def _fimtre6(I0, I1, TR0, TR1):
ctr0 = gs_recon(I0, pc_axis=-1, second_pass=False)
# find where circle intersects line, this is ctr1 (or there-abouts)
# use equation of centered circle and line:
# x^2 + y^2 = r^2
# r = |GS(I0)|, where GS(I) is the geometric center of ellipse I
# (x0, y0) = 0 deg phase cycle of second TR ellipse
# (x1, y1) = 180 deg phase cycle of second TR ellipse
# y = mx + b
# m = (y1 - y0)/(x1 - x0)
# b = y - mx = y0 - m*x0 = y1 - m*x1
# x^2 + (mx + b)^2 = r^2
# => x = (sqrt((m^2 + 1)r^2 - b^2) - bm)/(m^2 + 1)
# OR -(sqrt((m^2 + 1)r^2 - b^2) + bm)/(m^2 + 1)
# Choose the smaller rotation, i.e. min |x|
# y = mx + b
# (x, y) is now GS(I1)
r2 = np.real(np.conj(ctr0)*ctr0)
I1 = I1*1e8 # scale I1 up to make sure endpoints don't miss ctr0 when rotating
m = (I1[..., 0].imag - I1[..., 1].imag)/(I1[..., 0].real - I1[..., 1].real)
m2 = m*m
# avg b = ((y_0 - mx_0) + (y_1 - mx_1))/2
b = (I1[..., 0].imag - m*I1[..., 0].real + I1[..., 1].imag - m*I1[..., 1].real)/2
vals = (m2 + 1)*r2 - b**2
nonneg = vals >= 0
x = np.zeros(r2.shape)
xalt = np.zeros(r2.shape)
x[nonneg] = (np.sqrt(vals[nonneg]) - b[nonneg]*m[nonneg])/(m2[nonneg] + 1)
xalt[nonneg] = -1*(np.sqrt(vals[nonneg]) + b[nonneg]*m[nonneg])/(m2[nonneg] + 1)
idx = np.abs(xalt) < np.abs(x)
x[idx] = xalt[idx]
y = m*x + b
return np.angle(ctr0*(x - 1j*y))
def _fimtre8(I0, I1, TR0, TR1):
ctr0 = gs_recon(I0, pc_axis=-1, second_pass=False)
ctr1 = gs_recon(I1, pc_axis=-1, second_pass=False)
return np.angle(ctr0*np.conj(ctr1))
pcs2 = np.linspace(0, 2*np.pi, 2, endpoint=False)
pcs4 = np.linspace(0, 2*np.pi, 4, endpoint=False)
many_pcs = np.linspace(0, 2*np.pi, 200, endpoint=True)
df = 1/(2*TRs[0])
e1 = pbssfp(TRs[0], pcs4)
e1_ext = pbssfp(TRs[0], many_pcs)
e2 = pbssfp(TRs[1], pcs2)
e2_ext = pbssfp(TRs[1], many_pcs)
df_est = fimtre(e1, e2, *TRs, rad=False)
rad_est = fimtre(e1, e2, *TRs, rad=True)
plt.plot(e1_ext.real, e1_ext.imag, 'b', label='TR1')
plt.plot(e1.real, e1.imag, 'bx')
plt.plot(e2_ext.real, e2_ext.imag, 'r', label='TR2')
plt.plot(e2.real, e2.imag, 'rx')
# Annotations
plt.plot(e2.real, e2.imag, 'r--')
xr, yr = rotate_points(e2.real, e2.imag, phi=rad_est)
plt.plot(xr, yr, 'rx')
plt.plot(xr, yr, 'r--')
ctr0 = gs_recon(np.atleast_2d(e1), pc_axis=-1, second_pass=False)
plt.plot(ctr0.real, ctr0.imag, 'bo')
plt.title(f'True: {df:.2f}Hz, Est: {df_est[0]:.2f}Hz')
plt.legend()
plt.xlabel('Real (a.u.)')
plt.ylabel('Imag (a.u.)')
plt.axis('square')
plt.show()
data = data[pdx2:-pdx2, ...]
sx, sy, npcs, nc = data.shape[:]
TR, alpha = 6e-3, np.deg2rad(70)
# MR params for bSSFP sim
pcs = np.linspace(0, 2 * np.pi, npcs, endpoint=False)
# Start timer
t0 = time()
# Coil combine (SOS + simple phase)
res_rcc_simple = robustcc(data, method='simple', coil_axis=-1, pc_axis=-2)
# Make a mask
gs = np.array([
gs_recon(res_rcc_simple[..., ii::4], pc_axis=-1)
for ii in range(npcs // 4)
])
gs = np.abs(np.mean(gs, axis=0))
thresh = threshold_li(gs)
mask = gs > thresh
# PLANET
_Meff, T1map, T2map, _df = planet(res_rcc_simple,
alpha,
TR,
T1_guess=1,
mask=mask)
# Stop timer
print('Recon took %g sec' % (time() - t0))
csm = _gaussian_csm(N, N, nc)
data = np.abs(csm[..., None, :]) * bssfp(
T1,
T2,
TR,
alpha,
field_map=df,
phase_cyc=pcs[None, None, :, None],
M0=M0,
phi_rf=np.angle(csm[..., None, :]))
# Do coil-by-coil recon
res_cbc = np.empty((N, N, nc), dtype=np.complex64)
t0 = time()
for ii in range(nc):
res_cbc[..., ii] = gs_recon(data[..., ii], pc_axis=-1)
res_cbc = np.sqrt(np.sum(np.abs(res_cbc)**2, axis=-1))
print('Took %g seconds to do coil-by-coil recon' % (time() - t0))
# Do robust coil combine then recon
t0 = time()
res_rcc_simple = robustcc(data, method='simple')
res_rcc_simple = np.abs(gs_recon(res_rcc_simple, pc_axis=-1))
print('Took %g seconds to do simple robustcc recon' % (time() - t0))
if include_full_robustcc:
t0 = time()
res_rcc_full = robustcc(data, method='full', mask=M0 > 0)
res_rcc_full = np.abs(gs_recon(res_rcc_full, pc_axis=-1))
print('Took %g seconds to do full robustcc recon' % (time() - t0))
T1, T2 = M0 * 2, M0 / 2
# Simulate bSSFP acquisition with linear off-resonance
TR, alpha = 3e-3, np.deg2rad(30)
pcs = np.linspace(0, 2 * np.pi, 4, endpoint=False)
df, _ = np.meshgrid(np.linspace(-1 / TR, 1 / TR, N),
np.linspace(-1 / TR, 1 / TR, N))
sig = bssfp(T1,
T2,
TR,
alpha,
field_map=df,
phase_cyc=pcs[None, None, :],
M0=M0)
# Show the phase-cycled images
nx, ny = 2, 2
plt.figure()
for ii in range(nx * ny):
plt.subplot(nx, ny, ii + 1)
plt.imshow(np.abs(sig[..., ii]))
plt.title('%d deg PC' % (ii * 90))
plt.show(block=False)
# Dhow the recon
recon = gs_recon(sig, pc_axis=-1)
plt.figure()
plt.imshow(np.abs(recon))
plt.title('GS Solution')
plt.show()