def loss(theta1, theta2, angles_rad, TE, TR):
T = len(angles_rad)
x1 = epg.FSE_signal(angles_rad, TE, theta1['T1'], theta1['T2']) * (1 - exp(-(TR - T * TE)/theta1['T1']))
x2 = epg.FSE_signal(angles_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 - np.dot(x1.ravel(), x2.ravel())
def loss_prime(theta1, theta2, angles_rad, TE, TR):
T = len(angles_rad)
x1 = epg.FSE_signal(angles_rad, TE, theta1['T1'], theta1['T2']).ravel() * (
1 - exp(-(TR - T * TE) / theta1['T1']))
x2 = epg.FSE_signal(angles_rad, TE, theta2['T1'], theta2['T2']).ravel() * (
1 - exp(-(TR - T * TE) / theta2['T1']))
T = len(angles_rad)
alpha_prime = np.zeros((T, ))
for i in range(T):
x1_prime = epg.FSE_signal_prime_alpha_idx(
angles_rad, TE, theta1['T1'], theta1['T2'],
i).ravel() * (1 - exp(-(TR - T * TE) / theta1['T1']))
x2_prime = epg.FSE_signal_prime_alpha_idx(
angles_rad, TE, theta2['T1'], theta2['T2'],
i).ravel() * (1 - exp(-(TR - T * TE) / theta2['T1']))
M1 = np.dot(x1, x1_prime)
M2 = np.dot(x2, x2_prime)
M3 = np.dot(x1, x2_prime)
M4 = np.dot(x2, x1_prime)
alpha_prime[i] = M1 + M2 - M3 - M4
return alpha_prime
def normalized_loss(theta1, theta2, angles_rad, TE, TR):
T = len(angles_rad)
x1 = epg.FSE_signal(angles_rad, TE, theta1['T1'], theta1['T2']) * (1 - exp(-(TR - T * TE)/theta1['T1']))
x2 = epg.FSE_signal(angles_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 - np.dot(x1.ravel(), x2.ravel())
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.TE, theta['T1'], theta['T2']))
plt.xlim((0,self.T))
plt.ylim((0,1))
def loss(thetas, angles_rad, TE, TR):
T = len(angles_rad)
l = 0
for theta in thetas:
x1 = epg.FSE_signal(angles_rad, TE, theta['T1'], theta['T2']) * (1 - exp(-(TR - T * TE)/theta['T1']))
l += 0.5 * np.dot(x1.ravel(), x1.ravel())
return l
def loss_prime(thetas, angles_rad, TE, TR):
T = len(angles_rad)
alpha_prime = np.zeros((T,))
for theta in thetas:
x1 = epg.FSE_signal(angles_rad, TE, theta['T1'], theta['T2']).ravel() * (1 - exp(-(TR - T * TE)/theta['T1']))
for i in range(T):
x1_prime = epg.FSE_signal_prime_alpha_idx(angles_rad, TE, theta['T1'], theta['T2'], i).ravel() * (1 - exp(-(TR - T * TE)/theta['T1']))
alpha_prime[i] += np.dot(x1, x1_prime)
return alpha_prime
def t1t2shuffle_prime_alpha_idx(angles_rad, TE, TRs, M0, T1, T2, idx):
"""derivative of signal equation w.r.t. the i'th flip angle"""
T = angles_rad.size
URs = TRs - (T+1)*TE # T + 1 to account for fast recovery
fi = epg.FSE_signal(angles_rad, TE, T1, T2)
fi_prime = epg.FSE_signal_prime_alpha_idx(angles_rad, TE, T1, T2, idx)
Ej = T1_recovery(URs, T1)[None,:]
a = 1 - Ej
b = 1 - Ej * fi[-1]
sig_prime = M0 * a / b**2 * (fi_prime * b + fi * Ej * fi_prime[-1])
return sig_prime.ravel(order='F')
def plot_vals(self, thetas):
plt.subplot(2,1,1)
plt.plot(range(self.T), RAD2DEG(self.angles_rad), 'o-')
plt.title('ETL={} POW={:.1f} MAX={:.0f} MIN={:.0f}'.format(
self.T, calc_power(self.angles_rad), RAD2DEG(np.max(self.angles_rad)), RAD2DEG(np.min(self.angles_rad))))
plt.xlim((0, self.T))
#plt.ylim((np.max((0,.9*np.min(RAD2DEG(self.angles_rad))), 180)))
plt.ylim((.5*np.min(RAD2DEG(self.angles_rad)), 180))
plt.ylabel('flip angles (deg)')
plt.subplot(2,1,2)
#leg_str = []
for theta in thetas:
plt.plot(range(self.T), epg.FSE_signal(self.angles_rad, self.TE, theta['T1'], theta['T2']) * (1 - exp(-(self.TR - self.T * self.TE)/theta['T1'])))
#leg_str.append('T1/T2={:.0f}/{:.0f}'.format(1000*theta['T1'], 1000*theta['T2']))
#plt.legend(leg_str)
plt.xlim((0,self.T))
plt.ylim((0, 1.))
plt.ylabel('signal level')
def forward(self, theta):
return epg.FSE_signal(self.angles_rad, TE, theta['T1'],
theta['T2']).ravel()
T = int(sys.argv[1])
else:
T = 10
angles = 150 * np.ones((T, ))
angles = read_angles('../data/flipangles.txt.408183520')
TT = len(angles)
if TT < T:
T = TT
else:
angles = angles[:T]
angles_rad = DEG2RAD(angles)
S = epg.FSE_signal(angles_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, TE, TR)
t2 = time.time()
LP = loss_prime(theta1, theta2, angles_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