def loss(theta1, theta2, angles_rad, phase_rad, TE, TR):
T = len(angles_rad)
x1 = epg.FSE_signal(
angles_rad, phase_rad, TE, theta1['T1'],
theta1['T2']) * (1 - exp(-(TR - T * TE) / theta1['T1']))
x2 = epg.FSE_signal(
angles_rad, phase_rad, TE, theta2['T1'],
theta2['T2']) * (1 - exp(-(TR - T * TE) / theta2['T1']))
return 0.5 * np.linalg.norm(x1, ord=2)**2 + 0.5 * np.linalg.norm(
x2, ord=2)**2 - 0.5 * np.vdot(x1.ravel(), x2.ravel()) - 0.5 * np.vdot(
x2.ravel(), x1.ravel())
def normalized_loss(theta1, theta2, angles_rad, phase_rad, TE, TR):
T = len(angles_rad)
x1 = epg.FSE_signal(
angles_rad, phase_rad, TE, theta1['T1'],
theta1['T2']) * (1 - exp(-(TR - T * TE) / theta1['T1']))
x2 = epg.FSE_signal(
angles_rad, phase_rad, TE, theta2['T1'],
theta2['T2']) * (1 - exp(-(TR - T * TE) / theta2['T1']))
x1 = x1 / np.linalg.norm(x1, ord=2)
x2 = x2 / np.linalg.norm(x2, ord=2)
return -0.5 * (np.vdot(x1.ravel(), x2.ravel()) +
np.vdot(x2.ravel(), x1.ravel()))
def loss_prime(theta1,
theta2,
angles_rad,
phase_rad,
TE,
TR,
excitation_dict=None,
inversion_dict=None):
T = len(angles_rad)
x1 = epg.FSE_signal(angles_rad,
phase_rad,
TE,
theta1['T1'],
theta1['T2'],
excitation_dict=excitation_dict,
inversion_dict=inversion_dict).ravel() * (
1 - exp(-(TR - T * TE) / theta1['T1']))
x2 = epg.FSE_signal(angles_rad,
phase_rad,
TE,
theta2['T1'],
theta2['T2'],
excitation_dict=excitation_dict,
inversion_dict=inversion_dict).ravel() * (
1 - exp(-(TR - T * TE) / theta1['T1']))
T = len(angles_rad)
alpha_prime = np.zeros((T, ))
angle_e_prime = 0.
angle_TI_prime = 0.
for i in range(T):
x1_prime = sig_prime_i(theta1, angles_rad, phase_rad, i).ravel() * (
1 - exp(-(TR - T * TE) / theta1['T1']))
x2_prime = sig_prime_i(theta2, angles_rad, phase_rad, i).ravel() * (
1 - exp(-(TR - T * TE) / theta2['T1']))
M1 = 0.5 * (np.vdot(x1, x1_prime) + np.vdot(x1_prime, x1))
M2 = 0.5 * (np.vdot(x2, x2_prime) + np.vdot(x2_prime, x2))
M3 = 0.5 * (np.vdot(x2_prime, x1) + np.vdot(x1, x2_prime))
M4 = 0.5 * (np.vdot(x1_prime, x2) + np.vdot(x2, x1_prime))
alpha_prime[i] = np.real(M1 + M2 - M3 - M4)
return alpha_prime
def plot_vals(self, thetas):
plt.subplot(2, 1, 1)
plt.plot(range(self.T), RAD2DEG(self.angles_rad), 'o-')
plt.xlim((0, self.T))
plt.subplot(2, 1, 2)
for theta in thetas:
plt.plot(
range(self.T),
epg.FSE_signal(self.angles_rad, self.phase_rad, self.TE,
theta['T1'], theta['T2']))
plt.xlim((0, self.T))
plt.ylim((0, 1))
def forward(self, theta):
return epg.FSE_signal(self.angles_rad, self.phase_rad, TE, theta['T1'],
theta['T2']).ravel()
angles = 150 * np.ones((T, ))
angles = read_angles('../data/flipangles.txt.408183520')
TT = len(angles)
if TT < T:
T = TT
else:
angles = angles[:T]
phases = np.zeros(angles.shape)
angles_rad = DEG2RAD(angles)
phase_rad = DEG2RAD(phases)
S = epg.FSE_signal(angles_rad, phase_rad, TE, T1, T2)
S2 = abs(S)
theta1 = {'T1': 1000e-3, 'T2': 20e-3}
theta2 = {'T1': 1000e-3, 'T2': 100e-3}
t1 = time.time()
NG = numerical_gradient(theta1, theta2, angles_rad, phase_rad, TE, TR)
t2 = time.time()
LP = loss_prime(theta1, theta2, angles_rad, phase_rad, TE, TR)
t3 = time.time()
NG_time = t2 - t1
LP_time = t3 - t2
print 'Numerical Gradient\t(%03.3f s)\t' % NG_time, NG