def calc_alphadot(alpha, basis, T=100, k=5):
"""
calculates derivative along the path alpha
:param alpha: numpy ndarray of shape (2,M) of M samples
:param basis: list of numpy ndarray of shape (2,M) of M samples
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return alphadot: derivative of alpha
"""
alphadot = zeros((2, T, k))
for tau in range(0, k):
if tau == 0:
v = (k - 1) * (alpha[:, :, tau + 1] - alpha[:, :, tau])
elif tau == (k - 1):
v = (k - 1) * (alpha[:, :, tau] - alpha[:, :, (tau - 1)])
else:
v = ((k - 1) / 2.0) * (alpha[:, :, tau + 1] - alpha[:, :, (tau - 1)])
alphadot[:, :, tau] = cf.project_tangent(v, alpha[:, :, tau], basis[tau])
return alphadot
def calc_alphadot(alpha, basis, T=100, k=5):
"""
calculates derivative along the path alpha
:param alpha: numpy ndarray of shape (2,M) of M samples
:param basis: list of numpy ndarray of shape (2,M) of M samples
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return alphadot: derivative of alpha
"""
alphadot = zeros((2, T, k))
for tau in range(0, k):
if tau == 0:
v = (k-1)*(alpha[:, :, tau+1] - alpha[:, :, tau])
elif tau == (k-1):
v = (k-1)*(alpha[:, :, tau] - alpha[:, :, (tau-1)])
else:
v = ((k-1)/2.0)*(alpha[:, :, tau+1] - alpha[:, :, (tau-1)])
alphadot[:, :, tau] = cf.project_tangent(v, alpha[:, :, tau],
basis[tau])
return(alphadot)
def karcher_calc(mu, q, basis, closed, rotation, method):
# Compute shooting vector from mu to q_i
qn_t, R, gamI = cf.find_rotation_and_seed_unique(mu, q, closed, rotation,
method)
qn_t = qn_t / sqrt(cf.innerprod_q2(qn_t, qn_t))
q1dotq2 = cf.innerprod_q2(mu, qn_t)
if (q1dotq2 > 1):
q1dotq2 = 1
d = arccos(q1dotq2)
u = qn_t - q1dotq2 * mu
normu = sqrt(cf.innerprod_q2(u, u))
if (normu > 1e-4):
w = u * arccos(q1dotq2) / normu
else:
w = zeros(qn_t.shape)
# Project to tangent space of manifold to obtain v_i
if closed == 0:
v = w
else:
v = cf.project_tangent(w, q, basis)
return (v, gamI, d)
def karcher_calc(beta, q, betamean, mu, basis, mode):
# Compute shooting vector from mu to q_i
w, d = cf.inverse_exp_coord(betamean, beta, mode)
# Project to tangent space of manifold to obtain v_i
if mode == 0:
v = w
else:
v = cf.project_tangent(w, q, basis)
return (v, d)
def karcher_calc(beta, q, betamean, mu, mode=0):
if mode == 1:
basis = cf.find_basis_normal(mu)
# Compute shooting vector from mu to q_i
w, d = cf.inverse_exp_coord(betamean, beta)
# Project to tangent space of manifold to obtain v_i
if mode == 0:
v = w
else:
v = cf.project_tangent(w, q, basis)
return(v, d)
def curve_karcher_cov(betamean, beta, mode='O'):
"""
This claculates the mean of a set of curves
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param beta: numpy ndarray of shape (n, M, N) describing N curves
in R^M
:param mode: Open ('O') or closed curve ('C') (default 'O')
:rtype: tuple of numpy array
:return K: Covariance Matrix
"""
n, T, N = beta.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
# Compute Karcher covariance of uniformly sampled mean
betamean = cf.resamplecurve(betamean, T)
mu = cf.curve_to_q(betamean)
if mode == 1:
mu = cf.project_curve(mu)
basis = cf.find_basis_normal(mu)
v = zeros((n, T, N))
for i in range(0, N):
beta1 = beta[:, :, i]
w, dist = cf.inverse_exp_coord(betamean, beta1)
# Project to the tangent sapce of manifold to obtain v_i
if mode == 0:
v[:, :, i] = w
else:
v[:, :, i] = cf.project_tangent(w, mu, basis)
K = zeros((2*T, 2*T))
for i in range(0, N):
w = v[:, :, i]
wtmp = w.reshape((T*n, 1), order='C')
K = K + wtmp.dot(wtmp.T)
K = K/(N-1)
return(K)
def curve_karcher_cov(betamean, beta, mode='O'):
"""
This claculates the mean of a set of curves
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param beta: numpy ndarray of shape (n, M, N) describing N curves
in R^M
:param mode: Open ('O') or closed curve ('C') (default 'O')
:rtype: tuple of numpy array
:return K: Covariance Matrix
"""
n, T, N = beta.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
# Compute Karcher covariance of uniformly sampled mean
betamean = cf.resamplecurve(betamean, T)
mu = cf.curve_to_q(betamean)
if mode == 1:
mu = cf.project_curve(mu)
basis = cf.find_basis_normal(mu)
v = zeros((n, T, N))
for i in range(0, N):
beta1 = beta[:, :, i]
w, dist = cf.inverse_exp_coord(betamean, beta1)
# Project to the tangent sapce of manifold to obtain v_i
if mode == 0:
v[:, :, i] = w
else:
v[:, :, i] = cf.project_tangent(w, mu, basis)
K = zeros((2*T, 2*T))
for i in range(0, N):
w = v[:, :, i]
wtmp = w.reshape((T*n, 1), order='C')
K = K + wtmp.dot(wtmp.T)
K = K/(N-1)
return(K)