def riemann_metric(df, w, v, a, b, c, d): # df,w,v: 3*2*m*n
A = torch.einsum("simn,sjmn->ijmn", [df, df]) # A = df^Tdf: 2*2*m*n
detA = A[0, 0] * A[1, 1] - A[0, 1] * A[1,
0] + 1e-7 # detA = det(df^Tdf): m*n
invA = torch.zeros(A.size())
invA[:, :, 1:df.size(-2) - 1] = multi_dim_inv_for_22mn(
A[:, :, 1:df.size(-2) - 1],
detA[1:df.size(-2) - 1]) # invA = (df^Tdf)^{-1} 2*2*m*n
df_dfplus = torch.einsum("is...,st...,jt...->ij...", [df, invA, df])
B1 = torch.einsum("ismn,stmn,ktmn,kimn->mn", [w, invA, v, df_dfplus])
B2 = torch.einsum("ismn,tsmn,tkmn,klmn,jlmn,jimn->mn",
[invA, df, w, invA, df, v])
B3 = torch.einsum("ismn,stmn,itmn->mn", [w, invA, df]) * torch.einsum(
"ismn,stmn,itmn->mn", [v, invA, df])
# 1: measures the change in metric
part1 = (B1 + B2 - B3) / 2
# 2: measure the change in volume density
part2 = B3 / 2
# 3: measures the change in normal direction
part3 = torch.einsum("ismn,stmn,itmn->mn", [w, invA, v]) - B1
# 4: the additional term which measures ?
part4 = (B1 - B2) / 2
inner_prod = (a * part1 + b * part2 + c * part3 + d * part4) * detA.sqrt()
return integral_over_s2(inner_prod)
def E_L2_SRNFS(X, f1, f2, idty, Basis_vecFields):
gamma = idty + torch.einsum("i, ijkl->jkl", [X, Basis_vecFields])
gamma = gamma / torch.einsum("ijk, ijk->jk",
[gamma, gamma]).sqrt() # project to S2
# get the spherical coordinate representation
gammaSph = torch.stack((Cartesian_to_spherical(gamma[0] + 1e-7, gamma[1],
(1 - 1e-7) * gamma[2])))
f1_gamma = compose_gamma(f1, gammaSph)
q1_gamma = f_to_q(f1_gamma)
q2 = f_to_q(f2)
Diff = q2 - q1_gamma
return integral_over_s2(torch.einsum("imn,imn->mn", [Diff, Diff]))
def opt(X):
X = torch.from_numpy(X).float().requires_grad_()
R = torch.einsum("ij,jmn->imn", [torch_expm(X), idty])
gamma = torch.stack((Cartesian_to_spherical(R[0] + 1e-7, R[1],
(1 - 1e-7) * R[2])))
f1_gamma = compose_gamma(f1, gamma)
q1_gamma = f_to_q(f1_gamma)
q2 = f_to_q(f2)
y = integral_over_s2(
torch.einsum("imn,imn->mn", [q2 - q1_gamma, q2 - q1_gamma]))
y.backward()
return np.double(y.data.numpy()), np.double(X.grad.data.numpy())
def rescale_surf(f):
df = f_to_df(f)
n_f = torch.cross(df[:, 0], df[:, 1], dim=0)
Norm_n = torch.einsum("imn,imn->mn", [n_f, n_f]).sqrt()
return f / integral_over_s2(Norm_n).sqrt()
def initialize_over_paraSO3_SRNF(f1, f2, idty):
# load the elements in the icosahedral group
XIco_mat = sio.loadmat('Bases/skewIcosahedral.mat')
XIco = torch.from_numpy(XIco_mat['X']).float()
q2 = f_to_q(f2)
EIco = torch.zeros(60)
f1_gammaIco = torch.zeros(60, *f1.size())
q1_gammaIco = torch.zeros(60, *f1.size())
for i in range(60):
RIco = torch.einsum("ij,jmn->imn", [torch_expm(XIco[i]), idty])
gammaIco = torch.stack((Cartesian_to_spherical(RIco[0] + 1e-7, RIco[1],
(1 - 1e-7) * RIco[2])))
f1_gammaIco[i] = compose_gamma(f1, gammaIco)
q1_gammaIco[i] = f_to_q(f1_gammaIco[i])
EIco[i] = integral_over_s2(
torch.einsum("imn,imn->mn",
[q2 - q1_gammaIco[i], q2 - q1_gammaIco[i]]))
# get the index of the smallest value
Ind = np.argmin(EIco)
X = XIco[Ind]
L2_ESO3 = []
def opt(X):
X = torch.from_numpy(X).float().requires_grad_()
R = torch.einsum("ij,jmn->imn", [torch_expm(X), idty])
gamma = torch.stack((Cartesian_to_spherical(R[0] + 1e-7, R[1],
(1 - 1e-7) * R[2])))
f1_gamma = compose_gamma(f1, gamma)
q1_gamma = f_to_q(f1_gamma)
q2 = f_to_q(f2)
y = integral_over_s2(
torch.einsum("imn,imn->mn", [q2 - q1_gamma, q2 - q1_gamma]))
y.backward()
return np.double(y.data.numpy()), np.double(X.grad.data.numpy())
def printx(x):
L2_ESO3.append(opt(x)[0])
res = optimize.minimize(opt,
X,
method='BFGS',
jac=True,
callback=printx,
options={
'gtol': 1e-02,
'disp': False
}) # True
X_opt = torch.from_numpy(res.x).float()
R_opt = torch.einsum("ij,jmn->imn", [torch_expm(X_opt), idty])
gamma_opt = torch.stack((Cartesian_to_spherical(R_opt[0] + 1e-7, R_opt[1],
(1 - 1e-7) * R_opt[2])))
f1_gamma = compose_gamma(f1, gamma_opt)
return f1_gamma, L2_ESO3, f1_gammaIco[Ind], EIco