def cov_integral(alpha, alphadot, basis, T=100, k=5):
"""
Calculates covariance along path alpha
:param alpha: numpy ndarray of shape (2,M) of M samples (first curve)
:param alphadot: numpy ndarray of shape (2,M) of M samples
:param basis: list 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 u: covariance
"""
u = zeros((2, T, k))
for tau in range(1, k):
w = u[:, :, tau-1]
q1 = alpha[:, :, tau-1]
q2 = alpha[:, :, tau]
b = basis[tau]
wbar = cf.parallel_translate(w, q1, q2, b)
u[:, :, tau] = (1./(k-1))*alphadot[:, :, tau]+wbar
return(u)
def back_parallel_transport(u1, alpha, basis, T=100, k=5):
"""
backwards parallel translates q1 and q2 along manifold
:param u1: numpy ndarray of shape (2,M) of M samples
:param alpha: numpy ndarray of shape (2,M) of M samples
:param basis: list 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 utilde: translated vector
"""
utilde = zeros((2, T, k))
utilde[:, :, k-1] = u1
for tau in arange(k-2, -1, -1):
w = utilde[:, :, tau+1]
q1 = alpha[:, :, tau+1]
q2 = alpha[:, :, tau]
b = basis[tau]
utilde[:, :, tau] = cf.parallel_translate(w, q1, q2, b)
return(utilde)
def back_parallel_transport(u1, alpha, basis, T=100, k=5):
"""
backwards parallel translates q1 and q2 along manifold
:param u1: numpy ndarray of shape (2,M) of M samples
:param alpha: numpy ndarray of shape (2,M) of M samples
:param basis: list 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 utilde: translated vector
"""
utilde = zeros((2, T, k))
utilde[:, :, k - 1] = u1
for tau in arange(k - 2, -1, -1):
w = utilde[:, :, tau + 1]
q1 = alpha[:, :, tau + 1]
q2 = alpha[:, :, tau]
b = basis[tau]
utilde[:, :, tau] = cf.parallel_translate(w, q1, q2, b)
return utilde
def cov_integral(alpha, alphadot, basis, T=100, k=5):
"""
Calculates covariance along path alpha
:param alpha: numpy ndarray of shape (2,M) of M samples (first curve)
:param alphadot: numpy ndarray of shape (2,M) of M samples
:param basis: list 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 u: covariance
"""
u = zeros((2, T, k))
for tau in range(1, k):
w = u[:, :, tau - 1]
q1 = alpha[:, :, tau - 1]
q2 = alpha[:, :, tau]
b = basis[tau]
wbar = cf.parallel_translate(w, q1, q2, b)
u[:, :, tau] = (1.0 / (k - 1)) * alphadot[:, :, tau] + wbar
return u
def sample_shapes(mu, K, mode='O', no=3, numSamp=10):
"""
Computes sample shapes from mean and covariance
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param mu: numpy ndarray of shape (n, M) describing the mean srvf
:param K: numpy ndarray of shape (M, M) describing the covariance
:param mode: Open ('O') or closed curve ('C') (default 'O')
:param no: number of direction (default 3)
:param numSamp: number of samples (default 10)
:rtype: tuple of numpy array
:return samples: sample shapes
"""
n, T = mu.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]
U, s, V = svd(K)
if mode == 0:
N = 2
else:
N = 10
epsilon = 1./(N-1)
samples = empty(numSamp, dtype=object)
for i in range(0, numSamp):
v = zeros((2, T))
for m in range(0, no):
v = v + randn()*sqrt(s[m])*vstack((U[0:T, m], U[T:2*T, m]))
q1 = mu
for j in range(0, N-1):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1+sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
samples[i] = cf.q_to_curve(q2)
return(samples)
def sample_shapes(mu, K, mode='O', no=3, numSamp=10):
"""
Computes sample shapes from mean and covariance
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param mu: numpy ndarray of shape (n, M) describing the mean srvf
:param K: numpy ndarray of shape (M, M) describing the covariance
:param mode: Open ('O') or closed curve ('C') (default 'O')
:param no: number of direction (default 3)
:param numSamp: number of samples (default 10)
:rtype: tuple of numpy array
:return samples: sample shapes
"""
n, T = mu.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]
U, s, V = svd(K)
if mode == 0:
N = 2
else:
N = 10
epsilon = 1./(N-1)
samples = empty(numSamp, dtype=object)
for i in range(0, numSamp):
v = zeros((2, T))
for m in range(0, no):
v = v + randn()*sqrt(s[m])*vstack((U[0:T, m], U[T:2*T, m]))
q1 = mu
for j in range(0, N-1):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1+sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
samples[i] = cf.q_to_curve(q2)
return(samples)
def sample_shapes(self, no=3, numSamp=10):
"""
Computes sample shapes from mean and covariance
:param no: number of direction (default 3)
:param numSamp: number of samples (default 10)
"""
n, T = self.q_mean.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == self.mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
U, s, V = svd(self.C)
if mode == 0:
N = 2
else:
N = 10
epsilon = 1. / (N - 1)
samples = empty(numSamp, dtype=object)
for i in range(0, numSamp):
v = zeros((2, T))
for m in range(0, no):
v = v + randn() * sqrt(s[m]) * vstack(
(U[0:T, m], U[T:2 * T, m]))
q1 = self.q_mean
for j in range(0, N - 1):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = self.q_mean
else:
q2 = cos(epsilon * normv) * q1 + sin(
epsilon * normv) * v / normv
if mode == 1:
q2 = cf.project_curve(q2)
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
samples[i] = cf.q_to_curve(q2)
self.samples = samples
return
def curve_principal_directions(betamean, mu, K, mode='O', no=3, N=5):
"""
Computes principal direction of variation specified by no. N is
Number of shapes away from mean. Creates 2*N+1 shape sequence
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param mu: numpy ndarray of shape (n, M) describing the mean srvf
:param K: numpy ndarray of shape (M, M) describing the covariance
:param mode: Open ('O') or closed curve ('C') (default 'O')
:param no: number of direction (default 3)
:param N: number of shapes (2*N+1) (default 5)
:rtype: tuple of numpy array
:return pd: principal directions
"""
n, T = betamean.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]
U, s, V = svd(K)
qarray = empty((no, 2*N+1), dtype=object)
qarray1 = empty(N, dtype=object)
qarray2 = empty(N, dtype=object)
pd = empty((no, 2*N+1), dtype=object)
pd1 = empty(N, dtype=object)
pd2 = empty(N, dtype=object)
for m in range(0, no):
princDir = vstack((U[0:T, m], U[T:2*T, m]))
v = sqrt(s[m]) * princDir
q1 = mu
epsilon = 2./N
# Forward direction from mean
for i in range(0, N):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1 + sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
qarray1[i] = q2
p = cf.q_to_curve(q2)
centroid1 = -1*cf.calculatecentroid(p)
beta_scaled, scale = cf.scale_curve(p + tile(centroid1, [T, 1]).T)
pd1[i] = beta_scaled
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
# Backward direction from mean
v = -sqrt(s[m])*princDir
q1 = mu
for i in range(0, N):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1+sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
qarray2[i] = q2
p = cf.q_to_curve(q2)
centroid1 = -1*cf.calculatecentroid(p)
beta_scaled, scale = cf.scale_curve(p + tile(centroid1, [T, 1]).T)
pd2[i] = beta_scaled
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
for i in range(0, N):
qarray[m, i] = qarray2[(N-1)-i]
pd[m, i] = pd2[(N-1)-i]
qarray[m, N] = mu
centroid1 = -1*cf.calculatecentroid(betamean)
beta_scaled, scale = cf.scale_curve(betamean +
tile(centroid1, [T, 1]).T)
pd[m, N] = beta_scaled
for i in range(N+1, 2*N+1):
qarray[m, i] = qarray1[i-(N+1)]
pd[m, i] = pd1[i-(N+1)]
return(pd)
def curve_principal_directions(betamean, mu, K, mode='O', no=3, N=5):
"""
Computes principal direction of variation specified by no. N is
Number of shapes away from mean. Creates 2*N+1 shape sequence
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param mu: numpy ndarray of shape (n, M) describing the mean srvf
:param K: numpy ndarray of shape (M, M) describing the covariance
:param mode: Open ('O') or closed curve ('C') (default 'O')
:param no: number of direction (default 3)
:param N: number of shapes (2*N+1) (default 5)
:rtype: tuple of numpy array
:return pd: principal directions
"""
n, T = betamean.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]
U, s, V = svd(K)
qarray = empty((no, 2*N+1), dtype=object)
qarray1 = empty(N, dtype=object)
qarray2 = empty(N, dtype=object)
pd = empty((no, 2*N+1), dtype=object)
pd1 = empty(N, dtype=object)
pd2 = empty(N, dtype=object)
for m in range(0, no):
princDir = vstack((U[0:T, m], U[T:2*T, m]))
v = sqrt(s[m]) * princDir
q1 = mu
epsilon = 2./N
# Forward direction from mean
for i in range(0, N):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1 + sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
qarray1[i] = q2
p = cf.q_to_curve(q2)
centroid1 = -1*cf.calculatecentroid(p)
beta_scaled, scale = cf.scale_curve(p + tile(centroid1, [T, 1]).T)
pd1[i] = beta_scaled
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
# Backward direction from mean
v = -sqrt(s[m])*princDir
q1 = mu
for i in range(0, N):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1+sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
qarray2[i] = q2
p = cf.q_to_curve(q2)
centroid1 = -1*cf.calculatecentroid(p)
beta_scaled, scale = cf.scale_curve(p + tile(centroid1, [T, 1]).T)
pd2[i] = beta_scaled
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
for i in range(0, N):
qarray[m, i] = qarray2[(N-1)-i]
pd[m, i] = pd2[(N-1)-i]
qarray[m, N] = mu
centroid1 = -1*cf.calculatecentroid(betamean)
beta_scaled, scale = cf.scale_curve(betamean +
tile(centroid1, [T, 1]).T)
pd[m, N] = beta_scaled
for i in range(N+1, 2*N+1):
qarray[m, i] = qarray1[i-(N+1)]
pd[m, i] = pd1[i-(N+1)]
return(pd)
