title('right', ax_rvec.index + 1)
plotWiggles(ax_rvec, ax_rvec.volume[:, :, ax_rvec.index], times,
-T, T, sstart, sinterval, sourcePoints, rightTitle,
'right', plotParams)
fig_vec.tight_layout()
fig_vec.canvas.mpl_connect(
'key_press_event', lambda event: process_key_vectors(
event, times, -T, T, rstart, sstart, rinterval, sinterval,
receiverPoints, sourcePoints, plotParams))
#==============================================================================
# plot the singular values
# figure and axis for singular values
fig_vals, ax_vals = setFigure(num_axes=1, mode=plotParams['view_mode'])
n = np.arange(1, k + 1, 1)
kappa = s[0] / s[-1] # condition number = max(s) / min(s)
ax_vals.plot(n,
s,
'.',
clip_on=False,
markersize=9,
label=r'Condition Number: %0.1e' % (kappa),
color=ax_vals.pointcolor)
ax_vals.set_xlabel('n', color=ax_vals.labelcolor)
ax_vals.set_ylabel('$\sigma_n$', color=ax_vals.labelcolor)
legend = ax_vals.legend(title='Singular Values',
loc='upper center',
bbox_to_anchor=(0.5, 1.25),
markerscale=0,
handlelength=0,
handletextpad=0,
fancybox=True,
shadow=True,
fontsize='large')
legend.get_title().set_fontsize('large')
ax_vals.set_xlim([1, k])
ax_vals.set_ylim(bottom=0)
ax_vals.locator_params(axis='y', nticks=6)
ax_vals.ticklabel_format(style='sci', axis='y', scilimits=(0, 0))
fig_vals.tight_layout()
fig_vals.savefig('singularValues.' + args.format,
format=args.format,
bbox_inches='tight',
facecolor=fig_vals.get_facecolor())
plt.show()